\newpage # I: Varieties :::{.remark} Some useful basic properties: - Properties of $V$: - $\intersect_{i\in I} V(\mfa_i) = V\qty{\sum_{i\in I} \mfa_i}$. - E.g. $V(x) \intersect V(y) = V(\gens{x} + \gens{y})= V(x, y) = \ts{0}$, the origin. - $\union_{i\leq n} V(\mfa_i) = V\qty{\prod_{i\leq n} \mfa_i}$. - E.g. $V(x) \union V(y) = V(\gens{x}\gens{y}) = V(xy)$, the union of coordinate axes. - $V(\mfa)^c = \union_{f\in \mfa} D(f)$ - $V(\mfa_1) \subseteq V(\mfa_2) \iff \sqrt{\mfa_1}\contains \sqrt{\mfa_2}$. - 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 $\AA^n/\CC$, 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 $\star$ :::{.remark} Summary: - $\AA^n\slice k = \ts{\tv{a_1,\cdots, a_n} \st a_i \in k}$, 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 \(k = \bar{k}, \mfa \in \Id(\kxn)\), 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 \(k=\bar{k}\): 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 $k=\bar{k}$, and that this fails for $k\neq \bar{k}$. - 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 $k = \bar{k}$ 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 \[ \dim Y = \dim \cl_{\AA^n} Y .\] ::: :::{.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 $\star$ :::{.remark} \envlist - **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 $(d, n) = (3, 2)$, ![](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 \newpage # II: Schemes > Note: there are many, many important notions tucked away in the exercises in this section. ## II.1: Sheaves $\star$ :::{.remark} \envlist - **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 $\ro{s_i}{V_{ij}} = \ro{s_j}{V_{ij}} \in F(V_{ij})$ 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 $A\in \Ab\Grp$, 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 presheaf $U\mapsto \colim_i F_i(U)$. 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 $\Et(F) = \disjoint_{p\in X} F_p$ and a projection $\pi: \Et(F) \to X$ 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 \[ i_p(A)(U) = \begin{cases} A & p\in U \\ 0 & \text{otherwise}. \end{cases} .\] Also denoted $\iota_*(A)$ where $\iota: \cl_X(\ts p) \injects X$ is the inclusion. - The stalks are \[ (i_p(A))_q = \begin{cases} A & q\in \cl_X(\ts{p}) \\ 0 & \text{otherwise}. \end{cases} .\] - **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 are \[ (\iota_* F)_p = \begin{cases} F_p & p\in Z \\ 0 & \text{otherwise}. \end{cases} .\] - 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 are \[ (j_! F)_p = \begin{cases} F_p & p\in U \\ 0 & \text{otherwise}. \end{cases} .\] - **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 \newpage # 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 \newpage # IV: Curves $\star$ :::{.remark} Summary of major results: - $p_a(X) \da 1 - P_X(0) = (-1)^r (1-\chi(\OO_X))$. - Note: $P_X(\ell)$ is defined as the Hilbert polynomial of the homogeneous coordinate ring $S(Y)$, and then defined for graded $S\dash$modules $M$ by setting $\phi_M(\ell) = \dim_k M_\ell$ and showing $\exists ! P_M(z) \in \QQ[z]$ with $\phi_M(\ell) = P_M(\ell)$ for $\ell \gg 0$. - $p_g(X) \da h^0(\omega_X) = h^0(\mcl(K_X))$. - Remembering these: ![](figures/2022-12-04_20-08-00.png) > [Link to Diagram](https://q.uiver.app/?q=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) - 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**: $\Div(X) = \zadjoin{X_\cl}$ - **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 $\deg(D) = 0 \implies D\sim 0 \implies p\sim q$. > 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 $\star$ :::{.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 $\deg(R) = \sum_{P\in X}(e_P - 1)$. ::: :::{.remark title="proof 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="?"} \envlist - 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="?"} \envlist - 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 $\star$ :::{.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 $\star$ :::{.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"} \[ \size \Aut(X, p) = \begin{cases} 2 & j(E) \neq 0,1728\\ 4 & j(E) = 1728, \characteristic k \neq 3 \\ 6 & j(E) = 0,\characteristic k \neq 3 \\ 12 & j(E) = 0,1728, \characteristic k = 3 \end{cases} .