# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, there is a local section $\bar{s}: U\to \Et(F)$ where $p\mapsto s_p$, its germ at $p$; this satisfies $\pi \circ \bar s = \id_U$.
  Give $\Et(F)$ the strongest topology such that the $\bar{s}$ are all continuous.
  Then $F^+(U) \da \Top(U, \Et(F))$ is the set of continuous sections of $\Et(F)$ over $U$.
- **Support**: for $s\in F(U)$, $\supp(s) \da \ts{p\in U \st s_p \neq 0}$ where $s_p$ is the germ of $s$ in $F_p$. This is closed.
  - This extends to $\supp(F) \da \ts{p\in X \st F_p \neq 0}$, which need not be closed.
- **Sheaf hom**: $U\mapsto \Hom(\ro{F}{U}, \ro{G}{U})$ forms a sheaf of local morphisms and is denoted $\sheafhom(F, G)$.
- **Flasque sheaves**: a sheaf is flasque iff $V\injects U \implies F(U) \surjects F(V)$.
- **Skyscraper sheaves**: for $A\in \Ab\Grp$ and $p\in X$, define $i_p(A)$ by $U\mapsto A$ if $p\in U$ and 0 otherwise. Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion.
  - The stalks are $(i_p(A))_q = A$ if $q\in \cl_X(\ts p)$ and 0 otherwise.

- **Extension by zero**: if $\iota: Z\injects X$ is the inclusion of a closed set and $U\da X\sm Z$ with $j: U\to X$, then for $F\in \Sh(Z)$, the sheaf $\iota_* F\in \Sh(X)$ is the extension of $F$ by zero outside of $Z$.
  The stalks $(\iota_* F)_p$ are $F_p$ is $p\in Z$ and 0 otherwise.
  - For the open $U$, extension by zero is $j_! F$ which has presheaf $V \mapsto F(V)$ if $V \subseteq U$ and 0 otherwise.
  The stalks $(j_! F)_p$ are $F_p$ if $p\in U$ and 0 otherwise.
- **Sheaf of ideals**: for $Y \subseteq X$ closed and $U \subseteq X$ open, $\mci_Y(U)$ has presheaf $U \mapsto$ the ideal in $\OO_X(U)$ of regular functions vanishing on all of $Y \intersect U$.
  This is a subsheaf of $\OO_X$.
- **Gluing sheaves**: given $\mcu \covers X$ and sheaves $F_i\in \Sh(U_i)$, one can glue to a unique $F\in \Sh(X)$ if one is given morphisms $\phi_{ij}\ro{F_i}{U_{ij}} \iso \ro{F_j}{U_{ij}}$ where $\phi_{ii} = \id$ and $\phi_{ik} = \phi_{jk} \circ \phi_{ij}$ on $U_{ijk}$.
:::

:::{.warnings}
Some common mistakes:

- Kernel presheaves are already sheaves, but not cokernels or images. See exercise below.
- $\phi: F\to G$ is injective iff injective on sections, but this is not true for surjectivity.
- The sheaves $f\inv G$ and $f^* G$ are different! See III.5 for the latter.
- Global sections need not be right-exact.

:::

:::{.exercise title="Regular functions on varieties form a sheaf"}
For $X\in \Var\slice k$, define the ring $\OO_X(U)$ of literal regular functions $f_i: U\to k$ where restriction morphisms are induced by literal restrictions of functions.
Show that $\OO_X$ is a sheaf of rings on $X$.

> Hint: Locally regular implies regular, and regular + locally zero implies zero.

:::

:::{.exercise title="?"}
Show that for every connected open subset $U \subseteq X$, the constant sheaf satisfies $\ul{A}(U) = A$, and if $U$ is open with open connected component so the $\ul{A}(U) = A\prodpower{\size \pi_0 U}$.
:::

:::{.exercise title="?"}
Show that if $X\in\Var\slice k$ and $\OO_X$ is its sheaf of regular functions, then the stalk $\OO_{X, p}$ is the *local ring of $p$* on $X$ as defined in Ch. I.
:::

:::{.exercise title="Prop 1.1"}
Let $\phi: F\to G$ be a morphism in $\Sh(X)$ and show that $\phi$ is an isomorphism iff $\phi_p$ is an isomorphism on stalks for all $p\in X$.
Show that this is false for presheaves.
:::

:::{.exercise title="?"}
Show that for $\phi\in \Mor_{\Sh(X)}(F, G)$, $\ker \phi$ is a sheaf, but $\coker \phi, \im \phi$ are not in general.
:::

:::{.exercise title="?"}
Show that if $\phi: F\to G$ is surjective then the maps on sections $\phi(U): F(U) \to G(U)$ need not all be surjective.
:::


## II.2: Schemes

## II.3: First Properties of Schemes

## II.4: Separated and Proper Morphisms

## II.5: Sheaves of Modules

## II.6: Divisors

## II.7: Projective Morphisms

## II.8: Differentials

## II.9: Formal Schemes

# III: Cohomology

## III.1: Derived Functors

## III.2: Cohomology of Sheaves

## III.3: Cohomology of a Noetherian Affine Scheme 

## III.4: Čech Cohomology

## III.5: The Cohomology of Projective Space

## III.6: Ext Groups and Sheaves

## III.7: Serre Duality 

## III.8: Higher Direct Images of Sheaves 

## III.9: Flat Morphisms

## III.10: Smooth Morphisms 

## III.11: The Theorem on Formal Functions 

## III.12: The Semicontinuity Theorem

# IV: Curves


:::{.remark}
Summary of major results:

- $p_a(X) \da 1 - P_X(0)$
- $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$
- For curves, $p_a(X) = p_g(X) = h^1(\OO_X)$ by setting $D\da K_C$ in RR.
  - $\deg K_C = 2g-2$.
- $D_1\sim D_2 \iff D_1-D_2 = (f)$ for $f\in K(X)$ rational, $\abs{D} = \ts{D'\sim D}$, and this bijects with points of $H^0(\mcl(D))\smz\over \GG_m$.
  - Thus $\dim \abs{D} = h^0(\mcl(D)) - 1 \da \ell(D) - 1$.
- $X$ smooth $\implies \Cl(X)\iso \Pic(X)$ via $D\mapsto \mcl(D)$.
- $h^0(\mcl(D)) >0 \implies \deg(D) \geq 0$, and if $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
- RR:
\[
\chi(\mcl(D) &= h^0(\mcl(D)) - h^1(\mcl(D)) \\ 
  &= h^0(\mcl(D)) - h^0(\mcl(K-D)) \\
  &= \deg(D) + (1-g)
.\]
  - How to remember: note $g= h^1(\OO_X) = h^1(\mcl(0))$, and $H^0(\OO_X) = k$ so $h^0(\OO_X) = 1$, thus
  \[
  \chi(\OO_X) = h^0(\OO_X) - h^1(\OO_X) = 1-g = \deg \mcl(0) + 1-g
  .\]
  - For $C \subseteq \PP^n, \deg(C) = d$ and $D = C \intersect H$ a hyperplane section defining $\mcl(D) = \OO_X(1)$,
  \[
 \chi(\mcl(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
 \]
- A curve is rational iff isomorphic to $\PP^1$ iff $g=0$.
- $K\sim 0$ on an elliptic curve since $\deg K = 2g-2 = 0$ and $\deg D = 0\implies D\sim 0$.
- For $X$ elliptic, $\Pic^0(X) \da \ts{D\in \Div(X) \st \deg D = 0}$ and $\abs{X} \iso \abs{\Pic^0(X)}$ via $p\mapsto \mcl(p-p_0)$ for any fixed $p_0\in X$, inducing its group structure. (This is proved with RR.)
:::


:::{.remark}
Comments from preface:

- The statement of Riemann-Roch is important; less so its proof.
- Representing curves: 
  - A branched covering of $\PP^1$, 
  - More generally a branched covering of another curve,
  - Nonsingular projective curves: admit embeddings into $\PP^3$, maps to $\PP^2$ birationally such that the image is at worst a nodal curve.
- The central result regarding representing curves: Hurwitz's theorem which compares $K_X, K_Y$ for a cover $Y\to X$ of curves.
- Curves of genus 1: elliptic curves.
- Later sections: the canonical embedding of a curve.
:::

## IV.1: Riemann-Roch

:::{.definition title="Curves"}
A **curve** over $k=\kbar$ is a scheme over $\spec k$ which is 

- Integral
- Dimension 1
- Proper over $k$
- With regular local rings

In particular, a curve is smooth, complete, and necessary projective.
A **point** on a curve is a closed point.
:::

:::{.definition title="Arithmetic genus"}
The **arithmetic genus** of a projective curve $X$ is
\[
p_a(X) \da 1 - P_X(0)
\]
where $P_X(t)$ is the **Hilbert polynomial** of $X$.
:::

:::{.definition title="Geometric genus"}
The **geometric genus** of a curve is 
\[
p+g(X) \da \dim_k H^0(X; \omega_X)
\]
where $\omega_X$ is the canonical sheaf.
:::

:::{.exercise title="?"}
Show that if $X$ is a curve, there is a single well-defined **genus** 
\[
g \da p_A(X) = p_G(X) = \dim_k H^1(X; \OO_X)
.\]

> Hint: see Ch. III Ex. 5.3, and use Serre duality for $p_g$.

:::

:::{.exercise title="?"}
Show that for any $g\geq 0$ there exists a curve of genus $g$.

> Hint: take a divisor of type $(g+1, 2)$ on a smooth quadric which is irreducible and smooth with $p_a = g$.

:::

:::{.definition title="Divisors on a curve"}
Reviewing divisors:

- The **divisor group**: $\Div(X) = \ZZ\adjoin{X_\cl}$
- **Degrees**: $\deg(\sum n_i D_i) \da \sum n_i$, and 
- **Linear equivalence**: $D_1\sim D_2 \iff D_1 - D_1 = \div(f)$ for some $f\in k(X)$ a rational function.
- $D$ is **effective** if $n_i \geq 0$ for all $i$.
- $\abs{D} \da \ts{D'\in \Div(X) \st D'\sim D}$ is the **complete linear system** of $D$.
- $\abs{D} \cong \PP H^0(X; \mcl(D))$
- **Dimensions of linear systems**: $\ell(D) \da \dim_k H^0(X; \mcl(D))$ and $\dim \abs{D} \da \ell(D) - 1$.
- **Relative differentials**: $\Omega_X \da \Omega_{X\slice k}$ is the sheaf of relative differentials on $X$.
  - The technical definition: $\Omega_{X\slice S} \da \Delta_{X\slice Y}^*(\mci/\mci^2)$ where $\mci$ is the sheaf of ideals defining the locally closed subscheme $\im(\Delta_{X\slice Y}) \subseteq X\fp{Y} X$.
  - On affine schemes: on the ring side, $\Omega_{B\slice A}\in \mods{B}$ equipped with a differential $d: B\to \Omega{B\slice A}$, defined as $\gens{db\st b\in B}_B / \gens{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}_B$.
  - On curves, $\Omega_{X\slice Y}$ measures the "difference" between $K_X$ and $K_Y$.

- **Canonical sheaf**: $\dim X = 1, \Omega_{X\slice k} \cong \omega_X$.
- **Canonical divisor**: $K_X$ 2is any divisor in the linear equivalence class corresponding to $\omega_X$ 
- $D$ is **special** iff its **index of speciality** $\ell(K-D) > 0$, otherwise $D$ is **nonspecial**.

:::

:::{.exercise title="?"}
Show that $D_1\sim D_2\implies\deg(D_1) = \deg(D_2)$.
:::

:::{.exercise title="?"}
Show that 
\[
\abs{D} \mapstofrom \PP H^0(X; \mcl(D))
,\]
so $\abs{D}$ has the structure of the closed points of some projective space.
:::

:::{.exercise title="Lemma 1.2"}
Show that if $D\in \Div(X)$ for $X$ a curve and $\ell(D) \neq 0$, then $\deg(D) \geq 0$.

Show that is $\ell(D) \neq 0$ and $\deg D = 0$ then $D\sim 0$ and $\mcl(D) \cong \OO_X$.
:::

:::{.theorem title="Riemann-Roch"}
\[
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\]

:::

:::{.exercise title="Ingredients for proof of RR"}
Show the following:

- The divisor $K-D$ corresponds to $\omega_X \tensor\mcl(D)\dual\in \Pic(X)$.
- $H^1(X; \mcl(D))\dual \cong H^0(X; \omega_X \tensor \mcl(D)\dual)$.

- If $X$ is any projective variety,
\[
H^0(X; \OO_X) = k
.\]
:::

:::{.exercise title="?"}
Show that if $X \subseteq \PP^n$ is a curve with $\deg X = d$ and $D = X \intersect H$ is a hyperplane section, then $\mcl(D) = \OO_X(1)$ and $\chi(\mcl(D)) = d + 1 - p_a$.
:::

:::{.exercise title="?"}
Show that if $g(X) = g$ then $\deg K_X = 2g-2$.

> Hint: set $D = K$ and use $\ell(K) = p_g = g$ and $\ell(0) = 1$.

:::

:::{.remark}
More definitions:

- $X$ is **rational** iff birational to $\PP^1$.
- $X$ is **elliptic** if $g=1$.
:::

:::{.exercise title="?"}
Show that

1. If $\deg D > 2g-2$ then $D$ is nonspecial.
2. $p_a(\PP^1) = 0$.
3. A complete nonsingular curve is rational iff $X\cong \PP^1$ iff $g(X) = 0$.
4. If $X$ is elliptic then $K\sim 0$

> Hint: for (3) apply RR to $D = p-q$ for points $p\neq q$, and use $\deg(K-D) = -2$ and $\deg(D) = 0 \implies D\sim 0 \simplies p\sim q$.
> For (4), show $\ell(K) = p_g = 1$.

:::

:::{.exercise title="?"}
If $X$ is elliptic and $p\in X$, then there is a bijection
\[
m_p: X &\iso \Pic(X) \\
x &\mapsto \mcl(x-p)
,\]
so $\Pic(X) \in \Grp$.

> Hint: show that if $\deg(D) = 0$ then there is some $x\in X$ such that $D\sim x-p$ and apply RR to $D+p$.

:::


## IV.2: Hurwitz


:::{.remark}
Summary of results:

- For curves, complete = projective.
- Riemann-Hurwitz: for $f:X\to Y$ finite separable,
\[
K_X \sim f^* K_Y + R \implies \deg(K_X) = \deg(f^*K_Y) + \deg(R) \implies \\ \\
\chi(X) = \deg (f)\cdot \chi(Y) + \deg R, \qquad \deg R = \sum_{p\in X} (e_p - 1)
.\]

- $\deg f \da [K(X): K(Y)]$ for finite morphisms of curves.
- $e_p \da v_p(f^\sharp_* t)$ where $t$ is uniformizer in $\OO_{f(p)}$ and $f^\sharp: \OO_{Y, f(p)}\to \OO_{X, p}$ for $f:X\to Y$.
  - $e_p > 1 \implies$ ramification.
  - Unramified everywhere implies etale (since automatically flat).
  - $p\divides e_{x_0}\implies$ wild ramification, otherwise tame.
- $\exists f^*: \Div(Y)\to \Div(X)$ where $q\mapsto \sum_{p\mapsto q} e_p p$.
- Pullback commutes with forming line bundles:
\[
f^* \mcl(D) \cong \mcl(f^* D)
\]
where the LHS $f^*: \Pic(Y) \to \Pic(X)$.
- The fundamental SES for relative differentials: if $f:X\to Y$ is finite separable,
\[
f^* \Omega_{Y} \injects \Omega_{X} \surjects \Omega_{X/Y}
.\]

