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\title{
\textbf{
Preview
}
\\ {\normalsize University of Georgia} \\
}
\begin{document}
\date{}
\maketitle
\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
\textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2022-12-07 }
\end{flushleft}
\newpage
% Note: addsec only in KomaScript
\addsec{Table of Contents}
\tableofcontents
\newpage
\newpage
\hypertarget{i-varieties}{%
\section{I: Varieties}\label{i-varieties}}
\begin{remark}
Some useful basic properties:
\begin{itemize}
\tightlist
\item
Properties of \(V\):
\begin{itemize}
\tightlist
\item
\(\cap_{i\in I} V({\mathfrak{a}}_i) = V\qty{\sum_{i\in I} {\mathfrak{a}}_i}\).
\begin{itemize}
\tightlist
\item
E.g.
\(V(x) \cap V(y) = V(\left\langle{x}\right\rangle + \left\langle{y}\right\rangle)= V(x, y) = \left\{{0}\right\}\),
the origin.
\end{itemize}
\item
\(\cup_{i\leq n} V({\mathfrak{a}}_i) = V\qty{\prod_{i\leq n} {\mathfrak{a}}_i}\).
\begin{itemize}
\tightlist
\item
E.g.
\(V(x) \cup V(y) = V(\left\langle{x}\right\rangle\left\langle{y}\right\rangle) = V(xy)\),
the union of coordinate axes.
\end{itemize}
\item
\(V({\mathfrak{a}})^c = \cup_{f\in {\mathfrak{a}}} D(f)\)
\item
\(V({\mathfrak{a}}_1) \subseteq V({\mathfrak{a}}_2) \iff \sqrt{{\mathfrak{a}}_1}\supseteq\sqrt{{\mathfrak{a}}_2}\).
\end{itemize}
\item
Properties of \(I\):
\begin{itemize}
\tightlist
\item
\(I(V({\mathfrak{a}})) = \sqrt{\mathfrak{a}}\) and
\(V(I(Y)) = { \operatorname{cl}}_{{\mathbf{A}}^n}(Y)\). The
containment correspondence is contravariant in both directions.
\item
\(I(\cup_i Y_i) = \cap_i I(Y_i)\).
\end{itemize}
\item
If \(F\) is a sheaf taking values in subsets of a giant ambient set,
then \(F(\cup U_i) = \cap F(U_i)\). For
\({\mathbf{A}}^n/{\mathbf{C}}\), take
\({\mathbf{C}}(x_1,\cdots, x_n)\), the field of rational functions, to
be the ambient set.
\item
Distinguished open
\(D(f) \coloneqq\left\{{p\in X {~\mathrel{\Big\vert}~}f(p) \neq 0}\right\}\):
\begin{itemize}
\tightlist
\item
\({\mathcal{O}}_X(D(f)) = A(X){ \left[ { \scriptstyle \frac{1}{f} } \right] } = \left\{{{g\over f^k} {~\mathrel{\Big\vert}~}g\in A(X), k\geq 0}\right\}\),
and taking \(f=1\) shows \({\mathcal{O}}_X(X) = A(X)\), i.e.~global
regular functions are polynomial.
\item
Generally \(D(fg) = D(f) \cap D(g)\)
\item
For affines:
\begin{align*}
{\mathcal{O}}_{\operatorname{Spec}R}(D(f)) = R{ \left[ { \scriptstyle \frac{1}{f} } \right] }
.\end{align*}
\item
For \({\mathbf{C}}^n\),
\begin{align*}
{\mathcal{O}}_{{\mathbf{C}}^n}(D(f)) = k[x_1, \cdots, x_{n}] { \left[ \scriptstyle {1/f} \right] }
\implies
{\mathcal{O}}_{{\mathbf{C}}^n}(V({\mathfrak{a}})^c) = \cap_{f\in {\mathfrak{a}}} {\mathcal{O}}_{{\mathbf{C}}^n}(D(f))
.\end{align*}
\end{itemize}
\end{itemize}
\end{remark}
\hypertarget{i.1-affine-varieties-star}{%
\subsection{\texorpdfstring{I.1: Affine Varieties
\(\star\)}{I.1: Affine Varieties \textbackslash star}}\label{i.1-affine-varieties-star}}
\begin{remark}
Summary:
\begin{itemize}
\tightlist
\item
\({\mathbf{A}}^n_{/ {k}} = \left\{{{\left[ {a_1,\cdots, a_n} \right]} {~\mathrel{\Big\vert}~}a_i \in k}\right\}\),
and elements \(f\in A \coloneqq k[x_1, \cdots, x_{n}]\) are functions
on it.
\item
\(Z(f) \coloneqq\left\{{p\in {\mathbf{A}}^n {~\mathrel{\Big\vert}~}f(p) = 0}\right\}\),
and for any \(T \subseteq A\) we set
\(Z(T) \coloneqq\cap_{f\in T} Z(f)\).
\begin{itemize}
\tightlist
\item
Note that
\(Z(T) = Z(\left\langle{T}\right\rangle_A) = Z(\left\langle{f_1,\cdots, f_r}\right\rangle)\)
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.
\end{itemize}
\item
\textbf{Algebraic}: \(Y \subseteq {\mathbf{A}}^n\) is algebraic iff
\(Y = Z(T)\) for some \(T \subseteq A\).
\item
The Zariski topology is generated by open sets of the form \(Z(T)^c\).
\item
\({\mathbf{A}}^1\) is a non-Hausdorff space with the cofinite
topology.
\item
\textbf{Irreducible}: \(Y\) is reducible iff \(Y = Y_1 \cup Y_2\) with
\(Y_1, Y_2\) proper subsets of \(Y\) which are closed in \(Y\).
\begin{itemize}
\tightlist
\item
Nonempty open subsets of irreducible spaces are both irreducible and
dense.
\item
If \(Y \subseteq X\) is irreducible then
\({ \operatorname{cl}}_X(Y) \subseteq X\) is again irreducible.
\end{itemize}
\item
\textbf{Affine (algebraic) varieties}: irreducible closed subsets of
\({\mathbf{A}}^n\).
\item
\textbf{Quasi-affine varieties}: open subsets of affine varieties.
\item
The ideal of a subset:
\(I(Y) \coloneqq\left\{{f\in A {~\mathrel{\Big\vert}~}f(p) = 0 \,\, \forall p\in Y}\right\}\).
\item
\textbf{Nullstellensatz}: if
\(k = \overline{k}, {\mathfrak{a}}\in \operatorname{Id}(k[x_1, \cdots, x_{n}])\),
and \(f\in k[x_1, \cdots, x_{n}]\) with \(f(p) = 0\) for all
\(p\in V({\mathfrak{a}})\), then \(f^r \in {\mathfrak{a}}\) for some
\(r>0\), so \(f\in \sqrt{\mathfrak{a}}\). Thus there is a
contravariant correspondence between radical ideals of
\(k[x_1, \cdots, x_{n}]\) and algebraic sets in
\({\mathbf{A}}^n_{/ {k}}\).
\item
\textbf{Irreducibility criterion}: \(Y\) is irreducible iff
\(I(Y) \in \operatorname{Spec}k[x_1, \cdots, x_{n}]\) (i.e.~it is
prime).
\item
\textbf{Affine curves}: if \(f\in k[x,y]^{\mathrm{irr}}\) then
\(\left\langle{f}\right\rangle \in \operatorname{Spec}k[x,y]\) (since
this is a UFD) so \(Z(f)\) is irreducible and defines an affine curve
of degree \(d= \deg(f)\).
\item
\textbf{Affine surfaces}: \(Z(f)\) for
\(f\in k[x_1, \cdots, x_{n}]^{\mathrm{irr}}\) defines a surface.
\item
\textbf{Coordinate rings}:
\(A(Y) \coloneqq k[x_1, \cdots, x_{n}]/I(Y)\).
\item
\textbf{Noetherian spaces}: \(X\in {\mathsf{Top}}\) is Noetherian iff
the DCC on closed subsets holds.
\item
\textbf{Unique decomposition into irreducible components}: if
\(X\in {\mathsf{Top}}\) is Noetherian then every closed nonempty
\(Y \subseteq X\) is of the form \(Y = \cup_{i=1}^r Y_i\) with \(Y_i\)
a uniquely determined closed irreducible with
\(Y_i \not\subseteq Y_j\) for \(i\neq j\), the \emph{irreducible
components} of \(Y\).
\item
\textbf{Dimension}: for \(X\in {\mathsf{Top}}\), the dimension is
\(\dim X \coloneqq\sup \left\{{n {~\mathrel{\Big\vert}~}\exists Z_0 \subset Z_1 \subset \cdots \subset Z_n}\right\}\)
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.
\item
\textbf{Height}: for \({\mathfrak{p}}\in\operatorname{Spec}A\) define
\(\operatorname{ht}({\mathfrak{p}}) \coloneqq\sup\left\{{n{~\mathrel{\Big\vert}~}\exists {\mathfrak{p}}_0 \subset {\mathfrak{p}}_1 \subset \cdots \subset {\mathfrak{p}}_n = {\mathfrak{p}}}\right\}\)
with \({\mathfrak{p}}_i \in \operatorname{Spec}A\) distinct prime
ideals.
\item
\textbf{Krull dimension}: define
\(\operatorname{krulldim}A \coloneqq\sup_{{\mathfrak{p}}\in \operatorname{Spec}A}\operatorname{ht}({\mathfrak{p}})\),
the supremum of heights of prime ideals.
\end{itemize}
\end{remark}
\begin{exercise}[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.
\end{exercise}
\begin{exercise}[The affine line]
\envlist
\begin{itemize}
\tightlist
\item
Show that \({\mathbf{A}}^1_{/ {k}}\) has the cofinite topology when
\(k=\overline{k}\): the closed (algebraic) sets are finite sets and
the whole space, so the opens are empty or complements of finite
sets.\footnote{Hint: \(k[x]\) is a PID and factor any \(f(x)\) into
linear factors using that \(k = \overline{k}\) to write
\(Z({\mathfrak{a}}) = Z(f) = \left\{{a_1,\cdots, a_k}\right\}\) for
some \(k\).}
\item
Show that this topology is not Hausdorff.
\item
Show that \({\mathbf{A}}^1\) is irreducible without using the
Nullstellensatz.
\item
Show that \({\mathbf{A}}^n\) is irreducible.
\item
Show that maximal ideals
\({\mathfrak{m}}\in \operatorname{mSpec}k[x_1, \cdots, x_{n}]\)
correspond to minimal irreducible closed subsets
\(Y \subseteq {\mathbf{A}}^n\), which must be points.
\item
Show that
\(\operatorname{mSpec}k[x_1, \cdots, x_{n}]= \left\{{\left\langle{x_1-a_1,\cdots, x_n-a_n}\right\rangle {~\mathrel{\Big\vert}~}a_1,\cdots, a_n\in k}\right\}\)
for \(k=\overline{k}\), and that this fails for
\(k\neq \overline{k}\).
\item
Show that \({\mathbf{A}}^n\) is Noetherian.
\item
Show \(\dim {\mathbf{A}}^1 = 1\).
\item
Show \(\dim {\mathbf{A}}^n = n\).
\end{itemize}
\end{exercise}
\begin{exercise}[Commutative algebra]
\envlist
\begin{itemize}
\tightlist
\item
Show that if \(Y\) is affine then \(A(Y)\) is an integral domain and
in \({}_{k} \mathsf{Alg}^{\mathrm{fg}}\).
\item
Show that every
\(B \in {}_{k} \mathsf{Alg}^{\mathrm{fg}}\cap\mathsf{Domain}\) is of
the form \(B = A(Y)\) for some
\(Y\in{\mathsf{Aff}}{\mathsf{Var}}_{/ {k}}\).
\item
Show that if \(Y\) is an affine algebraic set then
\(\dim Y = \operatorname{krulldim}A(Y)\).
\end{itemize}
\end{exercise}
\begin{theorem}[Results from commutative algebra]
\envlist
\begin{itemize}
\tightlist
\item
If
\(k\in \mathsf{Field}, B\in {}_{k} \mathsf{Alg}^{\mathrm{fg}}\cap\mathsf{Domain}\),
\begin{itemize}
\tightlist
\item
\(\operatorname{krulldim}B = [K(B) : B]_{\mathrm{tr}}\) is the
transcendence degree of the quotient field of \(B\) over \(B\).
\item
If \({\mathfrak{p}}\in \operatorname{Spec}B\) then
\(\operatorname{ht}{\mathfrak{p}}+ \operatorname{krulldim}(B/{\mathfrak{p}}) = \operatorname{krulldim}B\).
\end{itemize}
\item
Krull's Hauptidealsatz:
\begin{itemize}
\tightlist
\item
If \(A \in \mathsf{CRing}^{ \mathrm{Noeth} }\) and
\(f\in A\setminus A^{\times}\) is not a zero divisor, then every
minimal \({\mathfrak{p}}\in \operatorname{Spec}A\) with
\({\mathfrak{p}}\ni f\) has height 1.
\end{itemize}
\item
If \(A \in \mathsf{CRing}^{ \mathrm{Noeth} }\cap\mathsf{Domain}\),
then \(A\) is a UFD iff every
\({\mathfrak{p}}\in \operatorname{Spec}(A)\) with
\(\operatorname{ht}({\mathfrak{p}}) = 1\) is principal.
\end{itemize}
\end{theorem}
\begin{exercise}[1.10]
Show that if \(Y\) is quasi-affine then
\begin{align*}
\dim Y = \dim { \operatorname{cl}}_{{\mathbf{A}}^n} Y
.\end{align*}
\end{exercise}
\begin{exercise}[1.13]
Show that if \(Y \subseteq {\mathbf{A}}^n\) then
\(\operatorname{codim}_{{\mathbf{A}}^n}(Y) = 1 \iff Y = Z(f)\) for a
single nonconstant \(f\in k[x_1, \cdots, x_{n}]^{\mathrm{irr}}\).
\end{exercise}
\begin{exercise}[?]
Show that if \({\mathfrak{p}}\in \operatorname{Spec}(A)\) and
\(\operatorname{ht}({\mathfrak{p}}) = 2\) then \({\mathfrak{p}}\) can
not necessarily be generated by two elements.
\end{exercise}
\hypertarget{i.2-projective-varieties-star}{%
\subsection{\texorpdfstring{I.2: Projective Varieties
\(\star\)}{I.2: Projective Varieties \textbackslash star}}\label{i.2-projective-varieties-star}}
\begin{remark}
\envlist
\begin{itemize}
\item
\textbf{Projective space}:
\(\left\{{\mathbf{a} \coloneqq{\left[ {a_0, \cdots, a_n} \right]} {~\mathrel{\Big\vert}~}a_i \in k}\right\}/\sim\)
where \(\mathbf{a} \sim \lambda \mathbf{a}\) for all
\(\lambda \in k\setminus\left\{{0}\right\}\), i.e.~lines in
\({\mathbf{A}}^{n+1}\) passing through \(\mathbf{0}\).
\item
\textbf{Graded rings}: a ring \(S\) with a decomposition
\(S = \oplus _{d\geq 0} S_d\) with each
\(S_d\in {\mathsf{Ab}}{\mathsf{Grp}}\) and
\(S_d S_e \subseteq S_{d+e}\); elements of \(S_d\) are
\textbf{homogeneous of degree \(d\)} and any element in \(S\) is a
finite sum of homogeneous elements of various degrees.
\item
\textbf{Homogeneous polynomials}: \(f\) is homogeneous of degree \(d\)
if
\(f(\lambda x_0, \cdots, \lambda x_n) = \lambda^d f(x_0, \cdots, x_n)\).
\item
\textbf{Homogeneous ideals}: \({\mathfrak{a}}\subseteq S\) is
homogeneous when it's of the form
\({\mathfrak{a}}= \bigoplus _{d\geq 0} ({\mathfrak{a}}\cap S_d)\).
\begin{itemize}
\tightlist
\item
\({\mathfrak{a}}\) is homogeneous iff generated by homogeneous
elements.
\item
The class of homogeneous ideals is closed under sums, products,
intersections, and radicals.
\item
Primality of homogeneous ideals can be tested on homogeneous
elements, i.e.~it STS
\(fg\in {\mathfrak{a}}\implies f,g\in {\mathfrak{a}}\) for \(f,g\)
homogeneous.
\end{itemize}
\item
\(k[x_1, \cdots, x_{n}]= \bigoplus _{d\geq 0} k[x_1, \cdots, x_{n}]_d\)
where the degree \(d\) part is generated by monomials of total weight
\(d\).
\begin{itemize}
\tightlist
\item
E.g.
\begin{align*}
k[x_1, \cdots, x_{n}]_1 &= \left\langle{x_1, x_2,\cdots, x_n}\right\rangle\\
k[x_1, \cdots, x_{n}]_2 &= \left\langle{x_1^2, x_1x^2, x_1x_3,\cdots, x_2^2,x_2x_3, x_2x_4,\cdots, x_n^2}\right\rangle
.\end{align*}
\item
Useful fact: by stars and bars,
\(\operatorname{rank}_k k[x_1, \cdots, x_{n}]_d = {d+n \choose n}\).
E.g. for \((d, n) = (3, 2)\),
\end{itemize}
\includegraphics{figures/2022-10-08_18-48-17.png}
\item
Arbitrary polynomials \(f\in k[x_0, \cdots, x_{n}]\) do not define
functions on \({\mathbf{P}}^n\) 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
\begin{align*}
\operatorname{ev}_f: {\mathbf{P}}^n &\to \left\{{0, 1}\right\} \\
p &\mapsto
\begin{cases}
0 & f(p) = 0
\\
1 & f(p) \neq 0.
\end{cases}
.\end{align*}
So
\(Z(f) \coloneqq\left\{{p\in {\mathbf{P}}^n {~\mathrel{\Big\vert}~}f(p) = 0}\right\}\)
makes sense.
\item
\textbf{Projective algebraic varieties}: \(Y\) is projective iff it is
an irreducible algebraic set in \({\mathbf{P}}^n\). Open subsets of
\({\mathbf{P}}^n\) are \textbf{quasi-projective varieties}.
\item
\textbf{Homogeneous ideals of varieties}:
\begin{align*}
I(Y) \coloneqq\left\{{f\in k[x_0, \cdots, x_{n}]^ { \mathrm{homog} }{~\mathrel{\Big\vert}~}f(p) =0 \, \forall p\in Y}\right\}
.\end{align*}
\item
\textbf{Homogeneous coordinate rings}:
\begin{align*}
S(Y) \coloneqq k[x_0, \cdots, x_{n}]/I(Y)
.\end{align*}
\item
\(Z(f)\) for \(f\) a linear homogeneous polynomial defines a
\textbf{hyperplane}.