\] ::: :::{.remark title="Proof idea"} 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, $\abs{\Jac(X)}\cong \abs{\Pic^0(X)}$ 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 $\PP^3$ \newpage # 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 # Toric Varieties ## Summaries ### Quick Criteria :::{.remark} Quick criteria: - **Normal** $\iff$ **Saturated**: For affines, $X = \spec \CC[S]$ where $S \subseteq M$ is a **saturated** semigroup. This is true for $S = S_\sigma = \sigma\dual \intersect M$ where $\sigma$ is any SCRPC. - **Complete/proper** $\iff$ **Full support**: $X_\Sigma$ is complete iff $\supp \Sigma = N_\RR$. - **Smooth** $\iff$ **Lattice basis**: - For a **cone** $\sigma = \Cone(S)$ is smooth iff $\det S = \pm 1$, the volume of the standard lattice $\ZZ^n$. - Consequences of smoothness: - $\CDiv(X) = \Div(X)$ - $\Cl(X) = \Pic(X)$ - Smooth implies simplicial, so non-simplicial cones are singular. - For $p_\sigma$ the $T\dash$fixed point corresponding to $\sigma$, $T_p X \cong H$ where $H$ is a Hilbert basis for $S_\sigma$. - **Simplicial** $\iff$ **Euclidean basis**: For $\sigma = \Cone(S)$, $\sigma$ is simplicial iff $\det(S) \neq 0$. - **Orbifold singularities** $\iff$ **Simplicial**: $X_\Sigma$ has at worst finite quotient singularities iff $\Sigma$ is simplicial. - **Projectivity** $\iff$ **Admits a strictly upper convex support function**: For $h$ a support function and $D_h$ its associated divisor, the linear system $\abs{D_h}$ defines an embedding $X(\Delta) \embeds \PP^N$ iff $h$ is strictly upper convex. - Alternatively, $X_\Sigma$ is projective iff $\Sigma$ arises as the normal fan of a polytope. - **Globally generated/basepoint free** $\iff$ **Upper convex support function**: $\OO(D)$ is globally generated iff $\psi_D$ is upper convex. - **Ample $\iff$ Strictly upper convex support function**: $D\in \CDiv_T(X)$ is ample iff $\psi_D$ is strictly upper convex. - **Very ample $\iff$ ample and semigroup generation**: for $\Sigma$ complete, $D$ is very ample iff $\psi_D$ is strictly upper convex **and** $S_\sigma$ is generated by $\ts{u-u(\sigma) \st u\in P_D \intersect M}$, or equivalently the semigroup $\ts{u-u' \st u'\in P \intersect M}$ is saturated in $M$. - For $\PP^n$: $D = \sum a_i D_i$ is globally generated iff $\sum a_i \geq 0$ and ample $\iff \sum a_i > 0$. - For $\FF_m$: $D = \sum a_i D_i$ is globally generated iff $a_2 + a_4 \geq 0,\, a_1 + a_3 \geq m a_1$, $\Pic(\FF_n) = \gens{D_1, D_4}$, and $D = aD_1 + bD_4$ is ample iff $a,b > 0$. - For $\dim X_\Sigma = 2$ and $X$ complete: ample $\iff$ very ample. - **$\QQ\dash$factorial $\iff$ simplicial**: iff every cone is simplicial. - **Fundamental groups**: - For $U_\sigma$ affine, $U_\sigma \cong \AA^k \times \GG_m^{n-k}$ so $\pi_1 U_\sigma \cong \ZZ^{n-k}$ since $\GG_m^{n-k}\homotopic (S^1)^{n-k}$. - Can write $\pi_1 U_\sigma = N/N_\sigma$ where $N_\sigma$ is the sublattice generated by $\sigma$. - By a Van Kampen argument, $\pi_1 X_\Sigma = N/N'$ where $N' = \gens{\sigma \intersect N \st \sigma \in \Sigma}$: \[ \pi_1 X_\Sigma = \pi_1 \union U_{\sigma} = \colim \pi_1 U_\sigma = \colim N/N_\sigma = N / \Sum N_\sigma = N/N' .\] - **Euler characteristic**: $\chi X_\Sigma = \size \Sigma(n)$. - Why: $H^i(U_\sigma; \ZZ) = \Extpower^i M(\sigma)$ where $M(\sigma ) \da \sigma\dual \intersect M$, so one gets a spectral sequence \[ E_1^{p, q} = \bigoplus _{I^p = i_0< \cdots < i_p} H^q(U_{\sigma_{I^p}}; \ZZ) \abuts H^{p+q}(X_\Sigma; \ZZ), \qquad \sigma_{I^p} = \sigma_{i_0} \intersect \cdots \sigma_{i_p}, \sigma_{i_j}\in \Sigma(n) \\ \\ \leadsto E_1^{p, q} = \bigoplus _{I^p} \Extpower^q M(\sigma_{I^p}) \abuts H^{p+q}(X_\Sigma; \ZZ) \\\\ \implies \chi X_\Sigma = \sum (-1)^{p+q} \rank_\ZZ E_1^{p, q} = \size \Sigma(n) ,\] using that \[ \sum (-1)^{q} \rank_\ZZ \extpower^q M(\tau) = \begin{cases} 0 & \dim \tau < n \\ 1 & \dim \tau = n. \end{cases} .\] - **Higher homology**: - If all maximal cones of $\Sigma$ are $n\dash$dimensional, $H^2(X_\Sigma; \ZZ) \cong \Pic(X_\Sigma)$. - **Global sections**: for $D\in \Div_T(X)$, $P_D$ its associated polyhedron, \[ H^0(X; \OO_X(D)) = \bigoplus _{m\in P_D \intersect M} \CC\, \chi^m .\] - **Betti numbers**: \[ \beta_{2k} = \sum_{i=k}^n (-1)^{i-k} {i\choose k} \size\Sigma(n-i) .\] - **Canonical bundles/divisors**: $\omega_{X_\Sigma} \da \det \Omega_{X_\Sigma/k} = \OO(K_{X_\Sigma})$ where $K_{X_\Sigma} = -\sum_{\rho_i} D_i$. - For a smooth complete surface with $D_i^2 = -d_i$, \[ K^2 = \sum D_i^2 + 2d = -\sum d_i + 2d = -(3d-12) + 2d = 12-d .