- $\dd{t}{u}$ for $t$ a uniformizer at $f(p)$ and $u$ a uniformizer at $p$ is defined by noting $\Omega{Y, f(p)} = \gens{\dt}, \Omega_{X, p} = \gens{\du}$, and there is some $g\in \OO_{X, p}$ such that $f^* \dt = g\du$; set $g \da \dd t u$.
- For finite separable morphisms of curves $f:X\to Y$,
  - $\supp \Omega_{X/Y} = \mathrm{Ram}(f)$ is the ramification locus, and $\Omega_{X/Y}$ is torsion so $\Ram(f)$ is finite.
  - $\length (\Omega_{X, Y})_p = v_p\qty{\dd t u}$ for any $p\in X$
  - Tamely ramified $\implies \length(\Omega_{X/Y})_p = e_p - 1$, and wild ramification increases this length.
  Recall that length is the largest size of chains of submodules.
- The ramification divisor:
\[
R \da \sum_{p\in X} \length (\Omega_{X/Y})_p p
.\]

- $K_X \sim f^*K_Y + R$



- $\PP^1$ can't admit an unramified cover: for $n\geq 1$,
\[
\chi(X) = n\chi(\PP^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\]
which forces $g(X) = 0, n=1, X = \PP^1, f=\id$.

- The Frobenius morphism on schemes is defined by taking $f^\sharp: \OO_X\to \OO_X$ to be the $p$th power map; pullback yields a definition of $X_p$, the Frobenius twist of $X$.
  - $F: X_p\to X$ is finite, $\deg F = p$, and corresponds to $K(X) \injects K(X)^{1\over p}$
- If $f:X\to Y$ induces a purely inseparable extension $K(X)/K(Y)$, then $X\iso Y$ as schemes, $g(X) = g(Y)$, and $f$ is a composition of Frobenii.
- Everywhere ramified extensions: $f:Y_p \to Y$, where $e_{q} = p$ for every $q\in X$.
  Induces $\Omega_{X/Y}\cong \Omega_{X}$.
- $\deg R$ is always even.
- Finite implies proper: finite implies separated, of finite type, closed by "going up" and universally closed by since finiteness is preserved under base change.
- $\PP^1$ no nontrivial etale covers.
- If $f:X\to Y$ then $g(X) \geq g(Y)$.
  - Thus $\exists \PP^1\to Y$ finite $\implies g(Y) = 0$.



:::


:::{.remark}
Preface:

- **Degree**: for a finite morphism of curves $X \mapsvia{f} Y$, set $\det(f) \da [k(X): k(Y)]$, the degree of the extension of function fields.
- **Ramification indices $e_p$**: for $p\in X$, let $q=f(p)$ and $t \in \OO_q$ a local coordinate. Pull back to $t\in \OO_p$ via $f^\sharp$ and define $e_p \da v_p(t)$ using the valuation $v_p$ for the DVR $\OO_p$.
- **Ramified**: $e_p > 1$, and **unramified** if $e_p = 1$.
- **Branch points** any $q = f(p)$ where $f$ is ramified.
- **Tame ramification**: for $\characteristic(k) = p$, tame if $p\notdivides e_P$.
- **Wild ramification**: when $p\divides e_P$.
- Pullback maps on divisor groups:
\[
f^*: \Div(Y) &\to \Div(X) \\
Q &\mapsto \sum_{P \mapsvia{f} q} e_P [P]
.\]
  - This commutes with taking line bundles (exercise), so induces a well-defined map $f^*: \Pic(X) \to \Pic(Y)$.
- $f$ is **separable** if $k(X) / k(Y)$ is a separable field extension.

:::

:::{.exercise title="?"}
Misc:

- Show that if $f$ is everywhere unramified then it is an étale morphism.
- Show that $f^* \mcl(D) = \mcl(f^* D)$
:::

:::{.exercise title="Prop 2.1"}
Show that if $X \mapsvia{f} Y$ is a finite separable morphism of curves, there is a SES
\[
f^* \Omega_Y \injects \Omega_X \surjects \Omega_{X\slice Y}
.\]
:::

:::{.remark}
Definitions: 

- **Derivatives**: for $f: X\to Y$, let $t$ be a parameter at $Q = f(P)$ and $u$ at $P$. Then $\Omega_{Y, Q} = \gens{dt}_{\OO_Q}$ and $\OO_{X, P} = \gens{du}_{\OO_P}$ and $\exists ! g\in \OO_P$ such that $f^* dt = du$ so we write $\dd{t}{u} \da g$.
- **Ramification divisor**: $R \da \sum_{P\in X} \length(\Omega_{X\slice Y})_P [P] \in \Div(X)$
:::

:::{.exercise title="Prop 2.2"}
For $X \mapsvia{f} Y$ a finite separable morphism of curves,

a. $\Omega_{X\slice Y}$ is a torsion sheaf on $X$ with support equal to the ramification locus of $f$.
Thus $f$ is ramified at finitely many points.
b. The stalks $(\Omega_{X\slice Y})_P$ are principal $\OO_P\dash$modules of finite length equal to $v_p\qty{\dd t u}$
c. 
\[
\length(\Omega_{X\slice Y})_P 

\begin{cases}
= e_p - 1 & f \text{ is tamely ramified at } P 
\\
 > e_p -1 & f \text{ is wildly ramified at } P.
\end{cases}
.\]

:::

:::{.exercise title="Prop 2.3"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
K_X \sim f^* K_Y + R
,\]
where $R$ is the ramification divisor of $f$.
:::

:::{.theorem title="Hurwitz"}
If $X \mapsvia{f} Y$ is a finite separable morphism of curves, then
\[
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\]
and if $f$ has only tame ramification then $\deg(R) = \sum_{P\in X}(e_P - 1)$.
:::

:::{.proof title="of Hurwitz"}
Take degrees of the divisor equation:
\[
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_\Top(X) &= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) \chi_\Top(Y) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &= \deg(f) (2g(Y) - 2) + \sum_{P\in X} (e_P - 1) \\
,\]
using tame ramification in the last step which implies $\length(\Omega_{X\slice Y})_P = (e_p - 1)$.


:::

:::{.remark}
Consider the purely inseparable case.

- **Frobenius morphism**: for $X \in\Sch$ where $\OO_P \contains \ZZ/p\ZZ$ for all $P$, define $\Frob: X\to X$ by $F(\abs{X}) = \abs{X}$ on spaces and $F^\sharp: \OO_X \to \OO_X$ is $f\mapsto f^p$.
  This is a morphism since $F^\sharp$ induces a morphism on all local rings, which are all characteristic $p$.
- **The $k\dash$linear Frobenius morphism**: define $X_p$ to be $X$ with the structure morphism $F\circ \pi$, so $k\actson \OO_{X_p}$ by $p$th powers and $F$ becomes a $k\dash$linear morphism $F': X_p\to X$.
    - Why this is necessary: $F$ as before is not a morphism in $\Sch\slice k$, and instead forms a commuting square involving $F: \spec k\to \spec k$ and the structure maps $X \mapsvia{\pi} \spec k$.
:::

:::{.exercise title="?"}
Find examples where

- $X_p \cong X \in \Sch\slice k$, and
- $X_p \not\cong X \in \Sch\slice k$.

> Hint: consider $X = \spec k[t]$ for $k$ perfect.

:::

:::{.exercise title="?"}
Show that if $X \mapsvia{f} Y$ is separable then $\deg(R)$ is always even.
:::

> Skipped some stuff around Example 2.4.2, I don't necessarily need characteristic $p$ things right now.