\end{itemize}
\end{remark}
\begin{exercise}[Cor. 2.3]
Show \({\mathbf{P}}^n\) admits an open covering by copies of
\({\mathbf{A}}^n\) by explicitly constructing open sets \(U_i\) and
well-defined homeomorphisms \(\phi_i :U_i\to {\mathbf{A}}^n\).
\end{exercise}
\hypertarget{i.3-morphisms}{%
\subsection{I.3: Morphisms}\label{i.3-morphisms}}
\hypertarget{i.4-rational-maps}{%
\subsection{I.4: Rational Maps}\label{i.4-rational-maps}}
\hypertarget{i.5-nonsingular-varieties}{%
\subsection{I.5: Nonsingular
Varieties}\label{i.5-nonsingular-varieties}}
\hypertarget{i.6-nonsingular-curves}{%
\subsection{I.6: Nonsingular Curves}\label{i.6-nonsingular-curves}}
\hypertarget{i.7-intersections-in-projective-space}{%
\subsection{I.7: Intersections in Projective
Space}\label{i.7-intersections-in-projective-space}}
\newpage
\hypertarget{ii-schemes}{%
\section{II: Schemes}\label{ii-schemes}}
\begin{quote}
Note: there are many, many important notions tucked away in the
exercises in this section.
\end{quote}
\hypertarget{ii.1-sheaves-star}{%
\subsection{\texorpdfstring{II.1: Sheaves
\(\star\)}{II.1: Sheaves \textbackslash star}}\label{ii.1-sheaves-star}}
\begin{remark}
\envlist
\begin{itemize}
\tightlist
\item
\textbf{Presheaves} \(F\) of abelian groups: contravariant functors
\(F\in {\mathsf{Fun}}({\mathsf{Open}}(X), {\mathsf{Ab}}{\mathsf{Grp}})\).
\begin{itemize}
\tightlist
\item
Assigns every open \(U \subseteq X\) some
\(F(U) \in {\mathsf{Ab}}{\mathsf{Grp}}\)
\item
For \(\iota_{VU}: V \subseteq U\), restriction morphisms
\(\phi_{UV}: F(U) \to F(V)\).
\item
\(F(\emptyset) = 0\), so
\(F({ \mathscr \emptyset^{\scriptscriptstyle \downarrow}}) = { 0_{\scriptscriptstyle \uparrow}}\).
\item
\(\phi_{UU} = \operatorname{id}_{F(U)}\)
\item
\(W \subseteq V \subseteq U \implies \phi_{UW} = \phi_{VW} \circ \phi_{UV}\).
\end{itemize}
\item
\textbf{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
\({ \left.{{s}} \right|_{{V}} } \coloneqq\phi_{UV}(s)\) for
\(s\in F(U)\).
\item
\textbf{Sheaves}: presheaves \(F\) which are completely determined by
local data. Additional requirements on open covers
\({\mathcal{V}}\rightrightarrows U\):
\begin{itemize}
\tightlist
\item
If \(s\in F(U)\) with \({ \left.{{s}} \right|_{{V_i}} } = 0\) for
all \(i\) then \(s\equiv 0 \in F(U)\).
\item
Given \(s_i\in F(V_i)\) where
\({ \left.{{s_i}} \right|_{{V_{ij}}} } = { \left.{{s_j}} \right|_{{V_{ij}}} } \in F(V_{ij})\)
then \(\exists s\in F(U)\) such that
\({ \left.{{s}} \right|_{{V_i}} } = s_i\) for each \(i\), which is
unique by the previous condition.
\end{itemize}
\item
\textbf{Constant sheaf}: for \(A\in {\mathsf{Ab}}{\mathsf{Grp}}\),
define the constant sheaf
\begin{align*}
\underline{A}(U) \coloneqq{\mathsf{Top}}(U, A^{\operatorname{disc}})
.\end{align*}
\item
\textbf{Stalks}: \(F_p \coloneqq\colim_{U\ni p} F(U)\) along the
system of restriction maps.
\begin{itemize}
\tightlist
\item
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
\(\exists W \subseteq U \cap V\) with
\({ \left.{{s}} \right|_{{w}} } = { \left.{{t}} \right|_{{w}} }\).
\end{itemize}
\item
\textbf{Germs}: a germ of a section of \(F\) at \(p\) is an elements
of the stalk \(F_p\).
\item
\textbf{Morphisms of presheaves}: natural transformations
\(\eta\in \mathop{\mathrm{Mor}}_{{\mathsf{Fun}}}(F, G)\), i.e.~for
every \(U, V\), components \(\eta_U, \eta_V\) fitting into a diagram
\end{itemize}
\begin{center}
\begin{tikzcd}
{{\mathsf{Open}}(X)} &&& {\mathsf{Ab}}{\mathsf{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}
\end{center}
\begin{quote}
\href{https://q.uiver.app/?q=WzAsOCxbMCwxLCJVIl0sWzAsMywiViJdLFsyLDEsIkYoVSkiXSxbNCwxLCJHKFUpIl0sWzIsMywiRihWKSJdLFs0LDMsIkcoVikiXSxbMCwwLCJcXE9wZW4oWCkiXSxbMywwLCJcXEFiXFxHcnAiXSxbNCw1LCJcXGV0YV9WIl0sWzIsMywiXFxldGFfVSJdLFsyLDQsIlxcbWF0aHJte1Jlc31fRihVLCBWKSJdLFszLDUsIlxcbWF0aHJte1Jlc31fRyhVLCBWKSJdLFsxLDAsIiIsMSx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEyLDEwLCJGLCBHIiwwLHsic2hvcnRlbiI6eyJzb3VyY2UiOjIwLCJ0YXJnZXQiOjIwfX1dXQ==}{Link
to Diagram}
\end{quote}
\begin{itemize}
\item
A morphism of sheaves is exactly a morphism of the underlying
presheaves.
\item
Morphisms of sheaves \(\eta: F\to G\) induce morphisms of rings on the
stalks \(\eta_p: F_p \to G_p\).
\item
Morphisms of sheaves are isomorphisms iff isomorphisms on all stalks,
see exercise below.
\item
\textbf{Kernels, cokernels, images}: for \(\phi: F\to G\), sheafify
the assignments to kernels/cokernels/images on open sets.
\item
\textbf{Sheafification}: for any
\(F\in \underset{ \mathsf{pre}} {\mathsf{Sh}}(X)\), there is a unique
\(F^+\in {\mathsf{Sh}}(X)\) and a morphism \(\theta: F\to F^+\) of
presheaves such that any sheaf presheaf morphism \(F\to G\) factors as
\(F\to F^+ \to G\).
\begin{itemize}
\tightlist
\item
The construction:
\(F^+(U) = {\mathsf{Top}}(U, {\textstyle\coprod}_{p\in U} F_p)\) 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 \(V\supseteq U \ni p\) and \(t\in F(V)\) such that
for all \(q\in V\), the germ \(t_q\) is equal to \(s(q)\).
\item
Note that the stalks are the same: \((F^+)_p = F_p\), and if \(F\)
is already a sheaf then \(\theta\) is an isomorphism.
\end{itemize}
\item
\textbf{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 \(F\).
\begin{itemize}
\tightlist
\item
If \(F'\leq F\) then \(F'_p \leq F_p\).
\item
\textbf{Injectivity}: \(\phi: F\to G\) is injective iff the sheaf
kernel \(\ker \phi = 0\) as a subsheaf of \(F\).
\begin{itemize}
\tightlist
\item
\(\phi\) is injective iff injective on all sections.
\end{itemize}
\item
\(\operatorname{im}\phi\leq G\) is a subsheaf.
\item
\textbf{Surjectivity}: \(\phi: F\to G\) is surjective iff
\(\operatorname{im}\phi = G\) as a subsheaf.
\end{itemize}
\item
\textbf{Exactness}: a sequence of sheaves
\((F_i, \phi_i:F_i\to F_{i+1})\) is exact iff
\(\ker \phi_i = \operatorname{im}\phi^{i-1}\) as subsheaves of
\(F_i\).
\begin{itemize}
\tightlist
\item
\(\phi:F\to G\) is injective iff \(0\to F \xrightarrow{\phi} G\) is
exact.
\item
\(\phi: F\to G\) is surjective iff \(F \xrightarrow{\phi} G \to 0\)
is exact.
\item
Sequences of sheaves are exact iff exact on stalks.
\end{itemize}
\item
\textbf{Quotient sheaves}: \(F/F'\) is the sheafification of
\(U\mapsto F(U) / F'(U)\).
\item
\textbf{Cokernels}: for \(\phi: F\to G\), \(\operatorname{coker}\phi\)
is sheafification of
\(U\mapsto \operatorname{coker}( F(U) \xrightarrow{\phi(U)} G(U))\).
\item
\textbf{Direct images}: for \(f \in {\mathsf{Top}}(X, Y)\), the sheaf
defined on sections by \((f_* F)(V) \coloneqq F(f^{-1}(V))\) for any
\(V \subseteq Y\). Yields a functor
\(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\).
\item
\textbf{Inverse images}: denoted \(f^{-1}G\), the sheafification of
\(U \mapsto \colim_{V\supseteq f(U)} G(V)\), i.e.~take the limit from
above of all open sets \(V\) of \(Y\) containing the image \(f(U)\).
Yields a functor \(f^{-1}: {\mathsf{Sh}}(Y) \to {\mathsf{Sh}}(X)\).
\item
\textbf{Restriction of a sheaf}: for \(F\in {\mathsf{Sh}}(X)\) and
\(Z \subseteq X\) with \(\iota:Z \hookrightarrow X\) the inclusion,
define \(i^{-1}F\in {\mathsf{Sh}}(Z)\) to be the restriction. Also
denoted \({ \left.{{F}} \right|_{{Z}} }\). This has the same stalks:
\(({ \left.{{F}} \right|_{{Z}} })_p = F_p\).
\item
For any \(U \subseteq X\), the global sections functor
\(\Gamma(U; {-}): {\mathsf{Sh}}(X)\to {\mathsf{Ab}}{\mathsf{Grp}}\) is
left-exact (proved in exercises).
\item
\textbf{Limits of sheaves}: for \(\left\{{F_i}\right\}\) 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:
\(\Gamma(X; \colim_i F_i) = \colim_i \Gamma(X; F_i)\).
\begin{itemize}
\tightlist
\item
Inverse limits exist and are defined similarly.
\end{itemize}
\item
\textbf{The espace étalé}: define
\(\text{Ét}(F) = {\textstyle\coprod}_{p\in X} F_p\) and a projection
\(\pi: \text{Ét}(F) \to X\) by sending \(s\in F_p\) to \(p\). For each
\(U \subseteq X\) and \(s\in F(U)\), there is a local section
\(\overline{s}: U\to \text{Ét}(F)\) where \(p\mapsto s_p\), its germ
at \(p\); this satisfies
\(\pi \circ \overline{s} = \operatorname{id}_U\). Give
\(\text{Ét}(F)\) the strongest topology such that the \(\overline{s}\)
are all continuous. Then
\(F^+(U) \coloneqq{\mathsf{Top}}(U, \text{Ét}(F))\) is the set of
continuous sections of \(\text{Ét}(F)\) over \(U\).
\item
\textbf{Support}: for \(s\in F(U)\),
\(\mathop{\mathrm{supp}}(s) \coloneqq\left\{{p\in U {~\mathrel{\Big\vert}~}s_p \neq 0}\right\}\)
where \(s_p\) is the germ of \(s\) in \(F_p\). This is closed.
\begin{itemize}
\tightlist
\item
This extends to
\(\mathop{\mathrm{supp}}(F) \coloneqq\left\{{p\in X {~\mathrel{\Big\vert}~}F_p \neq 0}\right\}\),
which need not be closed.
\end{itemize}
\item
\textbf{Sheaf hom}:
\(U\mapsto \mathop{\mathrm{Hom}}({ \left.{{F}} \right|_{{U}} }, { \left.{{G}} \right|_{{U}} })\)
forms a sheaf of local morphisms and is denoted
\(\mathop{\mathcal{H}\! \mathit{om}}(F, G)\).
\item
\textbf{Flasque sheaves}: a sheaf is flasque iff
\(V\hookrightarrow U \implies F(U) \twoheadrightarrow F(V)\).
\item
\textbf{Skyscraper sheaves}: for \(A\in {\mathsf{Ab}}{\mathsf{Grp}}\)
and \(p\in X\), define
\begin{align*}
i_p(A)(U) =
\begin{cases}
A & p\in U
\\
0 & \text{otherwise}.
\end{cases}
.\end{align*}
Also denoted \(\iota_*(A)\) where
\(\iota: { \operatorname{cl}}_X(\left\{{p}\right\}) \hookrightarrow X\)
is the inclusion.
\begin{itemize}
\tightlist
\item
The stalks are
\begin{align*}
(i_p(A))_q =
\begin{cases}
A & q\in { \operatorname{cl}}_X(\left\{{p}\right\})
\\
0 & \text{otherwise}.
\end{cases}
.\end{align*}
\end{itemize}
\item
\textbf{Extension by zero}: if \(\iota: Z\hookrightarrow X\) is the
inclusion of a closed set and \(U\coloneqq X\setminus Z\) with
\(j: U\to X\), then for \(F\in {\mathsf{Sh}}(Z)\), the sheaf
\(\iota_* F\in {\mathsf{Sh}}(X)\) is the extension of \(F\) by zero
outside of \(Z\). The stalks are
\begin{align*}
(\iota_* F)_p =
\begin{cases}
F_p & p\in Z
\\
0 & \text{otherwise}.
\end{cases}
.\end{align*}
\begin{itemize}
\tightlist
\item
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
\begin{align*}
(j_! F)_p
=
\begin{cases}
F_p & p\in U
\\
0 & \text{otherwise}.
\end{cases}
.\end{align*}
\end{itemize}
\item
\textbf{Sheaf of ideals}: for \(Y \subseteq X\) closed and
\(U \subseteq X\) open, \({\mathcal{I}}_Y(U)\) has presheaf
\(U \mapsto\) the ideal in \({\mathcal{O}}_X(U)\) of regular functions
vanishing on all of \(Y \cap U\). This is a subsheaf of
\({\mathcal{O}}_X\).
\item
\textbf{Gluing sheaves}: given \({\mathcal{U}}\rightrightarrows X\)
and sheaves \(F_i\in {\mathsf{Sh}}(U_i)\), one can glue to a unique
\(F\in {\mathsf{Sh}}(X)\) if one is given morphisms
\(\phi_{ij}{ \left.{{F_i}} \right|_{{U_{ij}}} } { \, \xrightarrow{\sim}\, }{ \left.{{F_j}} \right|_{{U_{ij}}} }\)
where \(\phi_{ii} = \operatorname{id}\) and
\(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\) on \(U_{ijk}\).
\end{itemize}
\end{remark}
\begin{warnings}
Some common mistakes:
\begin{itemize}
\tightlist
\item
Kernel presheaves are already sheaves, but not cokernels or images.
See exercise below.
\item
\(\phi: F\to G\) is injective iff injective on sections, but this is
not true for surjectivity.
\item
The sheaves \(f^{-1}G\) and \(f^* G\) are different! See III.5 for the
latter.
\item
Global sections need not be right-exact.
\end{itemize}
\end{warnings}
\begin{exercise}[Regular functions on varieties form a sheaf]
For \(X\in {\mathsf{Var}}_{/ {k}}\), define the ring
\({\mathcal{O}}_X(U)\) of literal regular functions \(f_i: U\to k\)
where restriction morphisms are induced by literal restrictions of
functions. Show that \({\mathcal{O}}_X\) is a sheaf of rings on \(X\).
\begin{quote}
Hint: Locally regular implies regular, and regular + locally zero
implies zero.
\end{quote}
\end{exercise}
\begin{exercise}[?]
Show that for every connected open subset \(U \subseteq X\), the
constant sheaf satisfies \(\underline{A}(U) = A\), and if \(U\) is open
with open connected component so the
\(\underline{A}(U) = A{ {}^{ \scriptscriptstyle\times^{{\sharp}\pi_0 U} } }\).
\end{exercise}
\begin{exercise}[?]
Show that if \(X\in{\mathsf{Var}}_{/ {k}}\) and \({\mathcal{O}}_X\) is
its sheaf of regular functions, then the stalk \({\mathcal{O}}_{X, p}\)
is the \emph{local ring of \(p\)} on \(X\) as defined in Ch. I.
\end{exercise}
\begin{exercise}[Prop 1.1]
Let \(\phi: F\to G\) be a morphism in \({\mathsf{Sh}}(X)\) 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.
\end{exercise}
\begin{exercise}[?]
Show that for
\(\phi\in \mathop{\mathrm{Mor}}_{{\mathsf{Sh}}(X)}(F, G)\),
\(\ker \phi\) is a sheaf, but
\(\operatorname{coker}\phi, \operatorname{im}\phi\) are not in general.
\end{exercise}
\begin{exercise}[?]
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.