\] - **Degree = $n! \cdot \Vol(P)$** (for $X_P$ projective) ::: :::{.remark} Some common counterexamples: - An ample divisor that is not very ample: $P \da \Conv(\tv{0,0,0}, \tv{0,1,1}, \tv{1,0,1}, \tv{1,1,0})$; then take $D_P$. $X_P$ is a double cover of $\PP^3$ branched along the 4 boundary divisors. - A Weil divisor that is not Cartier: ???? - A complete variety that is not projective: ??? ::: ### Cones and Lattices :::{.remark} \envlist - **Characters**: for groups $G$, a map $\chi\in \Grp(G, \cstar)$. For $G= T = (\cstar)^n$, there is an isomorphism \[ \ZZ^n &\iso \Grp(T, \cstar) \\ m = \tv{m_1,\cdots, m_n} &\mapsto \chi_m: \tv{t_1,\cdots, t_n} \mapsto \prod t_i^{m_i} .\] Generally set $M \da \Grp(T, \cstar)$, the character lattice. - $M$ is a lattice, $M_\RR \da M\tensor_\ZZ \RR$ is its associated Euclidean space. - **Cocharacters / one-parameter subgroups**: for groups $G$, a map $\lambda \in \Grp(\cstar, G)$. For $G = T = \cstar$, there is again an isomorphism \[ \ZZ^n &\mapsto \Grp(\cstar, T) \\ u =\tv{u_1,\cdots, u_n} &\mapsto \lambda^u: t\mapsto \tv{t^{u_1}, \cdots, t^{u_n}} .\] Define $N \da \Grp(\cstar, T)$ the cocharacter lattice. - $N$ is a lattice, $N_\RR \da N\tensor_\ZZ \RR$ its associated euclidean space. - There is a perfect pairing \[ \inp\wait\wait: M\times N &\to \ZZ \\ ,\] defined using the fact that if $m\in M, n\in N$ then $\chi^m \circ \lambda^n \in \Grp(\cstar, \cstar)$ is of the form $t\mapsto t^\ell$, so set $\inp{m}{n} \da \ell$. - Thus $M = \Grp(M, \ZZ)$ and $N = \Grp(N, \ZZ)$. - How to recover the torus: \[ N \tensor_\ZZ \cstar &\to T \\ u\tensor t &\mapsto \lambda^u(t) .\] - $\Delta$ is a **fan**, a collection of **strongly convex rational polyhedral cones**: - **Cone**: $0\in \sigma$ and $\RR_{\geq 0} \sigma \subseteq \sigma$. - **Strongly convex**: contains no nonzero subspace, i.e. no line through $\vector 0 \in N_\RR$. Equivalently, $\dim \sigma\dual = n$. - **Rational**: generated by $\ts{v_i} \subseteq N$, i.e. of the form $\Cone(S)$ for $S \subseteq N$. - **Dual cones**: \[ \sigma\dual &\da \ts{ u\in M \st \inp u v \geq 0 \,\,\forall v\in M_\RR } .\] - If $\sigma\dual = \Intersect_{i=1}^s H_{m_i}^+$ for $m_i \subseteq \sigma\dual$ then $\sigma\dual = \Cone(m_1,\cdots, m_s)$. - **Hyperplanes** and **closed half-spaces**: \[ H_m &\da \ts{u\in N_\RR \st \inp m u = 0} \subseteq N_\RR \\ H_m^+ &\da \ts{u\in N_\RR \st \inp m u \geq 0} \subseteq N_\RR .\] - **Face**: $\tau \leq \sigma$ is a face iff $\tau$ is of the form $\tau = H_m \intersect \sigma$ for some $m\in \sigma\dual \subseteq M_\RR$. - **Facet**: codimension one faces, $\Sigma(n-1)$ where $n\da \dim N$. - **Ray**: dimension 1 faces, $\Sigma(1)$. - The **semigroup** of a cone: \[ S_\sigma &\da \sigma\dual \intersect M = \ts{ u\in M \st \inp u v \geq 0 \,\,\forall v\in \sigma } .\] - The **semigroup algebra** of a semigroup: \[ \CC[S] \da \ts{\sum_{s\in S} c_s \chi^s \st c_s \in \CC, c_s = 0 \ae}, \qquad \chi^{m_1}\cdot \chi^{m_2} \da \chi^{m_1 + m_2} .\] - **Simplicial**: the generators can be extended to an $\RR\dash$basis of $N_\RR$. E.g. not simplicial: ![](figures/2022-10-19_18-23-05.png) - **Smooth**: the minimal generators can be extended to a $\ZZ\dash$basis of $N$. - Checking $T_p X$: $m$ is **decomposable** in $S_ \sigma$ iff $m = m_1 + m_2$ with $m_i\in S_{ \sigma}$; the maximal ideal at $p$ corresponding to $\sigma$ is $\mfm_p = \ts{\chi^m \st m\in S_ \sigma}$, and $\mfm_p/\mfm_p^2 = \ts{\chi^m \st m \text{ is indecomposable in } S_ \sigma}$. This exactly corresponds to a Hilbert basis. - **Facet**: face of codimension 1. - **Edge**: face of dimension 1. Note that facets = edges in $\dim N = 2$. - **Saturated**: $S$ is saturated if for all $k\in \NN\smz$ and all $m\in M$, $km\in S \implies m\in S$. Any SCRPC is saturated. - E.g. $S = \ts{(4,0), (3,1), (1,3), (0, 4)}$ is not saturated since $2\cdot(2,2) = (4, 4) \in \NN S$ but $(2,2)\not\in S$. - **Normalization**: in the affine case, write $X = \spec \CC[S]$ with torus character lattice $M = \ZZ S$, take a finite generating set $S'$, and set $\sigma = \Cone(S')\dual$. Then $\spec \CC[\sigma\dual \intersect M]\to X$ is the normalization. - **Distinguished points**: each strongly convex $\sigma \leadsto \gamma_\sigma \in U_\sigma$ a unique point corresponding to the semigroup morphism $m\mapsto \indic(m\in \sigma\dual \intersect M)$, which is $T\dash$fixed iff $\sigma$ is full-dimensional. - **Orbits**: $\Orb( \sigma) = T. \gamma_\sigma$, and $V(\sigma)\da \cl \Orb( \sigma)$. - **Orbit-Cone correspondence**: there is a correspondence \[ \ts{\text{Cones } \sigma \in \Sigma} &\mapstofrom \ts{T\dash\text{orbits in } X_\Sigma} \\ \sigma &\mapsto \Orb(\sigma) \da T.