:::{.remark}
Definitions:

- **Étale covers**: $X \mapsvia{f} Y$ is an étale cover if $f$ is a finite étale morphism,, i.e. $f$ is flat and $\Omega^1_{X\slice Y} = 0$.
- $Y$ is a **trivial** cover if $X \cong \disjoint_{i\in I} Y$ a finite disjoint union of copies of $Y$, 
- $Y$ is **simply connected** if there are no nontrivial étale covers.
:::

:::{.exercise title="?"}
In parts:

- Show that a connected regular curve is irreducible.
- Show that if $f$ is etale then $X$ is smooth over $k$.
- Show that if $f$ is finite, $X$ must be a curve.
- Show that if $f$ is étale, then $f$ must be separable.
- Show that $\pi_1^\et(\PP^1) = 0$.

> Hint: use Hurwitz and that when $f$ is unramified, $R = 0$.

:::

:::{.exercise title="?"}
In parts:

- Show that the genus of a curve doesn't change under purely inseparable extensions.
- Show that if $f:X\to Y$ is a finite morphism of curves then $g(X) \geq g(Y)$.

:::

:::{.exercise title="Lüroth"}
Show that if $L$ is a subfield of a purely transcendental extension $k(t) / k$ where $k = \kbar$, then $L$ is also purely transcendental.[^more_luroth_stuff]



> Hint: Assume $[L: k]_\tr = 1$, so $L = k(X)$ for $Y$ a curve and $L \subseteq k(t)$ corresponds to a finite morphism $f: \PP^1\to Y$. Conclude $g(Y) = 0$ so $Y\cong \PP^1$ and $L\cong k(u)$ for some $u$.

[^more_luroth_stuff]: 
This is true over any field $k$ in dimension 1, over $k=\kbar$ in dimension 2, and false in dimension 3 by the existence of nonrational unirational threefolds.

:::



## IV.3: Embeddings in Projective Space

:::{.remark}
A summary of major results:

- For $D\in \Div(C)$ with $g = g(C)$, 
  - $D$ is ample iff $\deg D > 0$.
  - $D$ is BPF iff $\deg D\geq 2g$.
  - $D$ is very ample iff $\deg D \geq 2g+1$.
- Being very ample is equivalent to being a hyperplane section under a projective embedding.
- Divisors $D\in \Div(\PP^n)$ are ample iff very ample iff $\deg D \geq 1$.
  - E.g. if $E$ is elliptic then $D$ is very ample if $\deg D \geq 3$, and for hyperelliptic, very ample if $\deg D\geq 5$.
- If $D$ is very ample then $\deg \phi(X) = \deg D$.
- Curves $C \subseteq \PP^n$ for $n\geq 4$ can be projected away from a point $p\not \in X$ to get a closed immersion into $\PP^m$ for some $m\leq n-1$.
  So any curve is birational to a nodal curve in $\PP^2$.
- Genus of normalizations of nodal curves: $g = {1\over 2}(d-1)(d-2)-\size\ts{\text{nodes}}$.
- Any curve embeds into $\PP^3$, and maps into $\PP^2$ with at worst nodal singularities.
:::

:::{.remark}
Main result: any curve can be embedded in $\PP^3$, and is birational to a nodal curve in $\PP^2$.
Some recollections:

- **Very ample line bundles**: $\mcl \in \Pic(X)$ is very ample if $\mcl \cong \OO_X(1)$ for some immersion of $f: X\embeds \PP^N$.
- **Ample**: $\mcl$ is ample when $\forall \mcf\in \Coh(X)$, the twist $\mcf \tensor \mcl^n$ is globally generated for $n \gg 0$.
- **(Very) ample divisors**: $D\in \Div(X)$ is (very) ample iff $\mcl(D)\in \Pic(X)$ is (very) ample.
- **Linear systems**: a linear system is any set $S \leq \abs{D}$ of effective divisors yielding a linear subspace.
- **Base points**: $P$ is a base point of $S$ iff $P \in \supp D$ for all $D\in S$.
- **Secant lines**: the secant line of $P, Q\in X$ is the line in $\PP^N$ joining them.
- **Tangent lines**: at $P\in X$, the unique line $L \subseteq \PP^N$ passing through $p$ such that $\T_P(L) = \T_P(X) \subseteq \T_P(\PP^N)$.
- **Nodes**: a singularity of multiplicity 2.
  - $y^2 = x^3 + x^2$ is a **node**.
  - $y^2 = x^3$ is a **cusp**.
  - $y^2 = x^4$ is a **tacnode**.
- **Multisecant**: for $X \subseteq \PP^3$, a line meeting $X$ in 3 or more distinct points.
- A **secant with coplanar tangent lines** is a secant through $P, Q$ whose tangent lines $L_P, L_Q$ lie in a common plane, or equivalently $L_P$ intersects $L_Q$.

:::

:::{.exercise title="II.8.20.2"}
Show that by Bertini's theorem there are irreducible smooth curves of degree $d$ in $\PP^2$ for any $d$.
:::

:::{.exercise title="?"}
\envlist
Show that

- $\mcl$ is ample iff $\mcl^n$ is very ample for $b \gg 0$.
- $\abs D$ is basepoint free iff $\mcl(D)$ is globally generated.
- If $D$ is very ample, then $\abs{D}$ is basepoint free.
- If $D$ is ample, $nD \sim H$ a hyperplane section for a projective embedding for some $n$.
- If $g(X) = 0$ then $D$ is ample iff very ample iff $\deg D > 0$.
- If $D$ is very ample and corresponds to a closed immersion $\phi: X\injects \PP^n$ then $\deg \phi(X) = \deg D$.
- If $XS$ is elliptic, any $D$ with $\deg D = 3$ is very ample and $\dim \abs{D} = 2$, and so can be embedded into $\PP^2$ as a cubic curve.
- Show that if $g(X) = 1$ then $D$ is very ample iff $\deg D \geq 3$.
- Show that if $g(X) = 2$ and $\deg D = 5$ then $D$ is very ample, so any genus 2 curve embeds in $\PP^3$ as a curve of degree 5.

:::

:::{.exercise title="Prop 3.1: when a linear system yields a closed immersion into $\PP^N$"}
Let $D\in \Div(X)$ for $X$ a curve and show

- $\abs{D}$ is basepoint free iff $\dim\abs{D-P} = \dim\abs{D} - 1$ for all points $p\in X$.
- $D$ is very ample iff $\dim\abs{D-P-Q} = \dim\abs{D} - 2$ for all points $P, Q\in X$.

> Hint: use the SES $\mcl(D-P)\injects \mcl(D) \surjects k(P)$ where $k(P)$ is the skyscraper sheaf at $P$.

:::

:::{.exercise title="Cor 3.2"}
Let $D\in \Div(X)$.

- If $\deg D \geq 2g(X)$ then $\abs{D}$ is basepoint free.
- If $\deg D \geq 2g(X) + 1$ then $D$ is very ample.
- $D$ is ample iff $\deg D > 0$
- This bounds is not sharp.

> Hint: apply RR. For the bound, consider a plane curve $X$ of degree 4 and $D = X.H$.

:::

:::{.remark}
Idea behind embedding in $\PP^3$: embed into $\PP^n$ and project away from a point in the complement.
:::

:::{.exercise title="3.4, 3.5, 3.6"}
Let $X \subseteq \PP^N$ be a curve and $O\not\in X$, let $\phi:X\to \PP^{n-1}$ be projection away from $O$.
Then $\phi$ is a closed immersion iff

- $O$ is not on any secant line of $X$, and
- $O$ is not on any tangent line of $X$.

Show that if $N\geq 4$ then there exists such a point $O$ yielding a closed immersion into $\PP^{N-1}$.
Conclude that any curve can be embedded into $\PP^3$.

> Hint: $\dim\mathrm{Sec}(X) \leq 3$ and $\dim \mathrm{Tan}(X) \leq 2$.

:::

:::{.proposition title="3.7"}
Let $X \subseteq \PP^3$, $O\not\in X$, and $\phi: X\to \PP^2$ be the projection from $O$.
Then $X\birational \phi(X)$ iff $\phi(X)$ is nodal iff the following hold:

- $O$ is only on finitely many secants of $X$,
- $O$ is on no tangents,
- $O$ is on no multisecant,
- $O$ is on no secant with coplanar tangent lines.