\end{exercise}
\hypertarget{ii.2-schemes}{%
\subsection{II.2: Schemes}\label{ii.2-schemes}}
\hypertarget{ii.3-first-properties-of-schemes}{%
\subsection{II.3: First Properties of
Schemes}\label{ii.3-first-properties-of-schemes}}
\hypertarget{ii.4-separated-and-proper-morphisms}{%
\subsection{II.4: Separated and Proper
Morphisms}\label{ii.4-separated-and-proper-morphisms}}
\hypertarget{ii.5-sheaves-of-modules}{%
\subsection{II.5: Sheaves of Modules}\label{ii.5-sheaves-of-modules}}
\hypertarget{ii.6-divisors}{%
\subsection{II.6: Divisors}\label{ii.6-divisors}}
\hypertarget{ii.7-projective-morphisms}{%
\subsection{II.7: Projective
Morphisms}\label{ii.7-projective-morphisms}}
\hypertarget{ii.8-differentials}{%
\subsection{II.8: Differentials}\label{ii.8-differentials}}
\hypertarget{ii.9-formal-schemes}{%
\subsection{II.9: Formal Schemes}\label{ii.9-formal-schemes}}
\newpage
\hypertarget{iii-cohomology}{%
\section{III: Cohomology}\label{iii-cohomology}}
\hypertarget{iii.1-derived-functors}{%
\subsection{III.1: Derived Functors}\label{iii.1-derived-functors}}
\hypertarget{iii.2-cohomology-of-sheaves}{%
\subsection{III.2: Cohomology of
Sheaves}\label{iii.2-cohomology-of-sheaves}}
\hypertarget{iii.3-cohomology-of-a-noetherian-affine-scheme}{%
\subsection{III.3: Cohomology of a Noetherian Affine
Scheme}\label{iii.3-cohomology-of-a-noetherian-affine-scheme}}
\hypertarget{iii.4-ux10dech-cohomology}{%
\subsection{III.4: ÄŒech Cohomology}\label{iii.4-ux10dech-cohomology}}
\hypertarget{iii.5-the-cohomology-of-projective-space}{%
\subsection{III.5: The Cohomology of Projective
Space}\label{iii.5-the-cohomology-of-projective-space}}
\hypertarget{iii.6-ext-groups-and-sheaves}{%
\subsection{III.6: Ext Groups and
Sheaves}\label{iii.6-ext-groups-and-sheaves}}
\hypertarget{iii.7-serre-duality}{%
\subsection{III.7: Serre Duality}\label{iii.7-serre-duality}}
\hypertarget{iii.8-higher-direct-images-of-sheaves}{%
\subsection{III.8: Higher Direct Images of
Sheaves}\label{iii.8-higher-direct-images-of-sheaves}}
\hypertarget{iii.9-flat-morphisms}{%
\subsection{III.9: Flat Morphisms}\label{iii.9-flat-morphisms}}
\hypertarget{iii.10-smooth-morphisms}{%
\subsection{III.10: Smooth Morphisms}\label{iii.10-smooth-morphisms}}
\hypertarget{iii.11-the-theorem-on-formal-functions}{%
\subsection{III.11: The Theorem on Formal
Functions}\label{iii.11-the-theorem-on-formal-functions}}
\hypertarget{iii.12-the-semicontinuity-theorem}{%
\subsection{III.12: The Semicontinuity
Theorem}\label{iii.12-the-semicontinuity-theorem}}
\newpage
\hypertarget{iv-curves-star}{%
\section{\texorpdfstring{IV: Curves
\(\star\)}{IV: Curves \textbackslash star}}\label{iv-curves-star}}
\begin{remark}
Summary of major results:
\begin{itemize}
\item
\(p_a(X) \coloneqq 1 - P_X(0) = (-1)^r (1-\chi({\mathcal{O}}_X))\).
\begin{itemize}
\tightlist
\item
Note: \(P_X(\ell)\) is defined as the Hilbert polynomial of the
homogeneous coordinate ring \(S(Y)\), and then defined for graded
\(S{\hbox{-}}\)modules \(M\) by setting
\(\phi_M(\ell) = \dim_k M_\ell\) and showing
\(\exists ! P_M(z) \in {\mathbf{Q}}[z]\) with
\(\phi_M(\ell) = P_M(\ell)\) for \(\ell \gg 0\).
\end{itemize}
\item
\(p_g(X) \coloneqq h^0(\omega_X) = h^0({\mathcal{L}}(K_X))\).
\item
Remembering these:
\includegraphics{figures/2022-12-04_20-08-00.png}
\end{itemize}
\begin{quote}
\href{https://q.uiver.app/?q=WzAsMTIsWzUsNCwiaF57MCwwfSA9IGheMChcXE9PX1gpIl0sWzQsMywiaF57MSwwfSA9IGheMChcXE9tZWdhXjEpIl0sWzYsMywiaF57MCwxfSA9IGheMShcXE9PX1gpIl0sWzUsMiwiaF57MSwxfSA9IGheMShcXE9tZWdhXjEpIl0sWzMsMiwicF9nXFxkYSBoXnsyLDB9PWheMChcXE9tZWdhXjIpID0gaF4wKEtfWCkiLFszMzgsNDEsNjAsMV1dLFs3LDIsImheezAsMn0gPSBoXjIoXFxPT19YKSJdLFs0LDEsImheezIsMX0gPSBoXjEoXFxPbWVnYV4yKSA9IGheMShLX1gpIl0sWzYsMSwiaF57MSwyfSA9IGheMihcXE9tZWdhXjEpIl0sWzUsMCwiaF57MiwyfSA9IGheMihcXE9tZWdhXjIpID0gaF4yKEtfWCkiXSxbMCwyLCJoXntwLCBxfSA9IEhecShcXE9tZWdhXnApIl0sWzQsNSwiXFxidWxsZXQiXSxbOCwxLCJcXCwiXSxbMTAsMTEsIlxccG0gcF9hID0gXFxjaGkoXFxPT19YKSAtIDEiLDAseyJsYWJlbF9wb3NpdGlvbiI6MTAwLCJvZmZzZXQiOjQsImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX0sWzI0MCw2MCw2MCwxXV1d}{Link
to Diagram}
\end{quote}
\begin{itemize}
\tightlist
\item
For curves, \(p_a(X) = p_g(X) = h^1({\mathcal{O}}_X)\) by setting
\(D\coloneqq K_C\) in RR.
\begin{itemize}
\tightlist
\item
\(\deg K_C = 2g-2\).
\end{itemize}
\item
\(D_1\sim D_2 \iff D_1-D_2 = (f)\) for \(f\in K(X)\) rational,
\({\left\lvert {D} \right\rvert} = \left\{{D'\sim D}\right\}\), and
this bijects with points of
\(H^0({\mathcal{L}}(D))\setminus\left\{{0}\right\}\over {\mathbf{G}}_m\).
\begin{itemize}
\tightlist
\item
Thus
\(\dim {\left\lvert {D} \right\rvert} = h^0({\mathcal{L}}(D)) - 1 \coloneqq\ell(D) - 1\).
\end{itemize}
\item
\(X\) smooth
\(\implies \operatorname{Cl}(X) { \, \xrightarrow{\sim}\, }\operatorname{Pic}(X)\)
via \(D\mapsto {\mathcal{L}}(D)\).
\item
\(h^0({\mathcal{L}}(D)) >0 \implies \deg(D) \geq 0\), and if
\(\deg D = 0\) then \(D\sim 0\) and
\({\mathcal{L}}(D) \cong {\mathcal{O}}_X\).
\item
RR:
\begin{align*}
\chi({\mathcal{L}}(D) &= h^0({\mathcal{L}}(D)) - h^1({\mathcal{L}}(D)) \\
&= h^0({\mathcal{L}}(D)) - h^0({\mathcal{L}}(K-D)) \\
&= \deg(D) + (1-g)
.\end{align*}
\begin{itemize}
\tightlist
\item
How to remember: note
\(g= h^1({\mathcal{O}}_X) = h^1({\mathcal{L}}(0))\), and
\(H^0({\mathcal{O}}_X) = k\) so \(h^0({\mathcal{O}}_X) = 1\), thus
\begin{align*}
\chi({\mathcal{O}}_X) = h^0({\mathcal{O}}_X) - h^1({\mathcal{O}}_X) = 1-g = \deg {\mathcal{L}}(0) + 1-g
.\end{align*}
\item
For \(C \subseteq {\mathbf{P}}^n, \deg(C) = d\) and \(D = C \cap H\)
a hyperplane section defining
\({\mathcal{L}}(D) = {\mathcal{O}}_X(1)\),
\begin{align*}
\chi({\mathcal{L}}(D)) = \deg(D) + (1-g) = d + (1-p_a(C))
\end{align*}
\end{itemize}
\item
A curve is rational iff isomorphic to \({\mathbf{P}}^1\) iff \(g=0\).
\item
\(K\sim 0\) on an elliptic curve since \(\deg K = 2g-2 = 0\) and
\(\deg D = 0\implies D\sim 0\).
\item
For \(X\) elliptic,
\(\operatorname{Pic}^0(X) \coloneqq\left\{{D\in \operatorname{Div}(X) {~\mathrel{\Big\vert}~}\deg D = 0}\right\}\)
and
\({\left\lvert {X} \right\rvert} { \, \xrightarrow{\sim}\, }{\left\lvert {\operatorname{Pic}^0(X)} \right\rvert}\)
via \(p\mapsto {\mathcal{L}}(p-p_0)\) for any fixed \(p_0\in X\),
inducing its group structure. (This is proved with RR.)
\end{itemize}
\end{remark}
\begin{remark}
Comments from preface:
\begin{itemize}
\tightlist
\item
The statement of Riemann-Roch is important; less so its proof.
\item
Representing curves:
\begin{itemize}
\tightlist
\item
A branched covering of \({\mathbf{P}}^1\),
\item
More generally a branched covering of another curve,
\item
Nonsingular projective curves: admit embeddings into
\({\mathbf{P}}^3\), maps to \({\mathbf{P}}^2\) birationally such
that the image is at worst a nodal curve.
\end{itemize}
\item
The central result regarding representing curves: Hurwitz's theorem
which compares \(K_X, K_Y\) for a cover \(Y\to X\) of curves.
\item
Curves of genus 1: elliptic curves.
\item
Later sections: the canonical embedding of a curve.
\end{itemize}
\end{remark}
\hypertarget{iv.1-riemann-roch}{%
\subsection{IV.1: Riemann-Roch}\label{iv.1-riemann-roch}}
\begin{definition}[Curves]
A \textbf{curve} over \(k={ \overline{k} }\) is a scheme over
\(\operatorname{Spec}k\) which is
\begin{itemize}
\tightlist
\item
Integral
\item
Dimension 1
\item
Proper over \(k\)
\item
With regular local rings
\end{itemize}
In particular, a curve is smooth, complete, and necessary projective. A
\textbf{point} on a curve is a closed point.
\end{definition}
\begin{definition}[Arithmetic genus]
The \textbf{arithmetic genus} of a projective curve \(X\) is
\begin{align*}
p_a(X) \coloneqq 1 - P_X(0)
\end{align*}
where \(P_X(t)\) is the \textbf{Hilbert polynomial} of \(X\).
\end{definition}
\begin{definition}[Geometric genus]
The \textbf{geometric genus} of a curve is
\begin{align*}
p+g(X) \coloneqq\dim_k H^0(X; \omega_X)
\end{align*}
where \(\omega_X\) is the canonical sheaf.
\end{definition}
\begin{exercise}[?]
Show that if \(X\) is a curve, there is a single well-defined
\textbf{genus}
\begin{align*}
g \coloneqq p_A(X) = p_G(X) = \dim_k H^1(X; {\mathcal{O}}_X)
.\end{align*}
\begin{quote}
Hint: see Ch. III Ex. 5.3, and use Serre duality for \(p_g\).
\end{quote}
\end{exercise}
\begin{exercise}[?]
Show that for any \(g\geq 0\) there exists a curve of genus \(g\).
\begin{quote}
Hint: take a divisor of type \((g+1, 2)\) on a smooth quadric which is
irreducible and smooth with \(p_a = g\).
\end{quote}
\end{exercise}
\begin{definition}[Divisors on a curve]
Reviewing divisors:
\begin{itemize}
\tightlist
\item
The \textbf{divisor group}:
\(\operatorname{Div}(X) = {\mathbf{Z}}\left[ {X_{ \operatorname{cl}}} \right]\)
\item
\textbf{Degrees}: \(\deg(\sum n_i D_i) \coloneqq\sum n_i\), and
\item
\textbf{Linear equivalence}:
\(D_1\sim D_2 \iff D_1 - D_1 = \operatorname{Div}(f)\) for some
\(f\in k(X)\) a rational function.
\item
\(D\) is \textbf{effective} if \(n_i \geq 0\) for all \(i\).
\item
\({\left\lvert {D} \right\rvert} \coloneqq\left\{{D'\in \operatorname{Div}(X) {~\mathrel{\Big\vert}~}D'\sim D}\right\}\)
is the \textbf{complete linear system} of \(D\).
\item
\({\left\lvert {D} \right\rvert} \cong {\mathbf{P}}H^0(X; {\mathcal{L}}(D))\)
\item
\textbf{Dimensions of linear systems}:
\(\ell(D) \coloneqq\dim_k H^0(X; {\mathcal{L}}(D))\) and
\(\dim {\left\lvert {D} \right\rvert} \coloneqq\ell(D) - 1\).
\item
\textbf{Relative differentials}:
\(\Omega_X \coloneqq\Omega_{X_{/ {k}}}\) is the sheaf of relative
differentials on \(X\).
\begin{itemize}
\tightlist
\item
The technical definition:
\(\Omega_{X_{/ {S}}} \coloneqq\Delta_{X_{/ {Y}}}^*({\mathcal{I}}/{\mathcal{I}}^2)\)
where \({\mathcal{I}}\) is the sheaf of ideals defining the locally
closed subscheme
\(\operatorname{im}(\Delta_{X_{/ {Y}}}) \subseteq X{ \operatorname{fp} }{Y} X\).
\item
On affine schemes: on the ring side,
\(\Omega_{B_{/ {A}}} \in {}_{B}{\mathsf{Mod}}\) equipped with a
differential \(d: B\to \Omega{B_{/ {A}}}\), defined as
\(\left\langle{db{~\mathrel{\Big\vert}~}b\in B}\right\rangle_B / \left\langle{d(b_1+b_2) =db_1 + db_2, d(b_1 b_2) = d(b_1)b_2 + b_1 d(b_2), da = 0\, \forall a\in A}\right\rangle_B\).
\item
On curves, \(\Omega_{X_{/ {Y}}}\) measures the ``difference''
between \(K_X\) and \(K_Y\).
\end{itemize}
\item
\textbf{Canonical sheaf}:
\(\dim X = 1, \Omega_{X_{/ {k}}} \cong \omega_X\).
\item
\textbf{Canonical divisor}: \(K_X\) 2is any divisor in the linear
equivalence class corresponding to \(\omega_X\)
\item
\(D\) is \textbf{special} iff its \textbf{index of speciality}
\(\ell(K-D) > 0\), otherwise \(D\) is \textbf{nonspecial}.
\end{itemize}
\end{definition}
\begin{exercise}[?]
Show that \(D_1\sim D_2\implies\deg(D_1) = \deg(D_2)\).
\end{exercise}
\begin{exercise}[?]
Show that
\begin{align*}
{\left\lvert {D} \right\rvert} \rightleftharpoons{\mathbf{P}}H^0(X; {\mathcal{L}}(D))
,\end{align*}
so \({\left\lvert {D} \right\rvert}\) has the structure of the closed
points of some projective space.
\end{exercise}
\begin{exercise}[Lemma 1.2]
Show that if \(D\in \operatorname{Div}(X)\) 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
\({\mathcal{L}}(D) \cong {\mathcal{O}}_X\).
\end{exercise}
\begin{theorem}[Riemann-Roch]
\begin{align*}
\ell(D) - \ell(K-D) = \deg(D) + (1-g)
.\end{align*}
\end{theorem}
\begin{exercise}[Ingredients for proof of RR]
Show the following:
\begin{itemize}
\item
The divisor \(K-D\) corresponds to
\(\omega_X \otimes{\mathcal{L}}(D) {}^{ \vee }\in \operatorname{Pic}(X)\).
\item
\(H^1(X; {\mathcal{L}}(D)) {}^{ \vee }\cong H^0(X; \omega_X \otimes{\mathcal{L}}(D) {}^{ \vee })\).
\item
If \(X\) is any projective variety,
\begin{align*}
H^0(X; {\mathcal{O}}_X) = k
.\end{align*}
\end{itemize}
\end{exercise}
\begin{exercise}[?]
Show that if \(X \subseteq {\mathbf{P}}^n\) is a curve with
\(\deg X = d\) and \(D = X \cap H\) is a hyperplane section, then
\({\mathcal{L}}(D) = {\mathcal{O}}_X(1)\) and
\(\chi({\mathcal{L}}(D)) = d + 1 - p_a\).
\end{exercise}
\begin{exercise}[?]
Show that if \(g(X) = g\) then \(\deg K_X = 2g-2\).
\begin{quote}
Hint: set \(D = K\) and use \(\ell(K) = p_g = g\) and \(\ell(0) = 1\).
\end{quote}
\end{exercise}
\begin{remark}
More definitions:
\begin{itemize}
\tightlist
\item
\(X\) is \textbf{rational} iff birational to \({\mathbf{P}}^1\).
\item
\(X\) is \textbf{elliptic} if \(g=1\).
\end{itemize}
\end{remark}
\begin{exercise}[?]
Show that
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
If \(\deg D > 2g-2\) then \(D\) is nonspecial.
\item
\(p_a({\mathbf{P}}^1) = 0\).
\item
A complete nonsingular curve is rational iff \(X\cong {\mathbf{P}}^1\)
iff \(g(X) = 0\).
\item
If \(X\) is elliptic then \(K\sim 0\)
\end{enumerate}
\begin{quote}
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
\(\ell(K) = p_g = 1\).
\end{quote}
\end{exercise}
\begin{exercise}[?]
If \(X\) is elliptic and \(p\in X\), then there is a bijection
\begin{align*}
m_p: X & { \, \xrightarrow{\sim}\, }\operatorname{Pic}(X) \\
x &\mapsto {\mathcal{L}}(x-p)
,\end{align*}
so \(\operatorname{Pic}(X) \in {\mathsf{Grp}}\).
\begin{quote}
Hint: show that if \(\deg(D) = 0\) then there is some \(x\in X\) such
that \(D\sim x-p\) and apply RR to \(D+p\).
\end{quote}
\end{exercise}
\hypertarget{iv.2-hurwitz-star}{%
\subsection{\texorpdfstring{IV.2: Hurwitz
\(\star\)}{IV.2: Hurwitz \textbackslash star}}\label{iv.2-hurwitz-star}}
\begin{remark}
Summary of results:
\begin{itemize}
\item
For curves, complete = projective.
\item
Riemann-Hurwitz: for \(f:X\to Y\) finite separable,
\begin{align*}
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)
.\end{align*}
\item
\(\deg f \coloneqq[K(X): K(Y)]\) for finite morphisms of curves.
\item
\(e_p \coloneqq v_p(f^\sharp_* t)\) where \(t\) is uniformizer in
\({\mathcal{O}}_{f(p)}\) and
\(f^\sharp: {\mathcal{O}}_{Y, f(p)}\to {\mathcal{O}}_{X, p}\) for
\(f:X\to Y\).
\begin{itemize}
\tightlist
\item
\(e_p > 1 \implies\) ramification.
\item
Unramified everywhere implies etale (since automatically flat).
\item
\(p\divides e_{x_0}\implies\) wild ramification, otherwise tame.
\end{itemize}
\item
\(\exists f^*: \operatorname{Div}(Y)\to \operatorname{Div}(X)\) where
\(q\mapsto \sum_{p\mapsto q} e_p p\).
\item
Pullback commutes with forming line bundles:
\begin{align*}
f^* {\mathcal{L}}(D) \cong {\mathcal{L}}(f^* D)
\end{align*}
where the LHS
\(f^*: \operatorname{Pic}(Y) \to \operatorname{Pic}(X)\).