\gamma_{\sigma} = \ts{\gamma: S_\sigma \to \CC\st \gamma(m) \neq 0 \iff m\in \sigma\dual \intersect M} \cong \Grp(\sigma \intersect M, \cstar) ,\] where $\dim \Orb( \sigma) = \codim_{N_\RR} \sigma$, and $\tau \leq \sigma \implies \cl \Orb(\tau) \contains \cl\Orb( \sigma)$ and in fact $\cl \Orb(\sigma) = \disjoint_{\tau\leq \sigma} \cl \Orb( \tau)$. - **Star**: define $N_\tau \da \ZZ \gens{\tau \intersect N}$ and $N(\tau)\RR \da N_\RR / (N_\tau)_\RR$ and $\bar\sigma$ for the image of $\sigma$ under the quotient map, then \[ \mathrm{Star}(\tau) \da \ts{\bar \sigma \subseteq N(\tau)_\RR \st \sigma\leq \tau } \subseteq N(\tau)_\RR .\] This is always a fan, and $V(\tau) = X_{\mathrm{Star}(\tau)}$. - **Star subdivision**: for $\sigma = \Cone(S)$ for $S \da \ts{u_1,\cdots, u_n}$, set $u_0 \da \sum u_i$ and take $\Sigma'(\sigma)$ defined as the cones generated by subsets of $\ts{u_0, u_1, \cdots, u_n}$ not containing $S$. The star subdivision of $\Sigma$ along $\sigma$ is $\Sigma^\star(\sigma) \da (\Sigma \smts{\sigma}) \union \Sigma'( \sigma)$. - **Blowups**: $\phi: X_{\Sigma^\star(\sigma)}\to X_{\Sigma}$ is the blowup at $\gamma_ \sigma$. ::: ### Divisors :::{.remark} \envlist - **(Weil) divisor**: $\Div(X) = \ts{\sum n_i V_i \st V_i \subseteq X, \codim V_i = 1}$. - $\OO_X(D)$: the (coherent) sheaf associated to a Weil divisor $D$. - **Cartier divisor**: $\CDiv(X) = H^0(X; \mck_X\units/\OO_X\units)$, the quotient of rational functions by regular functions. For $X$ normal, equivalently locally principal (Weil) divisors, so $D \leadsto \ts{(U_i, f_i)}$ where $\ro{D}{U_i} = \div(f_i)$. - **$\QQ\dash$Cartier divisor**: A $\QQ\dash$divisor $D =\sum n_i D_i$ with $n_i\in \QQ$ is $\QQ\dash$Cartier when $mD$ is Cartier for some $m\in \ZZ_{\geq 0}$. - **$\QQ\dash$factorial**: every prime divisor is $\QQ\dash$Cartier. - **Ray divisors**: every $\rho\in \Sigma(1)$ defines a divisor $D_\rho \da V(\rho) \da \cl\Orb( \rho)$. - **Very Ample**: $\mcl$ which defines a morphism into $\PP H^0(X; \mcl) \cong \PP^N$. - **Ample**: $\mcl$ is basepoint free and some power $\mcl^n$ is very ample. - $D$ is (very) ample iff $\OO_X(D)$ is (very) ample, i.e. $D$ is ample iff $nD$ is very ample for some $n$. - **Upper convex**: $f(n_1 + n_2) \leq f(n_1) + f(n_2)$. - **Strictly upper convex**: $\sigma_1\neq \sigma_2 \implies f_{\sigma_1} \neq f_{\sigma_2}$. - **Linearly equivalent divisors**: $D_1\sim D_2 \iff D_1 - D_2 = \div(f)$ for some $f$. - **Complete linear systems**: $\abs{D} = \ts{D'\in \Div(X) \st D'\sim D}$. - **Support function**: $\phi: \supp \Sigma \to \RR$ where $\ro{\phi}{\sigma}$ is linear for each cone $\sigma$. - **Integral** with respect to $N$ iff $\phi(\supp \Sigma \intersect N) \subseteq \ZZ$. Defines a set of integral support functions $\SF(\Sigma, N)$. - The class group complement exact sequence: for $D_1,\cdots, D_n \in \Div(X)$ distinct, \[ \ZZ^n &\to \Cl(X) \surjects \Cl(X\sm\union D_i) \\ e_1 &\mapsto [D_i] .\] - $\OO_X(D)$ is the sheaf \[ U\mapsto \ts{f\in \mck(X)\units(U) \st \div(f) + \ro{D}{U} \geq 0 \in \Cl(U) } .\] Then $D\in \CDiv(X) \iff \OO_X(D) \in \Pic(X)$. - The toric class group exact sequence: \[ M &\to \Div_T(X) \surjects \Cl(X) \\ m &\mapsto \div(\chi^m) = \sum_\rho \inp{m}{u_\rho} [D_\rho] \] where $u_\rho$ are minimal ray generators. ::: ### Polytopes :::{.remark} \envlist - **Supporting hyperplanes**: the positive side of an affine hyperplane \[ H_{u, b} &\da \ts{m\in M_\RR \st \inp m u = b} \\ H_{u, b}^+ &\da \ts{m\in M_\RR \st \inp m u \geq b} .\] - If $P$ is full dimensional and $F\leq P$ is a facet, then $F = P \intersect H_{u_F, -a_F}$ for a unique pair $(u_F, a_F) \in N_\RR\times \RR$. - **Polytope**: the convex hull of a finite set $S \subseteq N_\RR$ or an intersection of half-spaces: \[ P = \ts{\sum_{v\in S} \lambda_v v \st \sum \lambda_v = 1} = \Intersect _{i=1}^s H_{u_i, b_i}^+ .