:::

> Skipped things around Prop 3.8.
> The hard part: showing not every secant is a multisecant, and not every secant has coplanar tangent lines.
> Skipped strange curves.

:::{.remark}
Classifying all curves: any curve is birational to a nodal plane curve, so study the family $\mcf_{d, r}$ of plane curves of degree $d$ and $r$ nodes.
The family $\mcf_d$ of all plane curves is a linear system of dimension
\[
\dim \abs{\mcf_d} = {d(d+3)\over 2}
.\]
For any such curve $X$, consider its normalization $\nu(X)$, then
\[
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\]
Thus for $\mcf_{d, r}$ to be nonempty, one needs
\[
0 \leq r \leq {(d-1)(d-2) \over 2}
.\]
Both extremes can occur: $r=0$ follows from Bertini, and $r = {(d-1)(d-2)\over 2}$ by embedding $\PP^1\injects \PP^d$ as a curve of degree $d$ and projecting down to a nodal curve in $\PP^2$ of genus zero.
Severi states and Harris proves that for every $r$ in this range $\mcf_{d, r}$ is irreducible, nonempty, and $\dim \mcf_{d, r} = {d(d+3)\over 2} - r$.
:::

## IV.4: Elliptic Curves


:::{.remark}
Curves $E$ with $g(E) = 1$; we'll assume $\characteristic k \neq 2$ throughout.
Outline:

- Define the $j\dash$invariant, classifies isomorphism classes of elliptic curves.
- Group structure on the curve.
- $E = \Jac(E)$.
- Results about elliptic functions over $\CC$.
- The Hasse invariant of $E/\FF_q$ in characteristic $p$.
- $E(\QQ)$.
:::

### The $j\dash$invariant

:::{.remark}
The $j\dash$invariant:

- $j(E) \in k$, so $\AA^1\slice k$ is a coarse moduli space for elliptic curves over $K$.
- Defining $j(E)$:
  - Let $p_0\in X$, consider the linear system $L\da \abs{2p_0}$.
  - Nonspecial, so RR shows $\dim(L) = 1$.
  - BPF, otherwise $E$ is rational.
  - Defines a morphism $\phi_L: E\to \PP^1\slice k$ with $\deg \phi_L = 2$.
  - Up to change of coordinates, $f(p_0) = \infty$.
  - By Hurwitz, $f$ is ramified at 4 branch points $a,b,c,p_0$.
  - Move $a\mapsto 0, b\mapsto 1$ by a Mobius transformation fixing $\infty$, so $f$ is branched over $0,1,\lambda,\infty$ where $\lambda \in k\smts{0,1}$.
  - Use $\lambda$ to define the invariant:
  \[
  j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
  .\]

- Theorem 4.1:
  - $j$ depends only on the curve $E$ and not $\lambda$.
  - $E\cong E'\iff j(E) = j(E')$.
  - Every element of $k$ occurs as $j(E)$ for some $E$.
  - So this yields a bijection
  \[
  \correspond{
    \text{Elliptic curves over }k
  }\modiso
  &\mapstofrom
  \AA^1\slice k \\
  E &\mapsto j(E)
  .\]

- Some facts that go into proving this:
  - $\forall p,q\in X\,\,\exists \sigma\in \Aut(X)$ such that $\sigma^2=1, \sigma(p) = q$, for any $r\in X$, one has $r + \sigma(r) \sim p + q$.
  - $\Aut(X)\actson X$ transitively.
  - Any two degree two maps $f_1,f_2: X\to \PP^1$ fit into a commuting square.
  - Under $S_3\actson \AA^1\slice k\smts{0, 1}$, the orbit of $\lambda$ is 
  \[
  S_3 . \lambda= \ts{ \lambda, \lambda\inv, s_1 = 1- \lambda, s_1\inv = (1- \lambda)\inv, s_2 = \lambda(\lambda-1)\inv, s_3 = \lambda\inv (\lambda- 1)}
  .\]
  - Fixing $p\in X$, there is a closed immersion $X\to \PP^2$ whose image is $y^2=x(x-1)(x- \lambda)$ where $p\mapsto \infty = \tv{0:1:0}$ and this $\lambda$ is either the $\lambda$ from above or one of $s_1^{\pm 1}, s_2^{\pm 1}$.
    - Idea of proof: embed $X\injects \PP^2$ by $L\da \abs{3p}$, use RR to compute $h^0(\OO(np)) = n$ so $h^0(\OO(6p)) = 6$.
    - So $\ts{1,x,y,x^2,xy,y^2,x^3}$ has a linear dependence where $x^3,y^2$ have nonzero coefficients since they have poles at $p$.
    - Rescale $x^3, y^2$ to coefficient 1 to get
  \[
  y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
  .\]
  - Do a change of variable to put in the desired form: complete the square on the LHS, factor as $y^2=(x-a)(x-b)(x-c)$, send $a\to 0, b\to 1$ by a Mobius transformation.
- Note that one can project from $p$ to the $x\dash$axis to get a finite degree 2 morphism ramified at $0,1, \lambda, \infty$.
:::

:::{.example title="?"}
An elliptic curve that is smooth over every field of non-2 characteristic:
\[
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\]

![](figures/2022-12-03_23-36-23.png)

One that is smooth over every $k$ with $\characteristic k \neq 3$: the Fermat curve
\[
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\]
:::

:::{.theorem title="Orders of automorphism groups of elliptic curves"}
\[
\size \Aut(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \characteristic k \neq 3 \\
6 & j(E) = 0,\characteristic k \neq 3 \\ 
12 & j(E) = 0,1728, \characteristic k = 3
\end{cases}
.\]
:::

:::{.proof title="?"}
Idea: take the degree 2 morphism $f:X\to \PP^1$ with $f(p) = \infty$ branched over $\ts{0,1, \lambda, \infty}$.
Produce two elements in $G$: for $\sigma\in G$, find $\tau\in \Aut(\PP^1)$ so $f\sigma = \tau f$; then either $\tau \neq \id$, so $\ts{\sigma, \tau} \subseteq G$, or $\tau = id$ and either $\sigma=\id$ or $\sigma$ exchanges the sheets of $f$.

If $\tau\neq \id$, it permutes $\ts{0, 1, \lambda}$ and sends \( \lambda\mapsto \lambda\inv, s_1^{\pm 1}, s_2^{\pm 1} \) from above.
Cases:

1. $j=1728:$ If \( \lambda= -1, 1/2, 2, \characteristic k \neq 3 \), then $\lambda$ coincides with *one* other element of $S_3. \lambda$, so $\size G = 4$.
2. $j=0$: If \( \lambda= -\zeta_3, -\zeta_3^2, \characteristic k \neq 3 \) then $\lambda$ coincides with *two* elements in $S_3 . \lambda$ so $\size G = 6$.
3. $j=0=1728$: If \( \lambda= -1, \characteristic k = 3 \) then $S_3 . \lambda= \ts{ \lambda}$ and $\size G = 12$.