\item
The fundamental SES for relative differentials: if \(f:X\to Y\) is
finite separable,
\begin{align*}
f^* \Omega_{Y} \hookrightarrow\Omega_{X} \twoheadrightarrow\Omega_{X/Y}
.\end{align*}
\item
\({\frac{\partial t}{\partial u}\,}\) for \(t\) a uniformizer at
\(f(p)\) and \(u\) a uniformizer at \(p\) is defined by noting
\(\Omega{Y, f(p)} = \left\langle{\,dt}\right\rangle, \Omega_{X, p} = \left\langle{\,du}\right\rangle\),
and there is some \(g\in {\mathcal{O}}_{X, p}\) such that
\(f^* \,dt= g\,du\); set
\(g \coloneqq{\frac{\partial t}{\partial u}\,}\).
\item
For finite separable morphisms of curves \(f:X\to Y\),
\begin{itemize}
\tightlist
\item
\(\mathop{\mathrm{supp}}\Omega_{X/Y} = \mathrm{Ram}(f)\) is the
ramification locus, and \(\Omega_{X/Y}\) is torsion so
\(\operatorname{Ram}(f)\) is finite.
\item
\(\mathop{\mathrm{length}}(\Omega_{X, Y})_p = v_p\qty{{\frac{\partial t}{\partial u}\,}}\)
for any \(p\in X\)
\item
Tamely ramified
\(\implies \mathop{\mathrm{length}}(\Omega_{X/Y})_p = e_p - 1\), and
wild ramification increases this length. Recall that length is the
largest size of chains of submodules.
\end{itemize}
\item
The ramification divisor:
\begin{align*}
R \coloneqq\sum_{p\in X} \mathop{\mathrm{length}}(\Omega_{X/Y})_p p
.\end{align*}
\item
\(K_X \sim f^*K_Y + R\)
\item
\({\mathbf{P}}^1\) can't admit an unramified cover: for \(n\geq 1\),
\begin{align*}
\chi(X) = n\chi({\mathbf{P}}^1) + \deg R \implies \chi(X) = -2n + \deg R \implies \chi(X) = -2n \leq -2
,\end{align*}
which forces
\(g(X) = 0, n=1, X = {\mathbf{P}}^1, f=\operatorname{id}\).
\item
The Frobenius morphism on schemes is defined by taking
\(f^\sharp: {\mathcal{O}}_X\to {\mathcal{O}}_X\) to be the \(p\)th
power map; pullback yields a definition of \(X_p\), the Frobenius
twist of \(X\).
\begin{itemize}
\tightlist
\item
\(F: X_p\to X\) is finite, \(\deg F = p\), and corresponds to
\(K(X) \hookrightarrow K(X)^{1\over p}\)
\end{itemize}
\item
If \(f:X\to Y\) induces a purely inseparable extension \(K(X)/K(Y)\),
then \(X { \, \xrightarrow{\sim}\, }Y\) as schemes, \(g(X) = g(Y)\),
and \(f\) is a composition of Frobenii.
\item
Everywhere ramified extensions: \(f:Y_p \to Y\), where \(e_{q} = p\)
for every \(q\in X\). Induces \(\Omega_{X/Y}\cong \Omega_{X}\).
\item
\(\deg R\) is always even.
\item
Finite implies proper: finite implies separated, of finite type,
closed by ``going up'' and universally closed by since finiteness is
preserved under base change.
\item
\({\mathbf{P}}^1\) no nontrivial etale covers.
\item
If \(f:X\to Y\) then \(g(X) \geq g(Y)\).
\begin{itemize}
\tightlist
\item
Thus \(\exists {\mathbf{P}}^1\to Y\) finite \(\implies g(Y) = 0\).
\end{itemize}
\end{itemize}
\end{remark}
\begin{remark}
Preface:
\begin{itemize}
\tightlist
\item
\textbf{Degree}: for a finite morphism of curves
\(X \xrightarrow{f} Y\), set
\(\operatorname{det}(f) \coloneqq[k(X): k(Y)]\), the degree of the
extension of function fields.
\item
\textbf{Ramification indices \(e_p\)}: for \(p\in X\), let \(q=f(p)\)
and \(t \in {\mathcal{O}}_q\) a local coordinate. Pull back to
\(t\in {\mathcal{O}}_p\) via \(f^\sharp\) and define
\(e_p \coloneqq v_p(t)\) using the valuation \(v_p\) for the DVR
\({\mathcal{O}}_p\).
\item
\textbf{Ramified}: \(e_p > 1\), and \textbf{unramified} if
\(e_p = 1\).
\item
\textbf{Branch points} any \(q = f(p)\) where \(f\) is ramified.
\item
\textbf{Tame ramification}: for \(\operatorname{ch}(k) = p\), tame if
\(p\notdivides e_P\).
\item
\textbf{Wild ramification}: when \(p\divides e_P\).
\item
Pullback maps on divisor groups:
\begin{align*}
f^*: \operatorname{Div}(Y) &\to \operatorname{Div}(X) \\
Q &\mapsto \sum_{P \xrightarrow{f} q} e_P [P]
.\end{align*}
\begin{itemize}
\tightlist
\item
This commutes with taking line bundles (exercise), so induces a
well-defined map
\(f^*: \operatorname{Pic}(X) \to \operatorname{Pic}(Y)\).
\end{itemize}
\item
\(f\) is \textbf{separable} if \(k(X) / k(Y)\) is a separable field
extension.
\end{itemize}
\end{remark}
\begin{exercise}[?]
Misc:
\begin{itemize}
\tightlist
\item
Show that if \(f\) is everywhere unramified then it is an étale
morphism.
\item
Show that \(f^* {\mathcal{L}}(D) = {\mathcal{L}}(f^* D)\)
\end{itemize}
\end{exercise}
\begin{exercise}[Prop 2.1]
Show that if \(X \xrightarrow{f} Y\) is a finite separable morphism of
curves, there is a SES
\begin{align*}
f^* \Omega_Y \hookrightarrow\Omega_X \twoheadrightarrow\Omega_{X_{/ {Y}}}
.\end{align*}
\end{exercise}
\begin{remark}
Definitions:
\begin{itemize}
\tightlist
\item
\textbf{Derivatives}: for \(f: X\to Y\), let \(t\) be a parameter at
\(Q = f(P)\) and \(u\) at \(P\). Then
\(\Omega_{Y, Q} = \left\langle{dt}\right\rangle_{{\mathcal{O}}_Q}\)
and
\({\mathcal{O}}_{X, P} = \left\langle{du}\right\rangle_{{\mathcal{O}}_P}\)
and \(\exists ! g\in {\mathcal{O}}_P\) such that \(f^* dt = du\) so we
write \({\frac{\partial t}{\partial u}\,} \coloneqq g\).
\item
\textbf{Ramification divisor}:
\(R \coloneqq\sum_{P\in X} \mathop{\mathrm{length}}(\Omega_{X_{/ {Y}}})_P [P] \in \operatorname{Div}(X)\)
\end{itemize}
\end{remark}
\begin{exercise}[Prop 2.2]
For \(X \xrightarrow{f} Y\) a finite separable morphism of curves,
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
\(\Omega_{X_{/ {Y}}}\) is a torsion sheaf on \(X\) with support equal
to the ramification locus of \(f\). Thus \(f\) is ramified at finitely
many points.
\item
The stalks \((\Omega_{X_{/ {Y}}})_P\) are principal
\({\mathcal{O}}_P{\hbox{-}}\)modules of finite length equal to
\(v_p\qty{{\frac{\partial t}{\partial u}\,}}\)
\item
\begin{align*}
\mathop{\mathrm{length}}(\Omega_{X_{/ {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}
.\end{align*}
\end{enumerate}
\end{exercise}
\begin{exercise}[Prop 2.3]
If \(X \xrightarrow{f} Y\) is a finite separable morphism of curves,
then
\begin{align*}
K_X \sim f^* K_Y + R
,\end{align*}
where \(R\) is the ramification divisor of \(f\).
\end{exercise}
\begin{theorem}[Hurwitz]
If \(X \xrightarrow{f} Y\) is a finite separable morphism of curves,
then
\begin{align*}
2g(X) -2 = \deg(f)(2g(Y) - 2) + \deg(R)
,\end{align*}
and if \(f\) has only tame ramification then
\(\deg(R) = \sum_{P\in X}(e_P - 1)\).
\end{theorem}
\begin{remark}[proof of Hurwitz]
Take degrees of the divisor equation:
\begin{align*}
\deg(K_X ) &= \deg(f^* K_Y + R) \\
\implies \chi_{\mathsf{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_{\mathsf{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) \\
,\end{align*}
using tame ramification in the last step which implies
\(\mathop{\mathrm{length}}(\Omega_{X_{/ {Y}}})_P = (e_p - 1)\).
\end{remark}
\begin{remark}
Consider the purely inseparable case.
\begin{itemize}
\tightlist
\item
\textbf{Frobenius morphism}: for \(X \in{\mathsf{Sch}}\) where
\({\mathcal{O}}_P \supseteq{\mathbf{Z}}/p{\mathbf{Z}}\) for all \(P\),
define \(\operatorname{Frob}: X\to X\) by
\(F({\left\lvert {X} \right\rvert}) = {\left\lvert {X} \right\rvert}\)
on spaces and \(F^\sharp: {\mathcal{O}}_X \to {\mathcal{O}}_X\) is
\(f\mapsto f^p\). This is a morphism since \(F^\sharp\) induces a
morphism on all local rings, which are all characteristic \(p\).
\item
\textbf{The \(k{\hbox{-}}\)linear Frobenius morphism}: define \(X_p\)
to be \(X\) with the structure morphism \(F\circ \pi\), so
\(k\curvearrowright{\mathcal{O}}_{X_p}\) by \(p\)th powers and \(F\)
becomes a \(k{\hbox{-}}\)linear morphism \(F': X_p\to X\).
\begin{itemize}
\tightlist
\item
Why this is necessary: \(F\) as before is not a morphism in
\({\mathsf{Sch}}_{/ {k}}\), and instead forms a commuting square
involving \(F: \operatorname{Spec}k\to \operatorname{Spec}k\) and
the structure maps \(X \xrightarrow{\pi} \operatorname{Spec}k\).
\end{itemize}
\end{itemize}
\end{remark}
\begin{exercise}[?]
Find examples where
\begin{itemize}
\tightlist
\item
\(X_p \cong X \in {\mathsf{Sch}}_{/ {k}}\), and
\item
\(X_p \not\cong X \in {\mathsf{Sch}}_{/ {k}}\).
\end{itemize}
\begin{quote}
Hint: consider \(X = \operatorname{Spec}k[t]\) for \(k\) perfect.
\end{quote}
\end{exercise}
\begin{exercise}[?]
Show that if \(X \xrightarrow{f} Y\) is separable then \(\deg(R)\) is
always even.
\end{exercise}
\begin{quote}
Skipped some stuff around Example 2.4.2, I don't necessarily need
characteristic \(p\) things right now.
\end{quote}
\begin{remark}
Definitions:
\begin{itemize}
\tightlist
\item
\textbf{Étale covers}: \(X \xrightarrow{f} Y\) is an étale cover if
\(f\) is a finite étale morphism,, i.e.~\(f\) is flat and
\(\Omega^1_{X_{/ {Y}}} = 0\).
\item
\(Y\) is a \textbf{trivial} cover if
\(X \cong {\textstyle\coprod}_{i\in I} Y\) a finite disjoint union of
copies of \(Y\),
\item
\(Y\) is \textbf{simply connected} if there are no nontrivial étale
covers.
\end{itemize}
\end{remark}
\begin{exercise}[?]
\envlist
\begin{itemize}
\tightlist
\item
Show that a connected regular curve is irreducible.
\item
Show that if \(f\) is etale then \(X\) is smooth over \(k\).
\item
Show that if \(f\) is finite, \(X\) must be a curve.
\item
Show that if \(f\) is étale, then \(f\) must be separable.
\item
Show that \(\pi_1^\text{ét}({\mathbf{P}}^1) = 0\).
\end{itemize}
\begin{quote}
Hint: use Hurwitz and that when \(f\) is unramified, \(R = 0\).
\end{quote}
\end{exercise}
\begin{exercise}[?]
\envlist
\begin{itemize}
\tightlist
\item
Show that the genus of a curve doesn't change under purely inseparable
extensions.
\item
Show that if \(f:X\to Y\) is a finite morphism of curves then
\(g(X) \geq g(Y)\).
\end{itemize}
\end{exercise}
\begin{exercise}[Lüroth]
Show that if \(L\) is a subfield of a purely transcendental extension
\(k(t) / k\) where \(k = { \overline{k} }\), then \(L\) is also purely
transcendental.\footnote{This is true over any field \(k\) in dimension
1, over \(k={ \overline{k} }\) in dimension 2, and false in dimension
3 by the existence of nonrational unirational threefolds.}
\begin{quote}
Hint: Assume \([L: k]_{\mathrm{tr}}= 1\), so \(L = k(X)\) for \(Y\) a
curve and \(L \subseteq k(t)\) corresponds to a finite morphism
\(f: {\mathbf{P}}^1\to Y\). Conclude \(g(Y) = 0\) so
\(Y\cong {\mathbf{P}}^1\) and \(L\cong k(u)\) for some \(u\).
\end{quote}
\end{exercise}
\hypertarget{iv.3-embeddings-in-projective-space-star}{%
\subsection{\texorpdfstring{IV.3: Embeddings in Projective Space
\(\star\)}{IV.3: Embeddings in Projective Space \textbackslash star}}\label{iv.3-embeddings-in-projective-space-star}}
\begin{remark}
A summary of major results:
\begin{itemize}
\tightlist
\item
For \(D\in \operatorname{Div}(C)\) with \(g = g(C)\),
\begin{itemize}
\tightlist
\item
\(D\) is ample iff \(\deg D > 0\).
\item
\(D\) is BPF iff \(\deg D\geq 2g\).
\item
\(D\) is very ample iff \(\deg D \geq 2g+1\).
\end{itemize}
\item
Being very ample is equivalent to being a hyperplane section under a
projective embedding.
\item
Divisors \(D\in \operatorname{Div}({\mathbf{P}}^n)\) are ample iff
very ample iff \(\deg D \geq 1\).
\begin{itemize}
\tightlist
\item
E.g. if \(E\) is elliptic then \(D\) is very ample if
\(\deg D \geq 3\), and for hyperelliptic, very ample if
\(\deg D\geq 5\).
\end{itemize}
\item
If \(D\) is very ample then \(\deg \phi(X) = \deg D\).
\item
Curves \(C \subseteq {\mathbf{P}}^n\) for \(n\geq 4\) can be projected
away from a point \(p\not \in X\) to get a closed immersion into
\({\mathbf{P}}^m\) for some \(m\leq n-1\). So any curve is birational
to a nodal curve in \({\mathbf{P}}^2\).
\item
Genus of normalizations of nodal curves:
\(g = {1\over 2}(d-1)(d-2)-{\sharp}\left\{{\text{nodes}}\right\}\).
\item
Any curve embeds into \({\mathbf{P}}^3\), and maps into
\({\mathbf{P}}^2\) with at worst nodal singularities.
\end{itemize}
\end{remark}
\begin{remark}
Main result: any curve can be embedded in \({\mathbf{P}}^3\), and is
birational to a nodal curve in \({\mathbf{P}}^2\). Some recollections:
\begin{itemize}
\tightlist
\item
\textbf{Very ample line bundles}:
\({\mathcal{L}}\in \operatorname{Pic}(X)\) is very ample if
\({\mathcal{L}}\cong {\mathcal{O}}_X(1)\) for some immersion of
\(f: X\hookrightarrow{\mathbf{P}}^N\).
\item
\textbf{Ample}: \({\mathcal{L}}\) is ample when
\(\forall {\mathcal{F}}\in {\mathsf{Coh}}(X)\), the twist
\({\mathcal{F}}\otimes{\mathcal{L}}^n\) is globally generated for
\(n \gg 0\).
\item
\textbf{(Very) ample divisors}: \(D\in \operatorname{Div}(X)\) is
(very) ample iff \({\mathcal{L}}(D)\in \operatorname{Pic}(X)\) is
(very) ample.
\item
\textbf{Linear systems}: a linear system is any set
\(S \leq {\left\lvert {D} \right\rvert}\) of effective divisors
yielding a linear subspace.
\item
\textbf{Base points}: \(P\) is a base point of \(S\) iff
\(P \in \mathop{\mathrm{supp}}D\) for all \(D\in S\).
\item
\textbf{Secant lines}: the secant line of \(P, Q\in X\) is the line in
\({\mathbf{P}}^N\) joining them.
\item
\textbf{Tangent lines}: at \(P\in X\), the unique line
\(L \subseteq {\mathbf{P}}^N\) passing through \(p\) such that
\({\mathbf{T}}_P(L) = {\mathbf{T}}_P(X) \subseteq {\mathbf{T}}_P({\mathbf{P}}^N)\).
\item
\textbf{Nodes}: a singularity of multiplicity 2.
\begin{itemize}
\tightlist
\item
\(y^2 = x^3 + x^2\) is a \textbf{node}.
\item
\(y^2 = x^3\) is a \textbf{cusp}.
\item
\(y^2 = x^4\) is a \textbf{tacnode}.
\end{itemize}
\item
\textbf{Multisecant}: for \(X \subseteq {\mathbf{P}}^3\), a line
meeting \(X\) in 3 or more distinct points.
\item
A \textbf{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 \(L_Q\).
\end{itemize}
\end{remark}
\begin{exercise}[II.8.20.2]
Show that by Bertini's theorem there are irreducible smooth curves of
degree \(d\) in \({\mathbf{P}}^2\) for any \(d\).
\end{exercise}
\begin{exercise}[?]
\envlist
Show that
\begin{itemize}
\tightlist
\item
\({\mathcal{L}}\) is ample iff \({\mathcal{L}}^n\) is very ample for
\(b \gg 0\).
\item
\({\left\lvert {D} \right\rvert}\) is basepoint free iff
\({\mathcal{L}}(D)\) is globally generated.
\item
If \(D\) is very ample, then \({\left\lvert {D} \right\rvert}\) is
basepoint free.
\item
If \(D\) is ample, \(nD \sim H\) a hyperplane section for a projective
embedding for some \(n\).
\item
If \(g(X) = 0\) then \(D\) is ample iff very ample iff \(\deg D > 0\).
\item
If \(D\) is very ample and corresponds to a closed immersion
\(\phi: X\hookrightarrow{\mathbf{P}}^n\) then
\(\deg \phi(X) = \deg D\).
\item
If \(XS\) is elliptic, any \(D\) with \(\deg D = 3\) is very ample and
\(\dim {\left\lvert {D} \right\rvert} = 2\), and so can be embedded
into \({\mathbf{P}}^2\) as a cubic curve.