\] - **Simplex** $\dim P = d$ and there are exactly $d+1$ vertices. - **Simple**: $\dim P = d$ and every vertex is the intersection of exactly $d$ facets. - **Simplicial**: all facets are simplices. - E.g. simple but not simplicial: the cube in $\RR^3$, since each vertex meets 3 edges but a square is not a simplex. -E.g. Simplicial but not simple: the octahedron in $\RR^3$, since each vertex meets 4 edges but each face is a triangle. - **Combinatorial equivalence**: $P_1\sim P_2$ iff there is a bijection $P_1\to P_2$ preserving intersections, inclusions, and dimensions of all faces. - **Polar dual**: for $P \subseteq M_\RR$, \[ P^\circ = \ts{u\in N_\RR \st \inp m u \geq - 1\,\, \forall m\in P} .\] - Trick: for $P \subseteq M_\RR$ with $0\in P$, \[ P = \ts{m\in M_\RR \st \inp m {u_F} \geq -a_F,\, F \in \mathrm{Facets}(P) } \\ \implies P^\circ = \Conv(\ts{ a_F\inv u_F }) \subseteq N_\RR .\] E.g. write the square as $\ts{\inp{m}{\pm e_i}\geq -1}$, then $a_F = 1$ for all $F$: ![](figures/2022-10-19_18-35-59.png) - **Cone on a polytope**: $C(P) \da \Cone(P\times \ts{1}) \subseteq M_\RR \times \RR$, the set of cones through all proper faces of $P$. - **Normal**: $\qty{kP \intersect M} + \qty{\ell P \intersect M} \subseteq (k+\ell)P \intersect M$, or equivalently $k\cdot (P \intersect M) = (kP) \intersect M$, or equivalently $(P \intersect M)\times\ts{1}$ generates $C(P) \intersect (M\times \ZZ)$ as a semigroup. - If $P \subseteq M_\RR$ is a full-dimensional lattice polytope with $\dim P \geq 2$, then $kP$ is normal for all $k\geq \dim P - 1$. - Normal implies very ample. - $P\leadsto \mcl_P \in \Pic(X_P)$ - $P \intersect M \leadsto H^0(X_P; \mcl_P)$. - **Reflexive**: a polytope $P$ with facet presentation \[ P = \ts{m\in M_\RR \st \inp{m}{\mu_F} \geq -1 \forall F\in \mathrm{Facets}(P)} .\] Implies that $\int(P) \intersect M = \ts{\vector 0}$, and $P^\circ = \Conv(\ts{u_F \st F\in \mathrm{Facets}(P)})$. - **Polyhedron of a divisor $P_D$**: write $D = \sum_{\rho} a_{\rho} D_{\rho}$, for any $m\in M$, $\div(\chi^m) + D \geq 0 \implies \inp{m}{\rho} \geq a_{\rho} \implies \inp{m}{\rho} \geq - a_\rho$, so set \[ P_D \da \ts{ m\in M_\RR \st \inp m \rho \geq a_\rho \, \forall \rho \in \Sigma(1)} .\] - **Divisor of a polytope**: $D_P = \sum_F a_F D_F$ where $P = \ts{m \st \inp m{u_F} \geq -a_F}$. - $D_P$ is always the pullback of $\OO_{\PP^N}(1)$ along the embedding. - **Very ample polytopes**: for every vertex $v$, the semigroup $\ts{m' - v \st m'\in P \intersect M}$ is saturated in $M$. - Gives an embedding $X \embeds \PP^N$ where $N = \size(P \intersect M) - 1$. - The **toric variety of a polytope**: if $P \intersect M = \ts{m_1,\cdots, m_s}$ and $P$ is full dimensional very ample, then writing $T_N$ for the torus of $N$, \[ X_{P} \da \cl \im \phi,\qquad \phi: T_N &\to \PP^{s-1} \\ t &\mapsto \tv{\chi^{m_1}(t) : \cdots : \chi^{m_s}(t)} .\] - Vertices $m_i$ correspond to $U_{\sigma_i}$ for $\sigma_i = \Cone(P \intersect M - m_i)\dual$: ![](figures/2022-10-19_19-08-06.png) - **Smooth**: $P$ is smooth iff for all vertices $v\in P$, $\ts{w_E - v\st E\text{ is an edge containing }v}$ can be extended to a $\ZZ\dash$basis of $M$, where $w_E$ is the first lattice point on $E$. ::: ### Singularities and Classification :::{.remark} \envlist - **Gorenstein**: $X$ normal where $K_X \in \CDiv(X)$ is Cartier. - **Normal**: all local rings are integrally closed domains. - **Complete**: proper over $k$. E.g. for varieties, just universally closed. - **Factorial**: all local rings are UFDs. - **Fano**: $-K_X$ is ample. - **del Pezzo**: a smooth Fano surface. ::: :::{.remark} Classification of smooth complete toric varieties: - $\dim \Sigma = 2, \size \Sigma(1) = 3$: without loss of generality $\rho_1 = e_1, \rho_2 = e_2$. Then $\rho_3 = a e_1 + be_2$ with $a,b< 0$ to ensure $\supp \Sigma = \RR^2$, and determinants for $\abs{a} = \abs{b} = 1$, so $(-1, 1)$. - $\dim \Sigma = 2, \size \Sigma(1) = 4$: without loss of generality $\rho_1 = e_1, \rho_2 = e_2$. Then determinant conditions for $\rho_3 = (-1, b)$ and $\rho_4 = (a, -1)$, and $\det\matt{-1}{a}{b}{-1} = 1-ab = \pm 1 \implies ab=0,2$, so $(a,b) = (2,1), (1,2), (-2, -1), (-1,-2)$. - $\dim \Sigma = 2, \size\Sigma(1) = d$, smooth: $\Bl_{p_1,\cdots, p_\ell} X$ for $X = \PP^2$ or $\FF_a$ for some $a$ and $p_i$ torus fixed points. ::: ### Examples :::{.question} Things you can figure out for every example: - Given $\Delta$, for $\sigma\in \Delta$, - What is $\sigma\dual$? - Generators for $S_\sigma$? - Describe $U_\sigma$ and $X(\Delta)$. - What are the transition functions for $U_{\sigma_1} \to U_{\sigma_2}$ when $\sigma_1 \intersect \sigma_2 = \tau$ intersect in a common face? - What are the $T\dash$invariant points? - What are the $T\dash$invariant divisors $D_{\rho_i}$? - What are all of the $T\dash$orbit closures of various dimensions? - Is $X(\Delta)$ smooth? - Which cones $\sigma\in \Delta$ are smooth? - What is the canonical resolution of singularities? - What is the tangent space at each $T\dash$invariant point? - What is the associated polytope $P_\Delta$? What is its polar dual $P_\Delta^\circ$? - What are the intersection numbers $D_{\rho_i} \cdot D_{\rho_j}$? - What are the self-intersection numbers $D_{\rho_i}^2$? - What is $\Div_T(X)$? $\CDiv_T(X)$? - Which divisors are ample? Very ample? Globally generated? - What is $\Cl(X)$? $\Pic(X)$? - What is $K_X$? - Is $K_X$ ample? - Is $X(\Delta)$ projective? - What is $H^0(X(\Delta); \OO(D) )$ for $D\in \Div_T(X)$? - What is the Poincaré polynomial of $X(\Delta)$? (I.e. what are the Betti numbers?) ::: :::{.example title="of varieties"} Some useful explicit varieties: - $V(x^3-y^2)$ with torus $T = \ts{\tv{t^2, t^3} \st t\in \cstar}$. - $V(xy-zw)$ with torus $T = \ts{\tv{a,b,c,abc\inv} \st a,b,c,d\in \cstar}$. - $V(xz-y^2)$, note $V(x, y)\in \Div(X) \sm \CDiv(X)$. - $\im([x:y] \mapsto [x^3: x^2y : xy^2 : y^3])$ the twisted cubic. Corresponds to $\sigma\dual = \ts{(3,0), (2,1), (1,2), (0, 3)}$. - The **rational normal scroll**: $V\qty{2\times 2\text{ minors of } \left[\begin{array}{lll} x_0 & x_1 & y_0 \\ x_1 & x_2 & y_1 \end{array}\right]}$ is the image of $\tv{s,t} \mapsto \tv{1:s:s^2:t:st}$. - The Segre variety: $\spec \CC[x_1y_1, x_1 y_2, \cdots, x_1 y_n, x_2 y_1, \cdots, x_m y_1, \cdots x_m y_n]$. ::: :::{.example title="of fans"} \envlist - $(\CC\units)^n$: Take $\Delta = \ts{ \sigma_0 = \NN\gens{0}} \subseteq N$ with $\dim N = n$ yields $S_{\sigma_0} = \NN\gens{\pm e_1\dual,\cdots, \pm e_n\dual} = M$ for so $X(\Delta) = \spec \CC[x_1^{\pm 1},\cdots, x_n^{\pm 1}] = (\GG_m)^n$. - $\CC^n$: Take $\Delta = \Cone(\sigma_0 = \NN\gens{ e_1,\cdots, e_n} )$ yields the positive orthant $S_{\sigma_0} = \NN\gens{e_1\dual,\cdots, e_n\dual} \subseteq M$, so $X(\Delta) = \spec \CC[x_1,\cdots, x_n] = \AA^n$. - The quadric cone: $\Delta = \Cone(\sigma_1 = \NN\gens{e_2, 2e_1 - e_2})$ yields $S_{\sigma_1} = \NN\gens{e_1\dual, e_1\dual + e_2\dual, e_1\dual + 2e_2\dual}$ so $X(\Delta) = \spec \CC[x, xy, xy^2] = \spec \CC[u,v,w]/(v^2-uw)$: ![](figures/2022-10-18_15-33-37.png) ![](figures/2022-10-18_15-33-47.png) - $\PP^1$: Take $\Delta = \ts{\RR_{\geq 0}, 0, \RR_{\leq 0}}$ and glue along overlaps to get $X(\Delta) = \PP^1$ with gluing maps $x\mapsto x\inv$: ![](figures/2022-10-18_15-36-53.png) - $\Bl_1 \CC^2$: Take $\sigma_0 = \NN\gens{ e_2, e_1+e_2}$ and $\sigma_1 = \NN\gens{e_1+e_2, e_1}$ to get $U_{ \sigma_0} = \spec \CC[x, x\inv y]$ and $U_{ \sigma_1} = \spec \CC[y, xy\inv]$, both copies of $\CC^2$: ![](figures/2022-10-18_15-39-10.png) Why this is a blowup of $\CC^2$: write $\Bl_1 \CC^2 = V(xt_1 - yt_0) \subseteq \CC^2\times \PP^1$ for $\PP^1 = \ts{\tv{t_0: t_1}}$. Take the open cover $U_i = D(t_i) \cong \CC^2$, where coordinates on $U_0$ are $x, t_1/t_0 = x\inv y$ and on $U_1$ are $y, t_0/t_1 = xy\inv$ and glue. - $\PP^2$: take $\Delta = \Cone(e_1, e_2, -e_1-e_2)$: ![](figures/2022-10-18_15-42-04.png) This has dual cone: ![](figures/2022-10-18_15-42-18.png) Each $U_{\sigma_i} \cong \CC^2$ with coordinates $(x,y), (x\inv, x\inv y), (y\inv, xy\inv)$ respectively for $U_i$. Glue to obtain $x=t_1/t_0, y=t_2/t_0$. - $F_a$ the Hirzebruch surface: take $\Cone(e_1, -e_2, -e_1, -e_1 + ae_2)$ to get - $U_{\sigma_1} = \spec \CC[x,y]$, - $U_{\sigma_2} = \spec \CC[x,y\inv]$, - $U_{\sigma_3} = \spec \CC[x\inv,x^{-a} y\inv]$, - $U_{\sigma_4} = \spec \CC[x\inv ,x^a y]$, which patch in the following way: ![](figures/2022-10-18_15-45-17.png) Project to $y=0$ to get the patching $x\mapsto x\inv$, so a copy of $\PP^1$. Patching in the