:::

### The group structure

:::{.remark}
The group structure:

- Fixing $p_o\in E$, the map $p\mapsto \mcl(p-p_0)$ induces a bijection $E \iso \Pic^0(E)$, so the group structure on $E$ is the pullback along this with $p_0 = \id$ and $p+q=r\iff p+q \sim r+p_0 \in \Div(E)$.
- Under the embedding of $\abs{3p_0}$, points $p,q,r$ are collinear iff $p+q+r \sim 3p_0$, so $p+q+r=0$ in the group structure.
- $E$ is a group variety, since $p\mapsto -p$ and $(p, q)\mapsto p+q$ are morphisms.
  Thus there is a morphism $[n]: E\to E$, multiplication by $n$, which is a finite morphism of degree $n^2$ with kernel $\ker [n] = C_n^2$ if $(n, \characteristic k) = 1$.and $\ker [n] = C_p, 0$ if $n=\characteristic k$, depending on the Hasse invariant.
- If $f:E_1 \to E_2$ is a morphism of curves with $f(p_1) = p_2$ then $f$ induces a group morphism.
- $\Endo(E, p_0)$ forms a ring under $f+g = \mu\circ (f\times g)$ and $f\cdot g \da f\circ g$. 
- The map $n \mapsto ([n]: E\to E)$ defines a finite ring morphism $\ZZ\to \Endo(E, p_0)$ for $n\neq 0$.
- $R \da \Endo(E, p_0)\units = \Aut(E)$, and if $j=0,1728$ then $R$ contains $\ts{\pm 1}$ and is thus bigger than $\ZZ$.
:::

:::{.remark}
The Jacobian: a variety that generalizes to make sense for any curve, a moduli space of degree zero divisor classes.

- For $X/k$ a curve and $T\in\Sch\slice k$, define
\[
\Pic^0(X\times T) \da \ts{\mcf \in \Pic(X\times T) \st \deg \ro{\mcf}{X_t} = 0 \, \forall t\in T },\qquad \Pic(X/T) \da \Pic^0(X\times T)/ p^* \Pic(T)
\]
where $p:X\times T\to T$ is the second projection.
Regard this as *families of sheaves of degree 0 on $X$ parameterized by $T$*.
- The Jacobian variety of a curve $X$: $\Jac(X) \in \Sch^\ft\slice k$ along with $\mcl \in \Pic^0(X/\Jac(X))$ such that for any $T\in\Sch^\ft\slice k$ and any $\mcm\in \Pic^0(X/T)$, $\exists ! \, f: T\to \Jac(X)$ such that $f^* \mcl = \mcm$.
Thus $J$ represents the functor $\Pic^0(X/\wait)$.
- For $E$ elliptic, $E = \Jac(E)$.
In general, $\abs{\Jac(X)}\cong \abs{\Pix^0(X)}$ on points, since points of $\Jac(X)$ are morphisms $\spec k\to \Jac(X)$, which correspond to elements in $\Pic^0(X/k) = \Pic^0(X)$.
- $\Jac(X) \in \Grp\Sch\slice k$ where $e: \spec k\to \Jac(X)$ corresponds to $0\in \Pic^0(X/k)$, $\rho: \Jac(X) \to \Jac(X)$ is $\mcl \mapsto \mcl\inv\in\Pic^0(X/\Jac(X))$, and $\mu: \Jac(X)\cartpower{2}\to \Jac(X)$ is $\mcl \mapsto p_1^* \mcl \tensor p_2^*\mcl \in\Pic^0(X/\Pic(X)\cartpower 2)$.

- $\T_0 \Jac(X) \cong H^1(X; \OO_X)$: giving an element of $\T_p X$ is the same as a morphism $T\da \spec k[\eps]/\eps^2\to X$ sending $\spec k \to p$.
  So $\T_0 \Jac(X)$, this means giving $\mcm\in \Pic^0(X/T)$ whose restriction to $\Pic^0(X/k)$ is zero.
  Use the SES $H^1(X;\OO_X)\injects \Pic X[\eps] \to \Pic(X)$.

- $\Jac(X)$ is proper over $k$ by the valuative criterion.
Just show that an invertible sheaf $\mcm$ on $X\times \spec K$ lifts unique to $\tilde \mcm$ on $X\times \spec R$, but $X\times \spec R$ is regular, so apply $\rm{II}.6.5$.
- For any $n$ there is a morphism 
\[
\phi^n: X\cartpower{n} &\to \Jac(X) \\
(p_1,\cdots, p_n) &\mapsto \mcl(\sum p_i - np_0)
.\]
  This is surjective for $n\geq g(X)$ by RR since every divisor class of degree $d\geq g$ has an effective representative.
  The fibers of $\phi^n$ are all tuples $(p_1,\cdots, p_n)$ such that $D = \sum p_i$ forms a complete linear system.
  - Most fibers are finite, so $\Jac(X)$ is irreducible of dimension $g$.
  - Smoothness: $\dim \T_0 \Jac(X) = \dim H^1(X;\OO_X) = g$, so smooth at zero, and group schemes are homogeneous so smooth everywhere.
:::


### Elliptic functions

> Stopped at elliptic functions.


## IV.5: The Canonical Embedding

## IV.6: Classification of Curves in $\PP^3$

# V: Surfaces

## V.1: Geometry on a Surface

## V.2: Ruled Surfaces

## V.3: Monoidal Transformations 

## V.4:  The Cubic Surface in $\PP^3$

## V.5: Birational Transformations 

## V.6: Classification of Surfaces 


# I: Varieties


:::{.remark}
Some useful basic properties:

- Properties of $V$:
  - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$.
  - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$.
  - $V(\mfa)^c = \union_{f\in \mfa} D(f)$
  - $V(\mfa_1) \subseteq V(\mfa_1) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$.
- Properties of $I$:
  - $I(V(\mfa)) = \sqrt\mfa$ and $V(I(Y)) = \cl_{\AA^n}(Y)$.
    The containment correspondence is contravariant in both directions.
  - $I(\union_i Y_i) = \intersect_i I(Y_i)$.
- If $F$ is a sheaf taking values in subsets of a giant ambient set, then $F(\union U_i) = \intersect F(U_i)$.
  For $\AA^n\slice \CC$, take $\CC(x_1,\cdots, x_n)$, the field of rational functions, to be the ambient set.
- Distinguished open $D(f) \da \ts{p\in X \st f(p) \neq 0}$:
  - $\OO_X(D(f)) = A(X)\invert{f} = \ts{{g\over f^k} \st g\in A(X), k\geq 0}$, and taking $f=1$ shows $\OO_X(X) = A(X)$, i.e. global regular functions are polynomial.
  - Generally $D(fg) = D(f) \intersect D(g)$
  - For affines:
  \[
  \OO_{\spec R}(D(f)) = R\invert{f}
  .\]
  - For $\CC^n$, 
  \[
  \OO_{\CC^n}(D(f)) = \kxn\adjoin{1/f}
  \implies
  \OO_{\CC^n}(V(\mfa)^c) = \intersect _{f\in \mfa} \OO_{\CC^n}(D(f))
  .\]
:::