\item
Show that if \(g(X) = 1\) then \(D\) is very ample iff
\(\deg D \geq 3\).
\item
Show that if \(g(X) = 2\) and \(\deg D = 5\) then \(D\) is very ample,
so any genus 2 curve embeds in \({\mathbf{P}}^3\) as a curve of degree
5.
\end{itemize}
\end{exercise}
\begin{exercise}[Prop 3.1: when a linear system yields a closed immersion into $\PP^N$]
Let \(D\in \operatorname{Div}(X)\) for \(X\) a curve and show
\begin{itemize}
\tightlist
\item
\({\left\lvert {D} \right\rvert}\) is basepoint free iff
\(\dim{\left\lvert {D-P} \right\rvert} = \dim{\left\lvert {D} \right\rvert} - 1\)
for all points \(p\in X\).
\item
\(D\) is very ample iff
\(\dim{\left\lvert {D-P-Q} \right\rvert} = \dim{\left\lvert {D} \right\rvert} - 2\)
for all points \(P, Q\in X\).
\end{itemize}
\begin{quote}
Hint: use the SES
\({\mathcal{L}}(D-P)\hookrightarrow{\mathcal{L}}(D) \twoheadrightarrow k(P)\)
where \(k(P)\) is the skyscraper sheaf at \(P\).
\end{quote}
\end{exercise}
\begin{exercise}[Cor 3.2]
Let \(D\in \operatorname{Div}(X)\).
\begin{itemize}
\tightlist
\item
If \(\deg D \geq 2g(X)\) then \({\left\lvert {D} \right\rvert}\) is
basepoint free.
\item
If \(\deg D \geq 2g(X) + 1\) then \(D\) is very ample.
\item
\(D\) is ample iff \(\deg D > 0\)
\item
This bounds is not sharp.
\end{itemize}
\begin{quote}
Hint: apply RR. For the bound, consider a plane curve \(X\) of degree 4
and \(D = X.H\).
\end{quote}
\end{exercise}
\begin{remark}
Idea behind embedding in \({\mathbf{P}}^3\): embed into
\({\mathbf{P}}^n\) and project away from a point in the complement.
\end{remark}
\begin{exercise}[3.4, 3.5, 3.6]
Let \(X \subseteq {\mathbf{P}}^N\) be a curve and \(O\not\in X\), let
\(\phi:X\to {\mathbf{P}}^{n-1}\) be projection away from \(O\). Then
\(\phi\) is a closed immersion iff
\begin{itemize}
\tightlist
\item
\(O\) is not on any secant line of \(X\), and
\item
\(O\) is not on any tangent line of \(X\).
\end{itemize}
Show that if \(N\geq 4\) then there exists such a point \(O\) yielding a
closed immersion into \({\mathbf{P}}^{N-1}\). Conclude that any curve
can be embedded into \({\mathbf{P}}^3\).
\begin{quote}
Hint: \(\dim\mathrm{Sec}(X) \leq 3\) and
\(\dim \mathrm{Tan}(X) \leq 2\).
\end{quote}
\end{exercise}
\begin{proposition}[3.7]
Let \(X \subseteq {\mathbf{P}}^3\), \(O\not\in X\), and
\(\phi: X\to {\mathbf{P}}^2\) be the projection from \(O\). Then
\(X\overset{\sim}{\dashrightarrow}\phi(X)\) iff \(\phi(X)\) is nodal iff
the following hold:
\begin{itemize}
\tightlist
\item
\(O\) is only on finitely many secants of \(X\),
\item
\(O\) is on no tangents,
\item
\(O\) is on no multisecant,
\item
\(O\) is on no secant with coplanar tangent lines.
\end{itemize}
\end{proposition}
\begin{quote}
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.
\end{quote}
\begin{remark}
Classifying all curves: any curve is birational to a nodal plane curve,
so study the family \({\mathcal{F}}_{d, r}\) of plane curves of degree
\(d\) and \(r\) nodes. The family \({\mathcal{F}}_d\) of all plane
curves is a linear system of dimension
\begin{align*}
\dim {\left\lvert {{\mathcal{F}}_d} \right\rvert} = {d(d+3)\over 2}
.\end{align*}
For any such curve \(X\), consider its normalization \(\nu(X)\), then
\begin{align*}
g(\nu(X)) = {(d-1)(d-2)\over 2} - r
.\end{align*}
Thus for \({\mathcal{F}}_{d, r}\) to be nonempty, one needs
\begin{align*}
0 \leq r \leq {(d-1)(d-2) \over 2}
.\end{align*}
Both extremes can occur: \(r=0\) follows from Bertini, and
\(r = {(d-1)(d-2)\over 2}\) by embedding
\({\mathbf{P}}^1\hookrightarrow{\mathbf{P}}^d\) as a curve of degree
\(d\) and projecting down to a nodal curve in \({\mathbf{P}}^2\) of
genus zero. Severi states and Harris proves that for every \(r\) in this
range \({\mathcal{F}}_{d, r}\) is irreducible, nonempty, and
\(\dim {\mathcal{F}}_{d, r} = {d(d+3)\over 2} - r\).
\end{remark}
\hypertarget{iv.4-elliptic-curves-star}{%
\subsection{\texorpdfstring{IV.4: Elliptic Curves
\(\star\)}{IV.4: Elliptic Curves \textbackslash star}}\label{iv.4-elliptic-curves-star}}
\begin{remark}
Curves \(E\) with \(g(E) = 1\); we'll assume
\(\operatorname{ch}k \neq 2\) throughout. Outline:
\begin{itemize}
\tightlist
\item
Define the \(j{\hbox{-}}\)invariant, classifies isomorphism classes of
elliptic curves.
\item
Group structure on the curve.
\item
\(E = \operatorname{Jac}(E)\).
\item
Results about elliptic functions over \({\mathbf{C}}\).
\item
The Hasse invariant of \(E/{ \mathbf{F} }_q\) in characteristic \(p\).
\item
\(E({\mathbf{Q}})\).
\end{itemize}
\end{remark}
\hypertarget{the-jhbox-invariant}{%
\subsubsection{\texorpdfstring{The
\(j{\hbox{-}}\)invariant}{The j\{\textbackslash hbox\{-\}\}invariant}}\label{the-jhbox-invariant}}
\begin{remark}
The \(j{\hbox{-}}\)invariant:
\begin{itemize}
\tightlist
\item
\(j(E) \in k\), so \({\mathbf{A}}^1_{/ {k}}\) is a coarse moduli space
for elliptic curves over \(K\).
\item
Defining \(j(E)\):
\begin{itemize}
\tightlist
\item
Let \(p_0\in X\), consider the linear system
\(L\coloneqq{\left\lvert {2p_0} \right\rvert}\).
\item
Nonspecial, so RR shows \(\dim(L) = 1\).
\item
BPF, otherwise \(E\) is rational.
\item
Defines a morphism \(\phi_L: E\to {\mathbf{P}}^1_{/ {k}}\) with
\(\deg \phi_L = 2\).
\item
Up to change of coordinates, \(f(p_0) = \infty\).
\item
By Hurwitz, \(f\) is ramified at 4 branch points \(a,b,c,p_0\).
\item
Move \(a\mapsto 0, b\mapsto 1\) by a Mobius transformation fixing
\(\infty\), so \(f\) is branched over \(0,1,\lambda,\infty\) where
\(\lambda \in k\setminus\left\{{ 0,1 }\right\}\).
\item
Use \(\lambda\) to define the invariant:
\begin{align*}
j(E) = j( \lambda) = 2^8\qty{(\lambda^2 - \lambda+ 1)^3 \over \lambda^2 (\lambda- 1)^2}
.\end{align*}
\end{itemize}
\item
Theorem 4.1:
\begin{itemize}
\tightlist
\item
\(j\) depends only on the curve \(E\) and not \(\lambda\).
\item
\(E\cong E'\iff j(E) = j(E')\).
\item
Every element of \(k\) occurs as \(j(E)\) for some \(E\).
\item
So this yields a bijection
\begin{align*}
\left\{{\substack{
\text{Elliptic curves over }k
}}\right\}{_{\scriptstyle / \sim} }
&\rightleftharpoons
{\mathbf{A}}^1_{/ {k}} \\
E &\mapsto j(E)
.\end{align*}
\end{itemize}
\item
Some facts that go into proving this:
\begin{itemize}
\tightlist
\item
\(\forall p,q\in X\,\,\exists \sigma\in \mathop{\mathrm{Aut}}(X)\)
such that \(\sigma^2=1, \sigma(p) = q\), for any \(r\in X\), one has
\(r + \sigma(r) \sim p + q\).
\item
\(\mathop{\mathrm{Aut}}(X)\curvearrowright X\) transitively.
\item
Any two degree two maps \(f_1,f_2: X\to {\mathbf{P}}^1\) fit into a
commuting square.
\item
Under
\(S_3\curvearrowright{\mathbf{A}}^1_{/ {k}}\setminus\left\{{ 0, 1 }\right\}\),
the orbit of \(\lambda\) is
\begin{align*}
S_3 . \lambda= \left\{{ \lambda, \lambda^{-1}, s_1 = 1- \lambda, s_1^{-1}= (1- \lambda)^{-1}, s_2 = \lambda(\lambda-1)^{-1}, s_3 = \lambda^{-1}(\lambda- 1)}\right\}
.\end{align*}
\item
Fixing \(p\in X\), there is a closed immersion
\(X\to {\mathbf{P}}^2\) whose image is \(y^2=x(x-1)(x- \lambda)\)
where \(p\mapsto \infty = {\left[ {0:1:0} \right]}\) and this
\(\lambda\) is either the \(\lambda\) from above or one of
\(s_1^{\pm 1}, s_2^{\pm 1}\).
\begin{itemize}
\tightlist
\item
Idea of proof: embed \(X\hookrightarrow{\mathbf{P}}^2\) by
\(L\coloneqq{\left\lvert {3p} \right\rvert}\), use RR to compute
\(h^0({\mathcal{O}}(np)) = n\) so \(h^0({\mathcal{O}}(6p)) = 6\).
\item
So \(\left\{{1,x,y,x^2,xy,y^2,x^3}\right\}\) has a linear
dependence where \(x^3,y^2\) have nonzero coefficients since they
have poles at \(p\).
\item
Rescale \(x^3, y^2\) to coefficient 1 to get
\begin{align*}
y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6
.\end{align*}
\end{itemize}
\item
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.
\end{itemize}
\item
Note that one can project from \(p\) to the \(x{\hbox{-}}\)axis to get
a finite degree 2 morphism ramified at \(0,1, \lambda, \infty\).
\end{itemize}
\end{remark}
\begin{example}[?]
An elliptic curve that is smooth over every field of non-2
characteristic:
\begin{align*}
E: y^2 = x^3-x, \qquad \lambda=-1,\, j(E) = 2^6 \cdot 3^3 = 1728
.\end{align*}
\includegraphics{figures/2022-12-03_23-36-23.png}
One that is smooth over every \(k\) with \(\operatorname{ch}k \neq 3\):
the Fermat curve
\begin{align*}
E: x^3 + y^3 = z^3,\qquad \lambda = \pm \zeta_3^{k},\, j(E) = 0
.\end{align*}
\end{example}
\begin{theorem}[Orders of automorphism groups of elliptic curves]
\begin{align*}
{\sharp}\mathop{\mathrm{Aut}}(X, p) =
\begin{cases}
2 & j(E) \neq 0,1728\\
4 & j(E) = 1728, \operatorname{ch}k \neq 3 \\
6 & j(E) = 0,\operatorname{ch}k \neq 3 \\
12 & j(E) = 0,1728, \operatorname{ch}k = 3
\end{cases}
.\end{align*}
\end{theorem}
\begin{remark}[Proof idea]
Idea: take the degree 2 morphism \(f:X\to {\mathbf{P}}^1\) with
\(f(p) = \infty\) branched over
\(\left\{{0,1, \lambda, \infty}\right\}\). Produce two elements in
\(G\): for \(\sigma\in G\), find
\(\tau\in \mathop{\mathrm{Aut}}({\mathbf{P}}^1)\) so
\(f\sigma = \tau f\); then either \(\tau \neq \operatorname{id}\), so
\(\left\{{\sigma, \tau}\right\} \subseteq G\), or \(\tau = id\) and
either \(\sigma=\operatorname{id}\) or \(\sigma\) exchanges the sheets
of \(f\).
If \(\tau\neq \operatorname{id}\), it permutes
\(\left\{{0, 1, \lambda}\right\}\) and sends
\(\lambda\mapsto \lambda^{-1}, s_1^{\pm 1}, s_2^{\pm 1}\) from above.
Cases:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
\(j=1728:\) If \(\lambda= -1, 1/2, 2, \operatorname{ch}k \neq 3\),
then \(\lambda\) coincides with \emph{one} other element of
\(S_3. \lambda\), so \({\sharp}G = 4\).
\item
\(j=0\): If
\(\lambda= -\zeta_3, -\zeta_3^2, \operatorname{ch}k \neq 3\) then
\(\lambda\) coincides with \emph{two} elements in \(S_3 . \lambda\) so
\({\sharp}G = 6\).
\item
\(j=0=1728\): If \(\lambda= -1, \operatorname{ch}k = 3\) then
\(S_3 . \lambda= \left\{{ \lambda}\right\}\) and \({\sharp}G = 12\).
\end{enumerate}
\end{remark}
\hypertarget{the-group-structure}{%
\subsubsection{The group structure}\label{the-group-structure}}
\begin{remark}
The group structure:
\begin{itemize}
\tightlist
\item
Fixing \(p_o\in E\), the map \(p\mapsto {\mathcal{L}}(p-p_0)\) induces
a bijection \(E { \, \xrightarrow{\sim}\, }\operatorname{Pic}^0(E)\),
so the group structure on \(E\) is the pullback along this with
\(p_0 = \operatorname{id}\) and
\(p+q=r\iff p+q \sim r+p_0 \in \operatorname{Div}(E)\).
\item
Under the embedding of \({\left\lvert {3p_0} \right\rvert}\), points
\(p,q,r\) are collinear iff \(p+q+r \sim 3p_0\), so \(p+q+r=0\) in the
group structure.
\item
\(E\) 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
\((n, \operatorname{ch}k) = 1\).and \(\ker [n] = C_p, 0\) if
\(n=\operatorname{ch}k\), depending on the Hasse invariant.
\item
If \(f:E_1 \to E_2\) is a morphism of curves with \(f(p_1) = p_2\)
then \(f\) induces a group morphism.
\item
\({ \operatorname{End} }(E, p_0)\) forms a ring under
\(f+g = \mu\circ (f\times g)\) and \(f\cdot g \coloneqq f\circ g\).
\item
The map \(n \mapsto ([n]: E\to E)\) defines a finite ring morphism
\({\mathbf{Z}}\to { \operatorname{End} }(E, p_0)\) for \(n\neq 0\).
\item
\(R \coloneqq{ \operatorname{End} }(E, p_0)^{\times}= \mathop{\mathrm{Aut}}(E)\),
and if \(j=0,1728\) then \(R\) contains \(\left\{{\pm 1}\right\}\) and
is thus bigger than \({\mathbf{Z}}\).
\end{itemize}
\end{remark}
\begin{remark}
The Jacobian: a variety that generalizes to make sense for any curve, a
moduli space of degree zero divisor classes.
\begin{itemize}
\item
For \(X/k\) a curve and \(T\in{\mathsf{Sch}}_{/ {k}}\), define
\begin{align*}
\operatorname{Pic}^0(X\times T) \coloneqq\left\{{{\mathcal{F}}\in \operatorname{Pic}(X\times T) {~\mathrel{\Big\vert}~}\deg { \left.{{{\mathcal{F}}}} \right|_{{X_t}} } = 0 \, \forall t\in T }\right\},\qquad \operatorname{Pic}(X/T) \coloneqq\operatorname{Pic}^0(X\times T)/ p^* \operatorname{Pic}(T)
\end{align*}
where \(p:X\times T\to T\) is the second projection. Regard this as
\emph{families of sheaves of degree 0 on \(X\) parameterized by
\(T\)}.
\item
The Jacobian variety of a curve \(X\):
\(\operatorname{Jac}(X) \in {\mathsf{Sch}}^{\mathrm{ft}}_{/ {k}}\)
along with
\({\mathcal{L}}\in \operatorname{Pic}^0(X/\operatorname{Jac}(X))\)
such that for any \(T\in{\mathsf{Sch}}^{\mathrm{ft}}_{/ {k}}\) and any
\({\mathcal{M}}\in \operatorname{Pic}^0(X/T)\),
\(\exists ! \, f: T\to \operatorname{Jac}(X)\) such that
\(f^* {\mathcal{L}}= {\mathcal{M}}\). Thus \(J\) represents the
functor \(\operatorname{Pic}^0(X/{-})\).
\item
For \(E\) elliptic, \(E = \operatorname{Jac}(E)\).
\begin{itemize}
\tightlist
\item
In general,
\({\left\lvert {\operatorname{Jac}(X)} \right\rvert}\cong {\left\lvert {\operatorname{Pic}^0(X)} \right\rvert}\)
on points, since points of \(\operatorname{Jac}(X)\) are morphisms
\(\operatorname{Spec}k\to \operatorname{Jac}(X)\), which correspond
to elements in
\(\operatorname{Pic}^0(X/k) = \operatorname{Pic}^0(X)\).
\end{itemize}
\item
\(\operatorname{Jac}(X) \in {\mathsf{Grp}}{\mathsf{Sch}}_{/ {k}}\)
where \(e: \operatorname{Spec}k\to \operatorname{Jac}(X)\) corresponds
to \(0\in \operatorname{Pic}^0(X/k)\),
\(\rho: \operatorname{Jac}(X) \to \operatorname{Jac}(X)\) is
\({\mathcal{L}}\mapsto {\mathcal{L}}^{-1}\in\operatorname{Pic}^0(X/\operatorname{Jac}(X))\),
and
\(\mu: \operatorname{Jac}(X){ {}^{ \scriptscriptstyle\times^{2} } }\to \operatorname{Jac}(X)\)
is
\({\mathcal{L}}\mapsto p_1^* {\mathcal{L}}\otimes p_2^*{\mathcal{L}}\in\operatorname{Pic}^0(X/\operatorname{Pic}(X){ {}^{ \scriptscriptstyle\times^{2} } })\).