fiber direction, e.g. $U_{\sigma_1}$ and $U_{\sigma_2}$, gives a copy of $\CC \times \PP^1$. Thus this is a bundle $\PP^1\to \mce \to \PP^1$. - $\CC\times \PP^1$: todo. - $\PP^1 \times \PP^1$: todo. - $\CC^a \times \PP^b$: todo. - $\PP^a \times \PP^b$: todo. ::: :::{.example title="of polytopes"} - Hirzebruch surfaces: ![](figures/2022-12-03_20-09-23.png) - $(\PP^2, \OO(1))$: take $P = \Conv(0, e_1, e_2)$, so $X_P = \cl \Phi_P$ where \[ \Phi_P: (\cstar)^2 &\to \PP^2 \\ (s,t) &\mapsto [1: s: t] ,\] which is the identity embedding corresponding to $\OO(1)$ on $\PP^2$. - $2P$ yields \[ \Phi_{2P}: (\cstar)^2 &\to \PP^5 \\ (s,t) &\mapsto [1: s: t : s^2: st: t^2] ,\] the Veronese embedding corresponding to $\OO(2)$ on $\PP^2$. ::: :::{.example title="Projective spaces"} Some useful facts about $\PP^n$: - The torus embedding is \[ (\cstar)^n &\embeds \PP^n \\ \tv{a_1,\cdots, a_n} &\mapsto \tv{1: a_1 : \cdots : a_n} .\] - The torus action is \[ (\cstar)^n &\actson \PP^n \\ \tv{t_1,\cdots, t_n} . \tv{x_0: x_1:\cdots:x_n} &= \tv{x_0: t_1 x_1:\cdots:t_n x_n} .\] ::: :::{.example title="of class groups and Picard groups"} ![](figures/2022-10-20_00-07-20.png) ![](figures/2022-10-20_00-10-35.png) ::: # I: Definitions and Examples ## 1.1: Introduction :::{.remark} Machinery used to study varieties: - Various cohomology theories - Resolutions of singularities - Intersection theory and cycles - Riemann-Roch theorems - Vanishing theorems - Linear systems (via line bundles and projective embeddings) Varieties that arise as examples - Grassmannians - Flag varieties - Veronese embeddings - Scrolls - Quadrics - Cubic surfaces - Toric varieties (of course) - Symmetric varieties and their compactifications Misc notes: - Toric varieties are always rational ::: :::{.remark} \envlist - Toric varieties: normal varieties $X$ with $T\injects X$ contained as a dense open subset where the torus action $T\times T\to T$ extends to $T\times X\to X$. - Any product of copies of $\AA^n, \PP^m$ are toric. - $S_\sigma$ is a finitely-generated semigroup, so $\CC[S_\sigma] \in \alg{\CC}^\fg$ corresponds to an affine variety $U_\sigma \da \spec \CC[S_\sigma]$. - If $\tau \leq \sigma$ is a face then there is a map of affine varieties $U_\tau \to U_\sigma$ where $U_\tau = D(u_\tau)$ is a principal open subset given by the function $u_\tau$ picked such that $\tau = \sigma \intersect u_\tau^\perp$, so $u_\tau$ corresponds to the orthogonal normal vector for the wall $\tau$. - These glue to a variety $X(\Delta)$. - Smaller cones correspond to smaller open subsets. - The geometry in $N$ is nicer than that in $M$, usually. - Rays $\rho$ correspond to curves $D_\rho$. ::: :::{.exercise title="?"} \envlist - Show $F_a\to \PP^1$ is isomorphic to $\PP(\OO(a) \oplus \OO(1))$. - Let $\tau$ be the ray through $e_2$ in $F_a$ and show $D_\tau^2 = -a$. - Show that the normal bundle to $D_\tau \injects F_a$ is $\OO(-a)$. ::: ## 1.2: Convex Polyhedral Cones :::{.remark} \envlist - **Convex polyhedral cones**: generated by vectors $\sigma = \RR_{\geq 0}\gens{v_1,\cdots, v_n}$. Can take minimal vectors along these rays, say $\rho_i$. ![](figures/2022-10-18_20-35-45.png) - $\dim \sigma \da \dim_\RR \RR \sigma \da \dim_\RR (-\sigma + \sigma)$ - $(\sigma\dual)\dual = \sigma$, which follows from a general theorem: for $\sigma$ a convex polyhedral cone and $v\not\in \sigma$, there is some support vector $u_v\in \sigma\dual$ such that $\inp{u}{v} < 0$. I.e. $v$ is on the negative side of some hyperplane defined in $\sigma\dual$. - Faces are again convex polyhedral cones, faces are closed under intersections and taking further faces. - If $\sigma$ spans $V$ and $\tau$ is a facet, there is a unique $u_\tau\in \sigma\dual$ such that $\tau = \sigma \intersect u_\tau^\perp$; this defines an equation for the hyperplane $H_\tau$ spanned by $\tau$. - If $\sigma$ spans $V$ and $\sigma\neq V$, then $\sigma = \intersect_{\tau\in \Delta} H_\tau^+$, the intersection of positive half-spaces. - An alternative presentation: picking $u_1,\cdots, u_t$ generators of $\sigma\dual$, one has $\sigma = \ts{v\in N \st \inp{u_1}{v} \geq 0, \cdots, \inp{u_t}{v}\geq 0}$. - If $\tau \leq \sigma$ then $\sigma\dual \intersect \tau\dual \leq \sigma\dual$ and $\dim \tau = \codim(\sigma\dual \intersect \tau\dual)$, so the faces