## I.1: Affine Varieties

:::{.remark}
\envlist

- $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, and elements $f\in A \da \kxn$ are functions on it.
- $Z(f) \da \ts{p\in \AA^n \st f(p) = 0}$, and for any $T \subseteq A$ we set $Z(T) \da \intersect_{f\in T} Z(f)$.
  - Note that $Z(T) = Z(\gens{T}_A) = Z(\gens{f_1,\cdots, f_r})$ for some generators $f_i$, using that $A$ is a Noetherian ring.
  So every $Z(T)$ is the set of common zeros of finitely many polynomials, i.e. the intersection of finitely many hypersurfaces.
- **Algebraic**: $Y \subseteq \AA^n$ is algebraic iff $Y = Z(T)$ for some $T \subseteq A$.
- The Zariski topology is generated by open sets of the form $Z(T)^c$.
- $\AA^1$ is a non-Hausdorff space with the cofinite topology.
- **Irreducible**: $Y$ is reducible iff $Y = Y_1 \union Y_2$ with $Y_1, Y_2$ proper subsets of $Y$ which are closed in $Y$.
  - Nonempty open subsets of irreducible spaces are both irreducible and dense.
  - If $Y \subseteq X$ is irreducible then $\cl_X(Y) \subseteq X$ is again irreducible.
- **Affine (algebraic) varieties**: irreducible closed subsets of $\AA^n$.
- **Quasi-affine varieties**: open subsets of affine varieties.
- The ideal of a subset: $I(Y) \da \ts{f\in A \st f(p) = 0 \,\, \forall p\in Y}$.
- **Nullstellensatz**: if $k = \kbar, \mfa \in \Id(\kxn)$, and $f\in \kxn$ with $f(p) = 0$ for all $p\in V(\mfa)$, then $f^r \in \mfa$ for some $r>0$, so $f\in \sqrt\mfa$.
  Thus there is a contravariant correspondence between radical ideals of $\kxn$ and algebraic sets in $\AA^n\slice k$.
- **Irreducibility criterion**: $Y$ is irreducible iff $I(Y) \in \spec \kxn$ (i.e. it is prime).
- **Affine curves**: if $f\in k[x,y]^\irr$ then $\gens{f} \in \spec k[x,y]$ (since this is a UFD) so $Z(f)$ is irreducible and defines an affine curve of degree $d= \deg(f)$.
- **Affine surfaces**: $Z(f)$ for $f\in \kxn^\irr$ defines a surface.
- **Coordinate rings**: $A(Y) \da \kxn/I(Y)$.
- **Noetherian spaces**: $X\in \Top$ is Noetherian iff the DCC on closed subsets holds.
- **Unique decomposition into irreducible components**: if $X\in \Top$ is Noetherian then every closed nonempty $Y \subseteq X$ is of the form $Y = \union_{i=1}^r Y_i$ with $Y_i$ a uniquely determined closed irreducible with $Y_i \not\subseteq Y_j$ for $i\neq j$, the *irreducible components* of $Y$.
- **Dimension**: for $X\in \Top$, the dimension is $\dim X \da \sup \ts{n \st \exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}$ with $Z_i$ distinct irreducible closed subsets of $X$.
  Note that the dimension is the number of "links" here, not the number of subsets in the chain.
- **Height**: for $\mfp\in\spec A$ define $\height(\mfp) \da \sup\ts{n\st \exists \mfp_0 \subset \mfp_1 \subset \cdots \subset \mfp_n = \mfp}$ with $\mfp_i \in \spec A$ distinct prime ideals.
- **Krull dimension**: define $\krulldim A \da \sup_{\mfp\in \spec A}\height(\mfp)$, the supremum of heights of prime ideals.
:::

:::{.exercise title="The Zariski topology"}
Show that the class of algebraic sets form the closed sets of a topology, i.e. they are closed under finite unions, arbitrary intersections, etc.
:::

:::{.exercise title="The affine line"}
\envlist

- Show that $\AA^1\slice k$ has the cofinite topology when $k=\kbar$: the closed (algebraic) sets are finite sets and the whole space, so the opens are empty or complements of finite sets.[^hint_affine_line]
- Show that this topology is not Hausdorff.
- Show that $\AA^1$ is irreducible without using the Nullstellensatz.
- Show that $\AA^n$ is irreducible.
- Show that maximal ideals $\mfm \in \mspec \kxn$ correspond to minimal irreducible closed subsets $Y \subseteq \AA^n$, which must be points.
- Show that $\mspec \kxn = \ts{\gens{x_1-a_1,\cdots, x_n-a_n} \st a_1,\cdots, a_n\in k}$ for $k=\kbar$, and that this fails for $k\neq \kbar$.
- Show that $\AA^n$ is Noetherian.
- Show $\dim \AA^1 = 1$.
- Show $\dim \AA^n = n$.

[^hint_affine_line]: 
Hint: $k[x]$ is a PID and factor any $f(x)$ into linear factors using that $k = \kbar$ to write $Z(\mfa) = Z(f) = \ts{a_1,\cdots, a_k}$ for some $k$.

:::

:::{.exercise title="Commutative algebra"}
\envlist

- Show that if $Y$ is affine then $A(Y)$ is an integral domain and in $\kalg^\fg$.
- Show that every $B \in \kalg^\fg \intersect \Domain$ is of the form $B = A(Y)$ for some $Y\in\Aff\Var\slice k$.
- Show that if $Y$ is an affine algebraic set then $\dim Y = \krulldim A(Y)$.
:::

:::{.theorem title="Results from commutative algebra"}
\envlist

- If $k\in \Field, B\in \kalg^\fg \intersect \Domain$, 
  - $\krulldim B = [K(B) : B]_\tr$ is the transcendence degree of the quotient field of $B$ over $B$.
  - If $\mfp\in \spec B$ then $\height \mfp + \krulldim (B/\mfp) = \krulldim B$.

- Krull's Hauptidealsatz:
  - If $A \in \CRing^\Noeth$ and $f\in A\sm A\units$ is not a zero divisor, then every minimal $\mfp \in \spec A$ with $\mfp \ni f$ has height 1.

- If $A \in \CRing^\Noeth \intersect \Domain$, then $A$ is a UFD iff every $\mfp\in \spec(A)$ with $\height(\mfp) = 1$ is principal.

:::

:::{.exercise title="1.10"}
Show that if $Y$ is quasi-affine then 
\[
\dim Y = \dim \cl_{\AA^n} 
.\]
:::

:::{.exercise title="1.13"}
Show that if $Y \subseteq \AA^n$ then $\codim_{\AA^n}(Y) = 1 \iff Y = Z(f)$ for a single nonconstant $f\in \kxn^\irr$.
:::

:::{.exercise title="?"}
Show that if $\mfp \in \spec(A)$ and $\height(\mfp) = 2$ then $\mfp$ can not necessarily be generated by two elements.
:::

## I.2: Projective Varieties

:::{.remark}
- **Projective space**: $\ts{\vector a \da \tv{a_0, \cdots, a_n} \st a_i \in k}/\sim$ where $\vector a \sim \lambda \vector a$ for all $\lambda \in k\smz$, i.e. lines in $\AA^{n+1}$ passing through $\vector 0$.
- **Graded rings**: a ring $S$ with a decomposition $S = \oplus _{d\geq 0} S_d$ with each $S_d\in \Ab\Grp$ and $S_d S_e \subseteq S_{d+e}$; elements of $S_d$ are **homogeneous of degree $d$** and any element in $S$ is a finite sum of homogeneous elements of various degrees.
- **Homogeneous polynomials**: $f$ is homogeneous of degree $d$ if $f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)$.
- **Homogeneous ideals**: $\mfa \subseteq S$ is homogeneous when it's of the form $\mfa = \bigoplus _{d\geq 0} (\mfa \intersect S_d)$.
  - $\mfa$ is homogeneous iff generated by homogeneous elements.
  - The class of homogeneous ideals is closed under sums, products, intersections, and radicals.
  - Primality of homogeneous ideals can be tested on homogeneous elements, i.e. it STS $fg\in \mfa \implies f,g\in \mfa$ for $f,g$ homogeneous.
- $\kxn = \bigoplus _{d\geq 0} \kxn_d$ where the degree $d$ part is generated by monomials of total weight $d$.
  - E.g. $\kxn_1 = \gens{x_1, x_2,\cdots, x_n}$, $\kxn_2 = \gens{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}$.
  - Useful fact: by stars and bars, $\rank_k \kxn_d = {d+n \choose n}$.
  E.g. for $(d, n) = (3, 2)$,

  ![](0_Study%20Guides/Reading%20Notes/1_Hartshorne/figures/2022-10-08_18-48-17.png)

- Arbitrary polynomials $f\in \kxnz$ do not define functions on $\PP^n$ because of non-uniqueness of coordinates due to scaling, but homogeneous polynomials $f$ being zero or not is well-defined and there is a function
\[
\ev_f: \PP^n &\to \ts{0, 1} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0 
\\
1 & f(p) \neq 0.
\end{cases}
.\]
  So $Z(f) \da \ts{p\in \PP^n \st f(p) = 0}$ makes sense.

- **Projective algebraic varieties**: $Y$ is projective iff it is an irreducible algebraic set in $\PP^n$. Open subsets of $\PP^n$ are **quasi-projective varieties**.
- **Homogeneous ideals of varieties**:
\[
I(Y) \da \ts{f\in \kxnz^\homog \st f(p) =0 \, \forall p\in Y}
.\]

- **Homogeneous coordinate rings**:
\[
S(Y) \da \kxnz/I(Y)
.\]

- $Z(f)$ for $f$ a linear homogeneous polynomial defines a **hyperplane**.