\item
\({\mathbf{T}}_0 \operatorname{Jac}(X) \cong H^1(X; {\mathcal{O}}_X)\):
giving an element of \({\mathbf{T}}_p X\) is the same as a morphism
\(T\coloneqq\operatorname{Spec}k[{\varepsilon}]/{\varepsilon}^2\to X\)
sending \(\operatorname{Spec}k \to p\). So
\({\mathbf{T}}_0 \operatorname{Jac}(X)\), this means giving
\({\mathcal{M}}\in \operatorname{Pic}^0(X/T)\) whose restriction to
\(\operatorname{Pic}^0(X/k)\) is zero. Use the SES
\(H^1(X;{\mathcal{O}}_X)\hookrightarrow\operatorname{Pic}X[{\varepsilon}] \to \operatorname{Pic}(X)\).
\item
\(\operatorname{Jac}(X)\) is proper over \(k\) by the valuative
criterion. Just show that an invertible sheaf \({\mathcal{M}}\) on
\(X\times \operatorname{Spec}K\) lifts unique to
\(\tilde {\mathcal{M}}\) on \(X\times \operatorname{Spec}R\), but
\(X\times \operatorname{Spec}R\) is regular, so apply \(\rm{II}.6.5\).
\item
For any \(n\) there is a morphism
\begin{align*}
\phi^n: X{ {}^{ \scriptscriptstyle\times^{n} } } &\to \operatorname{Jac}(X) \\
(p_1,\cdots, p_n) &\mapsto {\mathcal{L}}(\sum p_i - np_0)
.\end{align*}
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.
\begin{itemize}
\tightlist
\item
Most fibers are finite, so \(\operatorname{Jac}(X)\) is irreducible
of dimension \(g\).
\item
Smoothness:
\(\dim {\mathbf{T}}_0 \operatorname{Jac}(X) = \dim H^1(X;{\mathcal{O}}_X) = g\),
so smooth at zero, and group schemes are homogeneous so smooth
everywhere.
\end{itemize}
\end{itemize}
\end{remark}
\hypertarget{elliptic-functions}{%
\subsubsection{Elliptic functions}\label{elliptic-functions}}
\begin{quote}
Stopped at elliptic functions.
\end{quote}
\hypertarget{iv.5-the-canonical-embedding}{%
\subsection{IV.5: The Canonical
Embedding}\label{iv.5-the-canonical-embedding}}
\hypertarget{iv.6-classification-of-curves-in-mathbfp3}{%
\subsection{\texorpdfstring{IV.6: Classification of Curves in
\({\mathbf{P}}^3\)}{IV.6: Classification of Curves in \{\textbackslash mathbf\{P\}\}\^{}3}}\label{iv.6-classification-of-curves-in-mathbfp3}}
\newpage
\hypertarget{v-surfaces}{%
\section{V: Surfaces}\label{v-surfaces}}
\hypertarget{v.1-geometry-on-a-surface}{%
\subsection{V.1: Geometry on a
Surface}\label{v.1-geometry-on-a-surface}}
\hypertarget{v.2-ruled-surfaces}{%
\subsection{V.2: Ruled Surfaces}\label{v.2-ruled-surfaces}}
\hypertarget{v.3-monoidal-transformations}{%
\subsection{V.3: Monoidal
Transformations}\label{v.3-monoidal-transformations}}
\hypertarget{v.4-the-cubic-surface-in-mathbfp3}{%
\subsection{\texorpdfstring{V.4: The Cubic Surface in
\({\mathbf{P}}^3\)}{V.4: The Cubic Surface in \{\textbackslash mathbf\{P\}\}\^{}3}}\label{v.4-the-cubic-surface-in-mathbfp3}}
\hypertarget{v.5-birational-transformations}{%
\subsection{V.5: Birational
Transformations}\label{v.5-birational-transformations}}
\hypertarget{v.6-classification-of-surfaces}{%
\subsection{V.6: Classification of
Surfaces}\label{v.6-classification-of-surfaces}}
\hypertarget{toric-varieties}{%
\section{Toric Varieties}\label{toric-varieties}}
\hypertarget{summaries}{%
\subsection{Summaries}\label{summaries}}
\hypertarget{quick-criteria}{%
\subsubsection{Quick Criteria}\label{quick-criteria}}
\begin{remark}
Quick criteria:
\begin{itemize}
\item
\textbf{Normal} \(\iff\) \textbf{Saturated}: For affines,
\(X = \operatorname{Spec}{\mathbf{C}}[S]\) where \(S \subseteq M\) is
a \textbf{saturated} semigroup. This is true for
\(S = S_\sigma = \sigma {}^{ \vee }\cap M\) where \(\sigma\) is any
SCRPC.
\item
\textbf{Complete/proper} \(\iff\) \textbf{Full support}: \(X_\Sigma\)
is complete iff \(\mathop{\mathrm{supp}}\Sigma = N_{\mathbf{R}}\).
\item
\textbf{Smooth} \(\iff\) \textbf{Lattice basis}:
\begin{itemize}
\tightlist
\item
For a \textbf{cone} \(\sigma = { \mathrm{Cone} }(S)\) is smooth iff
\(\operatorname{det}S = \pm 1\), the volume of the standard lattice
\({\mathbf{Z}}^n\).
\begin{itemize}
\tightlist
\item
Consequences of smoothness:
\begin{itemize}
\tightlist
\item
\(\operatorname{CDiv}(X) = \operatorname{Div}(X)\)
\item
\(\operatorname{Cl}(X) = \operatorname{Pic}(X)\)
\end{itemize}
\end{itemize}
\item
Smooth implies simplicial, so non-simplicial cones are singular.
\item
For \(p_\sigma\) the \(T{\hbox{-}}\)fixed point corresponding to
\(\sigma\), \(T_p X \cong H\) where \(H\) is a Hilbert basis for
\(S_\sigma\).
\end{itemize}
\item
\textbf{Simplicial} \(\iff\) \textbf{Euclidean basis}: For
\(\sigma = { \mathrm{Cone} }(S)\), \(\sigma\) is simplicial iff
\(\operatorname{det}(S) \neq 0\).
\item
\textbf{Orbifold singularities} \(\iff\) \textbf{Simplicial}:
\(X_\Sigma\) has at worst finite quotient singularities iff \(\Sigma\)
is simplicial.
\item
\textbf{Projectivity} \(\iff\) \textbf{Admits a strictly upper convex
support function}: For \(h\) a support function and \(D_h\) its
associated divisor, the linear system
\({\left\lvert {D_h} \right\rvert}\) defines an embedding
\(X(\Delta) \hookrightarrow{\mathbf{P}}^N\) iff \(h\) is strictly
upper convex.
\begin{itemize}
\tightlist
\item
Alternatively, \(X_\Sigma\) is projective iff \(\Sigma\) arises as
the normal fan of a polytope.
\end{itemize}
\item
\textbf{Globally generated/basepoint free} \(\iff\) \textbf{Upper
convex support function}: \({\mathcal{O}}(D)\) is globally generated
iff \(\psi_D\) is upper convex.
\item
\textbf{Ample \(\iff\) Strictly upper convex support function}:\\
\(D\in \operatorname{CDiv}_T(X)\) is ample iff \(\psi_D\) is strictly
upper convex.
\item
\textbf{Very ample \(\iff\) ample and semigroup generation}: for
\(\Sigma\) complete, \(D\) is very ample iff \(\psi_D\) is strictly
upper convex \textbf{and} \(S_\sigma\) is generated by
\(\left\{{u-u(\sigma) {~\mathrel{\Big\vert}~}u\in P_D \cap M}\right\}\),
or equivalently the semigroup
\(\left\{{u-u' {~\mathrel{\Big\vert}~}u'\in P \cap M}\right\}\) is
saturated in \(M\).
\begin{itemize}
\tightlist
\item
For \({\mathbf{P}}^n\): \(D = \sum a_i D_i\) is globally generated
iff \(\sum a_i \geq 0\) and ample \(\iff \sum a_i > 0\).
\item
For \({ \mathbf{F} }_m\): \(D = \sum a_i D_i\) is globally generated
iff \(a_2 + a_4 \geq 0,\, a_1 + a_3 \geq m a_1\),
\(\operatorname{Pic}({ \mathbf{F} }_n) = \left\langle{D_1, D_4}\right\rangle\),
and \(D = aD_1 + bD_4\) is ample iff \(a,b > 0\).
\item
For \(\dim X_\Sigma = 2\) and \(X\) complete: ample \(\iff\) very
ample.
\end{itemize}
\item
\textbf{\({\mathbf{Q}}{\hbox{-}}\)factorial \(\iff\) simplicial}: iff
every cone is simplicial.
\item
\textbf{Fundamental groups}:
\begin{itemize}
\tightlist
\item
For \(U_\sigma\) affine,
\(U_\sigma \cong {\mathbf{A}}^k \times {\mathbf{G}}_m^{n-k}\) so
\(\pi_1 U_\sigma \cong {\mathbf{Z}}^{n-k}\) since
\({\mathbf{G}}_m^{n-k}\simeq(S^1)^{n-k}\).
\item
Can write \(\pi_1 U_\sigma = N/N_\sigma\) where \(N_\sigma\) is the
sublattice generated by \(\sigma\).
\item
By a Van Kampen argument, \(\pi_1 X_\Sigma = N/N'\) where
\(N' = \left\langle{\sigma \cap N {~\mathrel{\Big\vert}~}\sigma \in \Sigma}\right\rangle\):
\begin{align*}
\pi_1 X_\Sigma = \pi_1 \cup U_{\sigma} = \colim \pi_1 U_\sigma = \colim N/N_\sigma = N / \sum N_\sigma = N/N'
.\end{align*}
\end{itemize}
\item
\textbf{Euler characteristic}: \(\chi X_\Sigma = {\sharp}\Sigma(n)\).
\begin{itemize}
\tightlist
\item
Why:
\(H^i(U_\sigma; {\mathbf{Z}}) = \bigwedge\nolimits^i M(\sigma)\)
where \(M(\sigma ) \coloneqq\sigma {}^{ \vee }\cap M\), so one gets
a spectral sequence
\begin{align*}
E_1^{p, q} = \bigoplus _{I^p = i_0< \cdots < i_p} H^q(U_{\sigma_{I^p}}; {\mathbf{Z}}) \Rightarrow H^{p+q}(X_\Sigma; {\mathbf{Z}}), \qquad \sigma_{I^p} = \sigma_{i_0} \cap\cdots \sigma_{i_p}, \sigma_{i_j}\in \Sigma(n) \\ \\
\leadsto E_1^{p, q} = \bigoplus _{I^p} \bigwedge\nolimits^q M(\sigma_{I^p}) \Rightarrow H^{p+q}(X_\Sigma; {\mathbf{Z}}) \\\\
\implies \chi X_\Sigma = \sum (-1)^{p+q} \operatorname{rank}_{\mathbf{Z}}E_1^{p, q} = {\sharp}\Sigma(n)
,\end{align*}
using that
\begin{align*}
\sum (-1)^{q} \operatorname{rank}_{\mathbf{Z}}\extpower^q M(\tau) =
\begin{cases}
0 & \dim \tau < n
\\
1 & \dim \tau = n.
\end{cases}
.\end{align*}
\end{itemize}
\item
\textbf{Higher homology}:
\begin{itemize}
\tightlist
\item
If all maximal cones of \(\Sigma\) are \(n{\hbox{-}}\)dimensional,
\(H^2(X_\Sigma; {\mathbf{Z}}) \cong \operatorname{Pic}(X_\Sigma)\).
\end{itemize}
\item
\textbf{Global sections}: for \(D\in \operatorname{Div}_T(X)\),
\(P_D\) its associated polyhedron,
\begin{align*}
H^0(X; {\mathcal{O}}_X(D)) = \bigoplus _{m\in P_D \cap M} {\mathbf{C}}\, \chi^m
.\end{align*}
\item
\textbf{Betti numbers}:
\begin{align*}
\beta_{2k} = \sum_{i=k}^n (-1)^{i-k} {i\choose k} {\sharp}\Sigma(n-i)
.\end{align*}
\item
\textbf{Canonical bundles/divisors}:
\(\omega_{X_\Sigma} \coloneqq\operatorname{det}\Omega_{X_\Sigma/k} = {\mathcal{O}}(K_{X_\Sigma})\)
where \(K_{X_\Sigma} = -\sum_{\rho_i} D_i\).
\begin{itemize}
\tightlist
\item
For a smooth complete surface with \(D_i^2 = -d_i\),
\begin{align*}
K^2 = \sum D_i^2 + 2d = -\sum d_i + 2d = -(3d-12) + 2d = 12-d
.\end{align*}
\end{itemize}
\item
\textbf{Degree = \(n! \cdot \Vol(P)\)} (for \(X_P\) projective)
\end{itemize}
\end{remark}
\begin{remark}
Some common counterexamples:
\begin{itemize}
\tightlist
\item
An ample divisor that is not very ample:
\(P \coloneqq\Conv({\left[ {0,0,0} \right]}, {\left[ {0,1,1} \right]}, {\left[ {1,0,1} \right]}, {\left[ {1,1,0} \right]})\);
then take \(D_P\). \(X_P\) is a double cover of \({\mathbf{P}}^3\)
branched along the 4 boundary divisors.
\item
A Weil divisor that is not Cartier: ????
\item
A complete variety that is not projective: ???
\end{itemize}
\end{remark}
\hypertarget{cones-and-lattices}{%
\subsubsection{Cones and Lattices}\label{cones-and-lattices}}
\begin{remark}
\envlist
\begin{itemize}
\item
\textbf{Characters}: for groups \(G\), a map
\(\chi\in {\mathsf{Grp}}(G, {{\mathbf{C}}^{\times}})\). For
\(G= T = ({{\mathbf{C}}^{\times}})^n\), there is an isomorphism
\begin{align*}
{\mathbf{Z}}^n & { \, \xrightarrow{\sim}\, }{\mathsf{Grp}}(T, {{\mathbf{C}}^{\times}}) \\
m = {\left[ {m_1,\cdots, m_n} \right]} &\mapsto \chi_m: {\left[ {t_1,\cdots, t_n} \right]} \mapsto \prod t_i^{m_i}
.\end{align*}
Generally set
\(M \coloneqq{\mathsf{Grp}}(T, {{\mathbf{C}}^{\times}})\), the
character lattice.
\begin{itemize}
\tightlist
\item
\(M\) is a lattice,
\(M_{\mathbf{R}}\coloneqq M\otimes_{\mathbf{Z}}{\mathbf{R}}\) is its
associated Euclidean space.
\end{itemize}
\item
\textbf{Cocharacters / one-parameter subgroups}: for groups \(G\), a
map \(\lambda \in {\mathsf{Grp}}({{\mathbf{C}}^{\times}}, G)\). For
\(G = T = {{\mathbf{C}}^{\times}}\), there is again an isomorphism
\begin{align*}
{\mathbf{Z}}^n &\mapsto {\mathsf{Grp}}({{\mathbf{C}}^{\times}}, T) \\
u ={\left[ {u_1,\cdots, u_n} \right]} &\mapsto \lambda^u: t\mapsto {\left[ {t^{u_1}, \cdots, t^{u_n}} \right]}
.\end{align*}
Define \(N \coloneqq{\mathsf{Grp}}({{\mathbf{C}}^{\times}}, T)\) the
cocharacter lattice.
\begin{itemize}
\tightlist
\item
\(N\) is a lattice,
\(N_{\mathbf{R}}\coloneqq N\otimes_{\mathbf{Z}}{\mathbf{R}}\) its
associated euclidean space.
\end{itemize}
\item
There is a perfect pairing
\begin{align*}
{\left\langle {{-}},~{{-}} \right\rangle}: M\times N &\to {\mathbf{Z}}\\
,\end{align*}
defined using the fact that if \(m\in M, n\in N\) then
\(\chi^m \circ \lambda^n \in {\mathsf{Grp}}({{\mathbf{C}}^{\times}}, {{\mathbf{C}}^{\times}})\)
is of the form \(t\mapsto t^\ell\), so set
\({\left\langle {m},~{n} \right\rangle} \coloneqq\ell\).
\begin{itemize}
\tightlist
\item
Thus \(M = {\mathsf{Grp}}(M, {\mathbf{Z}})\) and
\(N = {\mathsf{Grp}}(N, {\mathbf{Z}})\).
\item
How to recover the torus:
\begin{align*}
N \otimes_{\mathbf{Z}}{{\mathbf{C}}^{\times}}&\to T \\
u\otimes t &\mapsto \lambda^u(t)
.\end{align*}
\end{itemize}
\item
\(\Delta\) is a \textbf{fan}, a collection of \textbf{strongly convex
rational polyhedral cones}:
\begin{itemize}
\tightlist
\item
\textbf{Cone}: \(0\in \sigma\) and
\({\mathbf{R}}_{\geq 0} \sigma \subseteq \sigma\).
\item
\textbf{Strongly convex}: contains no nonzero subspace, i.e.~no line
through \(\mathbf{0} \in N_{\mathbf{R}}\). Equivalently,
\(\dim \sigma {}^{ \vee }= n\).
\item
\textbf{Rational}: generated by
\(\left\{{v_i}\right\} \subseteq N\), i.e.~of the form
\({ \mathrm{Cone} }(S)\) for \(S \subseteq N\).
\end{itemize}
\item
\textbf{Dual cones}:
\begin{align*}
\sigma {}^{ \vee }&\coloneqq\left\{{ u\in M {~\mathrel{\Big\vert}~}{\left\langle {u},~{v} \right\rangle} \geq 0 \,\,\forall v\in M_{\mathbf{R}}}\right\}
.\end{align*}
\begin{itemize}
\tightlist
\item
If \(\sigma {}^{ \vee }= \bigcap_{i=1}^s H_{m_i}^+\) for
\(m_i \subseteq \sigma {}^{ \vee }\) then
\(\sigma {}^{ \vee }= { \mathrm{Cone} }(m_1,\cdots, m_s)\).
\end{itemize}
\item
\textbf{Hyperplanes} and \textbf{closed half-spaces}:
\begin{align*}
H_m &\coloneqq\left\{{u\in N_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} = 0}\right\} \subseteq N_{\mathbf{R}}\\
H_m^+ &\coloneqq\left\{{u\in N_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} \geq 0}\right\} \subseteq N_{\mathbf{R}}
.\end{align*}
\item
\textbf{Face}: \(\tau \leq \sigma\) is a face iff \(\tau\) is of the
form \(\tau = H_m \cap\sigma\) for some
\(m\in \sigma {}^{ \vee }\subseteq M_{\mathbf{R}}\).
\item
\textbf{Facet}: codimension one faces, \(\Sigma(n-1)\) where
\(n\coloneqq\dim N\).
\item
\textbf{Ray}: dimension 1 faces, \(\Sigma(1)\).