of $\sigma, \sigma\dual$ biject contravariantly. - If $\tau = \sigma \intersect u_\tau^\perp$ then $S_\tau = S_\sigma + \NN\gens{-u_\tau}$. ::: # Singularities and Compactness ## 2.1 :::{.remark} - Any cone $\sigma\in \Sigma$ has a distinguished point $x_\sigma$ corresponding to $\Hom_{\semigroup}(S_\sigma, \CC)$ where $u\mapsto \chi_{u\in \sigma^\perp}$. - Note $S_\sigma \da \sigma\dual \intersect M$. - Define $A_\sigma \da \CC[S_\sigma]$. - Finding singular points: - Easy case: $\sigma$ spans $N_\RR$ so $\sigma^\perp = 0$; consider $\mfm \in \mspec A_\sigma$ be the maximal ideal at $x_\sigma$, then $\mfm = \gens{\chi^u \st u\in S_ \sigma}$ and $\mfm^2 = \gens{\chi^u \st u \in S_\sigma\smz + S_\sigma\smz}$, so $\T_{x_\sigma}\dual U_\sigma = \mfm/\mfm^2 = \ts{\chi^u \st u\not \in S_{\sigma}\smz + S_\sigma\smz}$, i.e. "primitive" elements $u$ which are not the sums of two other vectors in $S_\sigma\smz$. - Nonsingular implies $\dim U_\sigma = n$, so $\sigma\dual$ has $\leq n$ edges since each minimal ray generator yields a primitive $u$ above. Also implies minimal edge generators must generate $S_\sigma$, thus must be a basis for $M$, so $\sigma$ must be a basis for $N$ and $U_\sigma \cong \AA^n$. - **Characterization of smoothness**: $U_\sigma$ is smooth iff $\sigma$ is generated by a subset of a lattice basis for $N$, in which case $U_\sigma \cong \AA^k \times \GG_m^{n-k}$. - All toric varieties are normal since each $A_\sigma$ is integrally closed. - If $\sigma = \gens{v_1,\cdots, v_r}$ then $\sigma\dual = \intersect_{i=1}^r \tau_i\dual$ where $\tau_i$ is the ray along $v_i$. Thus $A_\sigma = \intersect A_{\tau_i}$, each of which is isomorphic to $\CC[x_1, x_2^{\pm 1}, \cdots, x_n^{\pm 1}$ which is integrally closed. - All toric varieties are **Cohen-Macaulay**: each local ring $R$ has depth $n$, i.e. contains a regular sequence of length $n = \dim R$. - All vector bundles on affine toric varieties are trivial, equivalently all projective modules over $A_\sigma$ are free. ::: ## 2.2 :::{.remark} - An example: $\Sigma = \Cone(me_1-e_2, e_2)$. Then $A_\sigma = \CC[x, xy, xy^2,\cdots,xy^m] = \CC[u^m u^{m-1}v,\cdots, uv^{m-1}, v^m]$ and $U_\sigma$ is the cone over the rational normal curve of degree $m$. - Note $A_\sigma = \CC[u,v]^{\mu_m}$ is the ring of invariants under the diagonal action $\zeta.\tv{u, v} = \tv{\zeta u, \zeta v}$. - If $\Sigma$ is simplicial, then $X_\Sigma$ is at worst an orbifold. ::: ## 2.3 :::{.remark} - $\Hom_{\Alg\Grp}(\GG_m, \GG_m) = \ZZ$ using $n\mapsto (z\mapsto z^n)$. - Cocharacters: - Pick a basis for $N$ to get $\Hom(\GG_m, T_N) = \Hom(\ZZ, N) = N$, then every cocharacter $\lambda \in \Hom(\GG_m, T_N)$ is given by a unique $v\in N$, so denote it $\lambda_v$. Then $\lambda_v(z)\in T_N = \Hom(M, \GG_m)$ for any $z\in \CCstar$, so \[ u\in M \implies \lambda_v(z)(u) = \chi^u(\lambda_v(z))= z^{\inp u v} .\] - Characters: $\chi \in \Hom(T_n, \GG_m) = \Hom(N, \ZZ) = M$ is given by a unique $u\in M$ and can be identified with $\chii^u \in \CC[M] = H^0(T_N, \OO_{T_N}\units)$. - $\lim_{z\to 0}\lambda_v(z) = \lim_{z\to 0} \tv{z^{m_1},\cdots, z^{m_n}} \in U_\sigma \iff m_i \geq 0$ for all $i$, and if $U_\sigma = \AA^k\times \GG_m^{n-k}$, $m_i = 0$ for $i > k$. This happens iff $v\in \sigma$, and the limit is $\tv{\delta_1,\cdots, \delta_n}$ where $\delta_i = 1\iff m_i = 0$ and $\delta_i = 0\iff m_i > 0$; each of which is a distinguished point $x_\tau$ for some face $\tau$ of $\sigma$. - Summary: $v\in \abs{\Sigma}$ and $v\in \tau\interior$ then $\lim_{z\to 0} \lambda_v(z) = x_\tau$, and the limit does not exist for $v\not\in\abs{\Sigma}$. ::: ## 2.4 :::{.remark} - Recall $X$ is compact in the Euclidean topology iff it is complete/proper in the Zariski topology, i.e. the map to a point is proper. - $X_\Sigma$ is compact iff $\abs{\Sigma} = N_\RR$, i.e. $\Sigma$ is complete. - Any morphism of lattices $\phi:N\to N'$ inducing a map of fans $\Sigma\to \Sigma'$ defines a morphism $X_{\Sigma}\to X_{\Sigma'}$ which is proper iff $\phi\inv(\abs{\Sigma'}) = \abs{\Sigma}$. Thus $X_\Sigma$ is compact iff $\phi: N\to 0$ is a proper morphism. - Blowing up at $x_\sigma$: take a basis $\ts{v_i}$, set $v_0\da \sum v_i$, and replace $\sigma$ by all subsets of $\ts{v_0,v_1,\cdots, v_n}$ not containing $\ts{v_1, \cdots, v_n}$. :::