:::

:::{.exercise title="Cor. 2.3"}
Show $\PP^n$ admits an open covering by copies of $\AA^n$ by explicitly constructing open sets $U_i$ and well-defined homeomorphisms $\phi_i :U_i\to \AA^n$.

:::


## I.3: Morphisms


## I.4: Rational Maps


## I.5: Nonsingular Varieties


## I.6: Nonsingular Curves


## I.7: Intersections in Projective Space


# II: Schemes

> Note: there are many, many important notions tucked away in the exercises in this section.

## II.1: Sheaves

:::{.remark}
- **Presheaves** $F$ of abelian groups: contravariant functors $F\in \Fun(\Open(X), \Ab\Grp)$.
  - Assigns every open $U \subseteq X$ some $F(U) \in \Ab\Grp$
  - For $\iota_{VU}: V \subseteq U$, restriction morphisms $\phi_{UV}: F(U) \to F(V)$.
  - $F(\emptyset) = 0$, so $F(\initial) = \terminal$.
  - $\phi{UU} = \id_{F(U)}$
  - $W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}$.
- **Sections**: elements $s\in F(U)$ are sections of $F$ over $U$.
  Also notation $\Gamma(U; F)$ and $H^0(U; F)$, and the restrictions are written $\ro s V \da \phi_{UV}(s)$ for $s\in F(U)$.
- **Sheaves**: presheaves $F$ which are completely determined by local data.
  Additional requirements on open covers $\mcv\covers U$:
  - If $s\in F(U)$ with $\ro{s}{V_i} = 0$ for all $i$ then $s\equiv 0 \in F(U)$.
  - Given $s_i\in F(V_i)$ where $\ro{s_i}{V_{ij}} = \so{s_j}{V_{ij}} \in F(V_{ij})$ then $\exists s\in F(U)$ such that $\ro{s}{V_i} = s_i$ for each $i$, which is unique by the previous condition.
- **Constant sheaf**: for $A\in \AbGrp$, define the constant sheaf
\[
\ul{A}(U) \da \Top(U, A^\disc)
.\]
- **Stalks**: $F_p \da \colim_{U\ni p} F(U)$ along the system of restriction maps.
  - These are represented by pairs $(U, s)$ with $U\ni p$ an open neighborhood and $s\in F(U)$, modulo $(U, s)\sim (V, t)$ when $\exists W \subseteq U \intersect V$ with $\ro s w = \ro t w$.
- **Germs**: a germ of a section of $F$ at $p$ is an elements of the stalk $F_p$.
- **Morphisms of presheaves**: natural transformations $\eta\in \Mor_{\Fun}(F, G)$, i.e. for every $U, V$, components $\eta_U, \eta_V$ fitting into a diagram

\begin{tikzcd}
	{\Open(X)} &&& \Ab\Grp \\
	U && {F(U)} && {G(U)} \\
	\\
	V && {F(V)} && {G(V)}
	\arrow["{\eta_V}", from=4-3, to=4-5]
	\arrow["{\eta_U}", from=2-3, to=2-5]
	\arrow[""{name=0, anchor=center, inner sep=0}, "{\mathrm{Res}_F(U, V)}", from=2-3, to=4-3]
	\arrow["{\mathrm{Res}_G(U, V)}", from=2-5, to=4-5]
	\arrow[""{name=1, anchor=center, inner sep=0}, hook, from=4-1, to=2-1]
	\arrow["{F, G}", shorten <=15pt, shorten >=15pt, Rightarrow, from=1, to=0]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==)

- A morphism of sheaves is exactly a morphism of the underlying presheaves.
- Morphisms of sheaves $\eta: F\to G$ induce morphisms of rings on the stalks $\eta_p: F_p \to G_p$.
- Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks, see exercise below.
- **Kernels, cokernels, images**: for $\phi: F\to G$, sheafify the assignments to kernels/cokernels/images on open sets.
- **Sheafification**: for any $F\in \Presh(X)$, there is a unique $F^+\in \Sh(X)$ and a morphism $\theta: F\to F^+$ of presheaves such that any sheaf presheaf morphism $F\to G$ factors as $F\to F^+ \to G$.
  - The construction: $F^+(U) = \Top(U, \disjoint_{p\in U} F_p)$ are all functions $s$ into the union of stalks, subject to $s(p) \in F_p$ for all $p\in U$ and for each $p\in U$, there is a neighborhood $V\contains U \ni p$ and $t\in F(V)$ such that for all $q\in V$, the germ $t_q$ is equal to $s(q)$.
  - Note that the stalks are the same: $(F^+)_p = F_p$, and if $F$ is already a sheaf then $\theta$ is an isomorphism.
- **Subsheaves**: $F'\leq F$ iff $F'(U) \leq F(U)$ is a subgroup for every $U$ and the restrictions on $F'$ are induced by restrictions from $F$.
  - If $F'\leq F$ then $F'_p \leq F_p$.
  - **Injectivity**: $\phi: F\to G$ is injective iff the sheaf kernel $\ker \phi = 0$ as a subsheaf of $F$.
    - $\phi$ is injective iff injective on all sections.
  - $\im \phi\leq G$ is a subsheaf.
  - **Surjectivity**: $\phi: F\to G$ is surjective iff $\im \phi = G$ as a subsheaf.

- **Exactness**: a sequence of sheaves $(F_i, \phi_i:F_i\to F_{i+1})$ is exact iff $\ker \phi_i = \im \phi^{i-1}$ as subsheaves of $F_i$.
  - $\phi:F\to G$ is injective iff $0\to F \mapsvia{\phi} G$ is exact.
  - $\phi: F\to G$ is surjective iff $F \mapsvia{\phi} G \to 0$ is exact.
  - Sequences of sheaves are exact iff exact on stalks.
- **Quotient sheaves**: $F/F'$ is the sheafification of $U\mapsto F(U) / F'(U)$.
- **Cokernels**: for $\phi: F\to G$, $\coker \phi$ is sheafification of $U\mapsto \coker( F(U) \mapsvia{\phi(U)} G(U))$.
- **Direct images**: for $f \in \Top(X, Y)$, the sheaf defined on sections by $(f_* F)(V) \da F(f\inv(V))$ for any $V \subseteq Y$.
  Yields a functor $f_*: \Sh(X) \to \Sh(Y)$.
- **Inverse images**: denoted $f\inv G$, the sheafification of $U \mapsto \colim_{V\contains f(U)} G(V)$, i.e. take the limit from above of all open sets $V$ of $Y$ containing the image $f(U)$.
  Yields a functor $f\inv: \Sh(Y) \to \Sh(X)$.
- **Restriction of a sheaf**: for $F\in \Sh(X)$ and $Z \subseteq X$ with $\iota:Z \injects X$ the inclusion, define $i\inv F\in \Sh(Z)$ to be the restriction.
  Also denoted $\ro{F}{Z}$.
  This has the same stalks: $(\ro{F}{Z})_p = F_p$.

- For any $U \subseteq X$, the global sections functor $\Gamma(U; \wait): \Sh(X)\to \Ab\Grp$ is left-exact (proved in exercises).
- **Limits of sheaves**: for $\ts{F_i}$ a direct system of sheaves, $\colim_{i} F_i$ has underlying presser $U\mapsto \colim_i F_i(U)$.
  If $X$ is Noetherian, then this is already a sheaf, and commutes with sections: $\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)$.
  - Inverse limits exist and are defined similarly.
- **The espace étalé**: define $\Et(F) = \disjoin_{p\in X} F_p$ and a projection $\ppi: \Et(F) \to X$ by sending $s\in F_p$ to $p$.
  For each $U \subseteq X$ and $s\in F(U)$, t