\item
The \textbf{semigroup} of a cone:
\begin{align*}
S_\sigma &\coloneqq\sigma {}^{ \vee }\cap M = \left\{{ u\in M {~\mathrel{\Big\vert}~}{\left\langle {u},~{v} \right\rangle} \geq 0 \,\,\forall v\in \sigma }\right\}
.\end{align*}
\item
The \textbf{semigroup algebra} of a semigroup:
\begin{align*}
{\mathbf{C}}[S] \coloneqq\left\{{\sum_{s\in S} c_s \chi^s {~\mathrel{\Big\vert}~}c_s \in {\mathbf{C}}, c_s = 0 { \text{a.e.} }}\right\}, \qquad \chi^{m_1}\cdot \chi^{m_2} \coloneqq\chi^{m_1 + m_2}
.\end{align*}
\item
\textbf{Simplicial}: the generators can be extended to an
\({\mathbf{R}}{\hbox{-}}\)basis of \(N_{\mathbf{R}}\). E.g. not
simplicial:
\end{itemize}
\includegraphics{figures/2022-10-19_18-23-05.png}
\begin{itemize}
\item
\textbf{Smooth}: the minimal generators can be extended to a
\({\mathbf{Z}}{\hbox{-}}\)basis of \(N\).
\begin{itemize}
\tightlist
\item
Checking \(T_p X\): \(m\) is \textbf{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
\({\mathfrak{m}}_p = \left\{{\chi^m {~\mathrel{\Big\vert}~}m\in S_ \sigma}\right\}\),
and
\({\mathfrak{m}}_p/{\mathfrak{m}}_p^2 = \left\{{\chi^m {~\mathrel{\Big\vert}~}m \text{ is indecomposable in } S_ \sigma}\right\}\).
This exactly corresponds to a Hilbert basis.
\end{itemize}
\item
\textbf{Facet}: face of codimension 1.
\item
\textbf{Edge}: face of dimension 1. Note that facets = edges in
\(\dim N = 2\).
\item
\textbf{Saturated}: \(S\) is saturated if for all
\(k\in {\mathbb{N}}\setminus\left\{{0}\right\}\) and all \(m\in M\),
\(km\in S \implies m\in S\). Any SCRPC is saturated.
\begin{itemize}
\tightlist
\item
E.g. \(S = \left\{{(4,0), (3,1), (1,3), (0, 4)}\right\}\) is not
saturated since \(2\cdot(2,2) = (4, 4) \in {\mathbb{N}}S\) but
\((2,2)\not\in S\).
\end{itemize}
\item
\textbf{Normalization}: in the affine case, write
\(X = \operatorname{Spec}{\mathbf{C}}[S]\) with torus character
lattice \(M = {\mathbf{Z}}S\), take a finite generating set \(S'\),
and set \(\sigma = { \mathrm{Cone} }(S') {}^{ \vee }\). Then
\(\operatorname{Spec}{\mathbf{C}}[\sigma {}^{ \vee }\cap M]\to X\) is
the normalization.
\item
\textbf{Distinguished points}: each strongly convex
\(\sigma \leadsto \gamma_\sigma \in U_\sigma\) a unique point
corresponding to the semigroup morphism
\(m\mapsto \indic(m\in \sigma {}^{ \vee }\cap M)\), which is
\(T{\hbox{-}}\)fixed iff \(\sigma\) is full-dimensional.
\item
\textbf{Orbits}: \({\mathrm{Orb}}( \sigma) = T. \gamma_\sigma\), and
\(V(\sigma)\coloneqq{ \operatorname{cl}}{\mathrm{Orb}}( \sigma)\).
\item
\textbf{Orbit-Cone correspondence}: there is a correspondence
\begin{align*}
\left\{{\text{Cones } \sigma \in \Sigma}\right\} &\rightleftharpoons\left\{{T{\hbox{-}}\text{orbits in } X_\Sigma}\right\} \\
\sigma &\mapsto {\mathrm{Orb}}(\sigma) \coloneqq T.\gamma_{\sigma} = \left\{{\gamma: S_\sigma \to {\mathbf{C}}{~\mathrel{\Big\vert}~}\gamma(m) \neq 0 \iff m\in \sigma {}^{ \vee }\cap M}\right\} \cong {\mathsf{Grp}}(\sigma \cap M, {{\mathbf{C}}^{\times}})
,\end{align*}
where
\(\dim {\mathrm{Orb}}( \sigma) = \operatorname{codim}_{N_{\mathbf{R}}} \sigma\),
and
\(\tau \leq \sigma \implies { \operatorname{cl}}{\mathrm{Orb}}(\tau) \supseteq{ \operatorname{cl}}{\mathrm{Orb}}( \sigma)\)
and in fact
\({ \operatorname{cl}}{\mathrm{Orb}}(\sigma) = {\textstyle\coprod}_{\tau\leq \sigma} { \operatorname{cl}}{\mathrm{Orb}}( \tau)\).
\item
\textbf{Star}: define
\(N_\tau \coloneqq{\mathbf{Z}}\left\langle{\tau \cap N}\right\rangle\)
and
\(N(\tau){\mathbf{R}}\coloneqq N_{\mathbf{R}}/ (N_\tau)_{\mathbf{R}}\)
and \(\overline{\sigma}\) for the image of \(\sigma\) under the
quotient map, then
\begin{align*}
\mathrm{Star}(\tau) \coloneqq\left\{{\overline{\sigma }\subseteq N(\tau)_{\mathbf{R}}{~\mathrel{\Big\vert}~}\sigma\leq \tau }\right\} \subseteq N(\tau)_{\mathbf{R}}
.\end{align*}
This is always a fan, and \(V(\tau) = X_{\mathrm{Star}(\tau)}\).
\item
\textbf{Star subdivision}: for \(\sigma = { \mathrm{Cone} }(S)\) for
\(S \coloneqq\left\{{u_1,\cdots, u_n}\right\}\), set
\(u_0 \coloneqq\sum u_i\) and take \(\Sigma'(\sigma)\) defined as the
cones generated by subsets of
\(\left\{{u_0, u_1, \cdots, u_n}\right\}\) not containing \(S\). The
star subdivision of \(\Sigma\) along \(\sigma\) is
\(\Sigma^\star(\sigma) \coloneqq(\Sigma \setminus\left\{{ \sigma }\right\}) \cup\Sigma'( \sigma)\).
\item
\textbf{Blowups}: \(\phi: X_{\Sigma^\star(\sigma)}\to X_{\Sigma}\) is
the blowup at \(\gamma_ \sigma\).
\end{itemize}
\end{remark}
\hypertarget{divisors}{%
\subsubsection{Divisors}\label{divisors}}
\begin{remark}
\envlist
\begin{itemize}
\item
\textbf{(Weil) divisor}:
\(\operatorname{Div}(X) = \left\{{\sum n_i V_i {~\mathrel{\Big\vert}~}V_i \subseteq X, \operatorname{codim}V_i = 1}\right\}\).
\begin{itemize}
\tightlist
\item
\({\mathcal{O}}_X(D)\): the (coherent) sheaf associated to a Weil
divisor \(D\).
\end{itemize}
\item
\textbf{Cartier divisor}:
\(\operatorname{CDiv}(X) = H^0(X; {\mathcal{K}}_X^{\times}/{\mathcal{O}}_X^{\times})\),
the quotient of rational functions by regular functions. For \(X\)
normal, equivalently locally principal (Weil) divisors, so
\(D \leadsto \left\{{(U_i, f_i)}\right\}\) where
\({ \left.{{D}} \right|_{{U_i}} } = \operatorname{Div}(f_i)\).
\begin{itemize}
\tightlist
\item
\textbf{\({\mathbf{Q}}{\hbox{-}}\)Cartier divisor}: A
\({\mathbf{Q}}{\hbox{-}}\)divisor \(D =\sum n_i D_i\) with
\(n_i\in {\mathbf{Q}}\) is \({\mathbf{Q}}{\hbox{-}}\)Cartier when
\(mD\) is Cartier for some \(m\in {\mathbf{Z}}_{\geq 0}\).
\item
\textbf{\({\mathbf{Q}}{\hbox{-}}\)factorial}: every prime divisor is
\({\mathbf{Q}}{\hbox{-}}\)Cartier.
\end{itemize}
\item
\textbf{Ray divisors}: every \(\rho\in \Sigma(1)\) defines a divisor
\(D_\rho \coloneqq V(\rho) \coloneqq{ \operatorname{cl}}{\mathrm{Orb}}( \rho)\).
\item
\textbf{Very Ample}: \({\mathcal{L}}\) which defines a morphism into
\({\mathbf{P}}H^0(X; {\mathcal{L}}) \cong {\mathbf{P}}^N\).
\item
\textbf{Ample}: \({\mathcal{L}}\) is basepoint free and some power
\({\mathcal{L}}^n\) is very ample.
\begin{itemize}
\tightlist
\item
\(D\) is (very) ample iff \({\mathcal{O}}_X(D)\) is (very) ample,
i.e.~\(D\) is ample iff \(nD\) is very ample for some \(n\).
\end{itemize}
\item
\textbf{Upper convex}: \(f(n_1 + n_2) \leq f(n_1) + f(n_2)\).
\begin{itemize}
\tightlist
\item
\textbf{Strictly upper convex}:
\(\sigma_1\neq \sigma_2 \implies f_{\sigma_1} \neq f_{\sigma_2}\).
\end{itemize}
\item
\textbf{Linearly equivalent divisors}:
\(D_1\sim D_2 \iff D_1 - D_2 = \operatorname{Div}(f)\) for some \(f\).
\item
\textbf{Complete linear systems}:
\({\left\lvert {D} \right\rvert} = \left\{{D'\in \operatorname{Div}(X) {~\mathrel{\Big\vert}~}D'\sim D}\right\}\).
\item
\textbf{Support function}:
\(\phi: \mathop{\mathrm{supp}}\Sigma \to {\mathbf{R}}\) where
\({ \left.{{\phi}} \right|_{{\sigma}} }\) is linear for each cone
\(\sigma\).
\begin{itemize}
\tightlist
\item
\textbf{Integral} with respect to \(N\) iff
\(\phi(\mathop{\mathrm{supp}}\Sigma \cap N) \subseteq {\mathbf{Z}}\).
Defines a set of integral support functions
\(\operatorname{SF}(\Sigma, N)\).
\end{itemize}
\item
The class group complement exact sequence: for
\(D_1,\cdots, D_n \in \operatorname{Div}(X)\) distinct,
\begin{align*}
{\mathbf{Z}}^n &\to \operatorname{Cl}(X) \twoheadrightarrow \operatorname{Cl}(X\setminus\cup D_i) \\
e_1 &\mapsto [D_i]
.\end{align*}
\item
\({\mathcal{O}}_X(D)\) is the sheaf
\begin{align*}
U\mapsto \left\{{f\in {\mathcal{K}}(X)^{\times}(U) {~\mathrel{\Big\vert}~}\operatorname{Div}(f) + { \left.{{D}} \right|_{{U}} } \geq 0 \in \operatorname{Cl}(U) }\right\}
.\end{align*}
Then
\(D\in \operatorname{CDiv}(X) \iff {\mathcal{O}}_X(D) \in \operatorname{Pic}(X)\).
\item
The toric class group exact sequence:
\begin{align*}
M &\to \operatorname{Div}_T(X) \twoheadrightarrow \operatorname{Cl}(X) \\
m &\mapsto \operatorname{Div}(\chi^m) = \sum_\rho {\left\langle {m},~{u_\rho} \right\rangle} [D_\rho]
\end{align*}
where \(u_\rho\) are minimal ray generators.
\end{itemize}
\end{remark}
\hypertarget{polytopes}{%
\subsubsection{Polytopes}\label{polytopes}}
\begin{remark}
\envlist
\begin{itemize}
\item
\textbf{Supporting hyperplanes}: the positive side of an affine
hyperplane
\begin{align*}
H_{u, b} &\coloneqq\left\{{m\in M_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} = b}\right\} \\
H_{u, b}^+ &\coloneqq\left\{{m\in M_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} \geq b}\right\}
.\end{align*}
\begin{itemize}
\tightlist
\item
If \(P\) is full dimensional and \(F\leq P\) is a facet, then
\(F = P \cap H_{u_F, -a_F}\) for a unique pair
\((u_F, a_F) \in N_{\mathbf{R}}\times {\mathbf{R}}\).
\end{itemize}
\item
\textbf{Polytope}: the convex hull of a finite set
\(S \subseteq N_{\mathbf{R}}\) or an intersection of half-spaces:
\begin{align*}
P = \left\{{\sum_{v\in S} \lambda_v v {~\mathrel{\Big\vert}~}\sum \lambda_v = 1}\right\} = \bigcap_{i=1}^s H_{u_i, b_i}^+
.\end{align*}
\item
\textbf{Simplex} \(\dim P = d\) and there are exactly \(d+1\)
vertices.
\item
\textbf{Simple}: \(\dim P = d\) and every vertex is the intersection
of exactly \(d\) facets.
\item
\textbf{Simplicial}: all facets are simplices.
\begin{itemize}
\tightlist
\item
E.g. simple but not simplicial: the cube in \({\mathbf{R}}^3\),
since each vertex meets 3 edges but a square is not a simplex. -E.g.
Simplicial but not simple: the octahedron in \({\mathbf{R}}^3\),
since each vertex meets 4 edges but each face is a triangle.
\end{itemize}
\item
\textbf{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.
\item
\textbf{Polar dual}: for \(P \subseteq M_{\mathbf{R}}\),
\begin{align*}
P^\circ = \left\{{u\in N_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} \geq - 1\,\, \forall m\in P}\right\}
.\end{align*}
\begin{itemize}
\tightlist
\item
Trick: for \(P \subseteq M_{\mathbf{R}}\) with \(0\in P\),
\begin{align*}
P = \left\{{m\in M_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u_F} \right\rangle} \geq -a_F,\, F \in \mathrm{Facets}(P) }\right\} \\
\implies P^\circ = \Conv(\left\{{ a_F^{-1}u_F }\right\}) \subseteq N_{\mathbf{R}}
.\end{align*}
E.g. write the square as
\(\left\{{{\left\langle {m},~{\pm e_i} \right\rangle}\geq -1}\right\}\),
then \(a_F = 1\) for all \(F\):
\includegraphics{figures/2022-10-19_18-35-59.png}
\end{itemize}
\item
\textbf{Cone on a polytope}:
\(C(P) \coloneqq{ \mathrm{Cone} }(P\times \left\{{1}\right\}) \subseteq M_{\mathbf{R}}\times {\mathbf{R}}\),
the set of cones through all proper faces of \(P\).
\item
\textbf{Normal}:
\(\qty{kP \cap M} + \qty{\ell P \cap M} \subseteq (k+\ell)P \cap M\),
or equivalently \(k\cdot (P \cap M) = (kP) \cap M\), or equivalently
\((P \cap M)\times\left\{{1}\right\}\) generates
\(C(P) \cap(M\times {\mathbf{Z}})\) as a semigroup.
\begin{itemize}
\tightlist
\item
If \(P \subseteq M_{\mathbf{R}}\) is a full-dimensional lattice
polytope with \(\dim P \geq 2\), then \(kP\) is normal for all
\(k\geq \dim P - 1\).
\item
Normal implies very ample.
\item
\(P\leadsto {\mathcal{L}}_P \in \operatorname{Pic}(X_P)\)
\item
\(P \cap M \leadsto H^0(X_P; {\mathcal{L}}_P)\).
\end{itemize}
\item
\textbf{Reflexive}: a polytope \(P\) with facet presentation
\begin{align*}
P = \left\{{m\in M_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{\mu_F} \right\rangle} \geq -1 \forall F\in \mathrm{Facets}(P)}\right\}
.\end{align*}
Implies that \(\int(P) \cap M = \left\{{\mathbf{0}}\right\}\), and
\(P^\circ = \Conv(\left\{{u_F {~\mathrel{\Big\vert}~}F\in \mathrm{Facets}(P)}\right\})\).
\item
\textbf{Polyhedron of a divisor \(P_D\)}: write
\(D = \sum_{\rho} a_{\rho} D_{\rho}\), for any \(m\in M\),
\(\operatorname{Div}(\chi^m) + D \geq 0 \implies {\left\langle {m},~{\rho} \right\rangle} \geq a_{\rho} \implies {\left\langle {m},~{\rho} \right\rangle} \geq - a_\rho\),
so set
\begin{align*}
P_D \coloneqq\left\{{ m\in M_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{\rho } \right\rangle}\geq a_\rho \, \forall \rho \in \Sigma(1)}\right\}
.\end{align*}
\item
\textbf{Divisor of a polytope}: \(D_P = \sum_F a_F D_F\) where
\(P = \left\{{m {~\mathrel{\Big\vert}~}{\left\langle {m},~{u_F} \right\rangle} \geq -a_F}\right\}\).
\begin{itemize}
\tightlist
\item
\(D_P\) is always the pullback of
\({\mathcal{O}}_{{\mathbf{P}}^N}(1)\) along the embedding.
\end{itemize}
\item
\textbf{Very ample polytopes}: for every vertex \(v\), the semigroup
\(\left\{{m' - v {~\mathrel{\Big\vert}~}m'\in P \cap M}\right\}\) is
saturated in \(M\).
\begin{itemize}
\tightlist
\item
Gives an embedding \(X \hookrightarrow{\mathbf{P}}^N\) where
\(N = {\sharp}(P \cap M) - 1\).
\end{itemize}
\item
The \textbf{toric variety of a polytope}: if
\(P \cap M = \left\{{m_1,\cdots, m_s}\right\}\) and \(P\) is full
dimensional very ample, then writing \(T_N\) for the torus of \(N\),
\begin{align*}
X_{P} \coloneqq{ \operatorname{cl}}\operatorname{im}\phi,\qquad \phi: T_N &\to {\mathbf{P}}^{s-1} \\
t &\mapsto {\left[ {\chi^{m_1}(t) : \cdots : \chi^{m_s}(t)} \right]}
.\end{align*}
\begin{itemize}
\tightlist
\item
Vertices \(m_i\) correspond to \(U_{\sigma_i}\) for
\(\sigma_i = { \mathrm{Cone} }(P \cap M - m_i) {}^{ \vee }\):
\end{itemize}
\includegraphics{figures/2022-10-19_19-08-06.png}
\item
\textbf{Smooth}: \(P\) is smooth iff for all vertices \(v\in P\),
\(\left\{{w_E - v{~\mathrel{\Big\vert}~}E\text{ is an edge containing }v}\right\}\)
can be extended to a \({\mathbf{Z}}{\hbox{-}}\)basis of \(M\), where
\(w_E\) is the first lattice point on \(E\).
\end{itemize}
\end{remark}
\hypertarget{singularities-and-classification}{%
\subsubsection{Singularities and
Classification}\label{singularities-and-classification}}
\begin{remark}
\envlist
\begin{itemize}
\tightlist
\item
\textbf{Gorenstein}: \(X\) normal where
\(K_X \in \operatorname{CDiv}(X)\) is Cartier.
\item
\textbf{Normal}: all local rings are integrally closed domains.
\item
\textbf{Complete}: proper over \(k\). E.g. for varieties, just
universally closed.
\item
\textbf{Factorial}: all local rings are UFDs.
\item
\textbf{Fano}: \(-K_X\) is ample.
\item
\textbf{del Pezzo}: a smooth Fano surface.
\end{itemize}
\end{remark}
\begin{remark}
Classification of smooth complete toric varieties:
\begin{itemize}
\tightlist
\item
\(\dim \Sigma = 2, {\sharp}\Sigma(1) = 3\): 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
\(\mathop{\mathrm{supp}}\Sigma = {\mathbf{R}}^2\), and determinants
for
\({\left\lvert {a} \right\rvert} = {\left\lvert {b} \right\rvert} = 1\),
so \((-1, 1)\).
\item
\(\dim \Sigma = 2, {\sharp}\Sigma(1) = 4\): 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
\(\operatorname{det}{ \begin{bmatrix} {-1} & {a} \\ {b} & {-1} \end{bmatrix} } = 1-ab = \pm 1 \implies ab=0,2\),
so \((a,b) = (2,1), (1,2), (-2, -1), (-1,-2)\).
\item
\(\dim \Sigma = 2, {\sharp}\Sigma(1) = d\), smooth:
\(\operatorname{Bl}_{p_1,\cdots, p_\ell} X\) for
\(X = {\mathbf{P}}^2\) or \({ \mathbf{F} }_a\) for some \(a\) and
\(p_i\) torus fixed points.
\end{itemize}
\end{remark}
\hypertarget{examples}{%
\subsubsection{Examples}\label{examples}}
\begin{question}
Things you can figure out for every example:
\begin{itemize}
\tightlist
\item
Given \(\Delta\), for \(\sigma\in \Delta\),
\begin{itemize}
\tightlist
\item
What is \(\sigma {}^{ \vee }\)?
\item
Generators for \(S_\sigma\)?
\item
Describe \(U_\sigma\) and \(X(\Delta)\).
\item
What are the transition functions for
\(U_{\sigma_1} \to U_{\sigma_2}\) when
\(\sigma_1 \cap\sigma_2 = \tau\) intersect in a common face?
\end{itemize}
\item
What are the \(T{\hbox{-}}\)invariant points?
\begin{itemize}
\tightlist
\item
What are the \(T{\hbox{-}}\)invariant divisors \(D_{\rho_i}\)?
\item
What are all of the \(T{\hbox{-}}\)orbit closures of various
dimensions?
\end{itemize}
\item
Is \(X(\Delta)\) smooth?
\begin{itemize}
\tightlist
\item
Which cones \(\sigma\in \Delta\) are smooth?
\item
What is the canonical resolution of singularities?
\item
What is the tangent space at each \(T{\hbox{-}}\)invariant point?
\end{itemize}
\item
What is the associated polytope \(P_\Delta\)? What is its polar dual
\(P_\Delta^\circ\)?
\item
What are the intersection numbers \(D_{\rho_i} \cdot D_{\rho_j}\)?
\begin{itemize}
\tightlist
\item
What are the self-intersection numbers \(D_{\rho_i}^2\)?
\end{itemize}
\item
What is \(\operatorname{Div}_T(X)\)? \(\operatorname{CDiv}_T(X)\)?
\begin{itemize}
\tightlist
\item
Which divisors are ample? Very ample? Globally generated?
\end{itemize}
\item
What is \(\operatorname{Cl}(X)\)? \(\operatorname{Pic}(X)\)?
\item
What is \(K_X\)?
\begin{itemize}
\tightlist
\item
Is \(K_X\) ample?
\end{itemize}
\item
Is \(X(\Delta)\) projective?
\item
What is \(H^0(X(\Delta); {\mathcal{O}}(D) )\) for
\(D\in \operatorname{Div}_T(X)\)?
\item
What is the Poincaré polynomial of \(X(\Delta)\)? (I.e. what are the
Betti numbers?)
\end{itemize}
\end{question}
\begin{example}[of varieties]
Some useful explicit varieties:
\begin{itemize}
\tightlist
\item
\(V(x^3-y^2)\) with torus
\(T = \left\{{{\left[ {t^2, t^3} \right]} {~\mathrel{\Big\vert}~}t\in {{\mathbf{C}}^{\times}}}\right\}\).
\item
\(V(xy-zw)\) with torus
\(T = \left\{{{\left[ {a,b,c,abc^{-1}} \right]} {~\mathrel{\Big\vert}~}a,b,c,d\in {{\mathbf{C}}^{\times}}}\right\}\).
\item
\(V(xz-y^2)\), note
\(V(x, y)\in \operatorname{Div}(X) \setminus\operatorname{CDiv}(X)\).
\item
\(\operatorname{im}([x:y] \mapsto [x^3: x^2y : xy^2 : y^3])\) the
twisted cubic. Corresponds to
\(\sigma {}^{ \vee }= \left\{{(3,0), (2,1), (1,2), (0, 3)}\right\}\).
\item
The \textbf{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
\({\left[ {s,t} \right]} \mapsto {\left[ {1:s:s^2:t:st} \right]}\).
\item
The Segre variety:
\(\operatorname{Spec}{\mathbf{C}}[x_1y_1, x_1 y_2, \cdots, x_1 y_n, x_2 y_1, \cdots, x_m y_1, \cdots x_m y_n]\).
\end{itemize}
\end{example}
\begin{example}[of fans]
\envlist
\begin{itemize}
\tightlist
\item
\(({\mathbf{C}}^{\times})^n\): Take
\(\Delta = \left\{{ \sigma_0 = {\mathbb{N}}\left\langle{0}\right\rangle}\right\} \subseteq N\)
with \(\dim N = n\) yields
\(S_{\sigma_0} = {\mathbb{N}}\left\langle{\pm e_1 {}^{ \vee },\cdots, \pm e_n {}^{ \vee }}\right\rangle = M\)
for so
\(X(\Delta) = \operatorname{Spec}{\mathbf{C}}[x_1^{\pm 1},\cdots, x_n^{\pm 1}] = ({\mathbf{G}}_m)^n\).
\item
\({\mathbf{C}}^n\): Take
\(\Delta = { \mathrm{Cone} }(\sigma_0 = {\mathbb{N}}\left\langle{ e_1,\cdots, e_n}\right\rangle )\)
yields the positive orthant
\(S_{\sigma_0} = {\mathbb{N}}\left\langle{e_1 {}^{ \vee },\cdots, e_n {}^{ \vee }}\right\rangle \subseteq M\),
so
\(X(\Delta) = \operatorname{Spec}{\mathbf{C}}[x_1,\cdots, x_n] = {\mathbf{A}}^n\).
\item
The quadric cone:
\(\Delta = { \mathrm{Cone} }(\sigma_1 = {\mathbb{N}}\left\langle{e_2, 2e_1 - e_2}\right\rangle)\)
yields
\(S_{\sigma_1} = {\mathbb{N}}\left\langle{e_1 {}^{ \vee }, e_1 {}^{ \vee }+ e_2 {}^{ \vee }, e_1 {}^{ \vee }+ 2e_2 {}^{ \vee }}\right\rangle\)
so
\(X(\Delta) = \operatorname{Spec}{\mathbf{C}}[x, xy, xy^2] = \operatorname{Spec}{\mathbf{C}}[u,v,w]/(v^2-uw)\):
\end{itemize}
\includegraphics{figures/2022-10-18_15-33-37.png}
\includegraphics{figures/2022-10-18_15-33-47.png}
\begin{itemize}
\tightlist
\item
\({\mathbf{P}}^1\): Take
\(\Delta = \left\{{{\mathbf{R}}_{\geq 0}, 0, {\mathbf{R}}_{\leq 0}}\right\}\)
and glue along overlaps to get \(X(\Delta) = {\mathbf{P}}^1\) with
gluing maps \(x\mapsto x^{-1}\):
\end{itemize}
\includegraphics{figures/2022-10-18_15-36-53.png}
\begin{itemize}
\tightlist
\item
\(\operatorname{Bl}_1 {\mathbf{C}}^2\): Take
\(\sigma_0 = {\mathbb{N}}\left\langle{ e_2, e_1+e_2}\right\rangle\)
and \(\sigma_1 = {\mathbb{N}}\left\langle{e_1+e_2, e_1}\right\rangle\)
to get \(U_{ \sigma_0} = \operatorname{Spec}{\mathbf{C}}[x, x^{-1}y]\)
and \(U_{ \sigma_1} = \operatorname{Spec}{\mathbf{C}}[y, xy^{-1}]\),
both copies of \({\mathbf{C}}^2\):
\end{itemize}
\includegraphics{figures/2022-10-18_15-39-10.png}
Why this is a blowup of \({\mathbf{C}}^2\): write
\(\operatorname{Bl}_1 {\mathbf{C}}^2 = V(xt_1 - yt_0) \subseteq {\mathbf{C}}^2\times {\mathbf{P}}^1\)
for \({\mathbf{P}}^1 = \left\{{{\left[ {t_0: t_1} \right]}}\right\}\).
Take the open cover \(U_i = D(t_i) \cong {\mathbf{C}}^2\), where
coordinates on \(U_0\) are \(x, t_1/t_0 = x^{-1}y\) and on \(U_1\) are
\(y, t_0/t_1 = xy^{-1}\) and glue.
\begin{itemize}
\item
\({\mathbf{P}}^2\): take
\(\Delta = { \mathrm{Cone} }(e_1, e_2, -e_1-e_2)\):
\includegraphics{figures/2022-10-18_15-42-04.png}
This has dual cone:
\includegraphics{figures/2022-10-18_15-42-18.png}
Each \(U_{\sigma_i} \cong {\mathbf{C}}^2\) with coordinates
\((x,y), (x^{-1}, x^{-1}y), (y^{-1}, xy^{-1})\) respectively for
\(U_i\). Glue to obtain \(x=t_1/t_0, y=t_2/t_0\).
\item
\(F_a\) the Hirzebruch surface: take
\({ \mathrm{Cone} }(e_1, -e_2, -e_1, -e_1 + ae_2)\) to get
\begin{itemize}
\tightlist
\item
\(U_{\sigma_1} = \operatorname{Spec}{\mathbf{C}}[x,y]\),
\item
\(U_{\sigma_2} = \operatorname{Spec}{\mathbf{C}}[x,y^{-1}]\),
\item
\(U_{\sigma_3} = \operatorname{Spec}{\mathbf{C}}[x^{-1},x^{-a} y^{-1}]\),
\item
\(U_{\sigma_4} = \operatorname{Spec}{\mathbf{C}}[x^{-1},x^a y]\),
\end{itemize}
which patch in the following way:
\includegraphics{figures/2022-10-18_15-45-17.png}
Project to \(y=0\) to get the patching \(x\mapsto x^{-1}\), so a copy
of \({\mathbf{P}}^1\). Patching in the fiber direction,
e.g.~\(U_{\sigma_1}\) and \(U_{\sigma_2}\), gives a copy of
\({\mathbf{C}}\times {\mathbf{P}}^1\). Thus this is a bundle
\({\mathbf{P}}^1\to {\mathcal{E}}\to {\mathbf{P}}^1\).
\item
\({\mathbf{C}}\times {\mathbf{P}}^1\): todo.
\item
\({\mathbf{P}}^1 \times {\mathbf{P}}^1\): todo.
\item
\({\mathbf{C}}^a \times {\mathbf{P}}^b\): todo.
\item
\({\mathbf{P}}^a \times {\mathbf{P}}^b\): todo.
\end{itemize}
\end{example}
\begin{example}[of polytopes]
\begin{itemize}
\tightlist
\item
Hirzebruch surfaces:
\end{itemize}
\includegraphics{figures/2022-12-03_20-09-23.png}
\begin{itemize}
\tightlist
\item
\(({\mathbf{P}}^2, {\mathcal{O}}(1))\): take
\(P = \Conv(0, e_1, e_2)\), so \(X_P = { \operatorname{cl}}\Phi_P\)
where
\begin{align*}
\Phi_P: ({{\mathbf{C}}^{\times}})^2 &\to {\mathbf{P}}^2 \\
(s,t) &\mapsto [1: s: t]
,\end{align*}
which is the identity embedding corresponding to \({\mathcal{O}}(1)\)
on \({\mathbf{P}}^2\).
\begin{itemize}
\tightlist
\item
\(2P\) yields
\begin{align*}
\Phi_{2P}: ({{\mathbf{C}}^{\times}})^2 &\to {\mathbf{P}}^5 \\
(s,t) &\mapsto [1: s: t : s^2: st: t^2]
,\end{align*}
the Veronese embedding corresponding to \({\mathcal{O}}(2)\) on
\({\mathbf{P}}^2\).
\end{itemize}
\end{itemize}
\end{example}
\begin{example}[Projective spaces]
Some useful facts about \({\mathbf{P}}^n\):
\begin{itemize}
\tightlist
\item
The torus embedding is
\begin{align*}
({{\mathbf{C}}^{\times}})^n &\hookrightarrow{\mathbf{P}}^n \\
{\left[ {a_1,\cdots, a_n} \right]} &\mapsto {\left[ {1: a_1 : \cdots : a_n} \right]}
.\end{align*}
\item
The torus action is
\begin{align*}
({{\mathbf{C}}^{\times}})^n &\curvearrowright{\mathbf{P}}^n \\
{\left[ {t_1,\cdots, t_n} \right]} . {\left[ {x_0: x_1:\cdots:x_n} \right]} &= {\left[ {x_0: t_1 x_1:\cdots:t_n x_n} \right]}
.\end{align*}
\end{itemize}
\end{example}
\begin{example}[of class groups and Picard groups]
\includegraphics{figures/2022-10-20_00-07-20.png}
\includegraphics{figures/2022-10-20_00-10-35.png}
\end{example}
\hypertarget{i-definitions-and-examples}{%
\section{I: Definitions and Examples}\label{i-definitions-and-examples}}
\hypertarget{introduction}{%
\subsection{1.1: Introduction}\label{introduction}}
\begin{remark}
Machinery used to study varieties:
\begin{itemize}
\tightlist
\item
Various cohomology theories
\item
Resolutions of singularities
\item
Intersection theory and cycles
\item
Riemann-Roch theorems
\item
Vanishing theorems
\item
Linear systems (via line bundles and projective embeddings)
\end{itemize}
Varieties that arise as examples
\begin{itemize}
\tightlist
\item
Grassmannians
\item
Flag varieties
\item
Veronese embeddings
\item
Scrolls
\item
Quadrics
\item
Cubic surfaces
\item
Toric varieties (of course)
\item
Symmetric varieties and their compactifications
\end{itemize}
Misc notes:
\begin{itemize}
\tightlist
\item
Toric varieties are always rational
\end{itemize}
\end{remark}
\begin{remark}
\envlist
\begin{itemize}
\tightlist
\item
Toric varieties: normal varieties \(X\) with \(T\hookrightarrow X\)
contained as a dense open subset where the torus action
\(T\times T\to T\) extends to \(T\times X\to X\).
\item
Any product of copies of \({\mathbf{A}}^n, {\mathbf{P}}^m\) are toric.
\item
\(S_\sigma\) is a finitely-generated semigroup, so
\({\mathbf{C}}[S_\sigma] \in \mathsf{Alg}{{\mathbf{C}}}^{\mathrm{fg}}\)
corresponds to an affine variety
\(U_\sigma \coloneqq\operatorname{Spec}{\mathbf{C}}[S_\sigma]\).
\item
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 \(\tau = \sigma \cap u_\tau^\perp\), so \(u_\tau\) corresponds to
the orthogonal normal vector for the wall \(\tau\).
\item
These glue to a variety \(X(\Delta)\).
\item
Smaller cones correspond to smaller open subsets.
\item
The geometry in \(N\) is nicer than that in \(M\), usually.
\item
Rays \(\rho\) correspond to curves \(D_\rho\).
\end{itemize}
\end{remark}
\begin{exercise}[?]
\envlist
\begin{itemize}
\tightlist
\item
Show \(F_a\to {\mathbf{P}}^1\) is isomorphic to
\({\mathbf{P}}({\mathcal{O}}(a) \oplus {\mathcal{O}}(1))\).
\item
Let \(\tau\) be the ray through \(e_2\) in \(F_a\) and show
\(D_\tau^2 = -a\).
\item
Show that the normal bundle to \(D_\tau \hookrightarrow F_a\) is
\({\mathcal{O}}(-a)\).
\end{itemize}
\end{exercise}
\hypertarget{convex-polyhedral-cones}{%
\subsection{1.2: Convex Polyhedral
Cones}\label{convex-polyhedral-cones}}
\begin{remark}
\envlist
\begin{itemize}
\tightlist
\item
\textbf{Convex polyhedral cones}: generated by vectors
\(\sigma = {\mathbf{R}}_{\geq 0}\left\langle{v_1,\cdots, v_n}\right\rangle\).
Can take minimal vectors along these rays, say \(\rho_i\).
\end{itemize}
\includegraphics{figures/2022-10-18_20-35-45.png}
\begin{itemize}
\tightlist
\item
\(\dim \sigma \coloneqq\dim_{\mathbf{R}}{\mathbf{R}}\sigma \coloneqq\dim_{\mathbf{R}}(-\sigma + \sigma)\)
\item
\((\sigma {}^{ \vee }) {}^{ \vee }= \sigma\), which follows from a
general theorem: for \(\sigma\) a convex polyhedral cone and
\(v\not\in \sigma\), there is some support vector
\(u_v\in \sigma {}^{ \vee }\) such that
\({\left\langle {u},~{v} \right\rangle} < 0\). I.e. \(v\) is on the
negative side of some hyperplane defined in \(\sigma {}^{ \vee }\).
\item
Faces are again convex polyhedral cones, faces are closed under
intersections and taking further faces.
\item
If \(\sigma\) spans \(V\) and \(\tau\) is a facet, there is a unique
\(u_\tau\in \sigma {}^{ \vee }\) such that
\(\tau = \sigma \cap u_\tau^\perp\); this defines an equation for the
hyperplane \(H_\tau\) spanned by \(\tau\).
\item
If \(\sigma\) spans \(V\) and \(\sigma\neq V\), then
\(\sigma = \cap_{\tau\in \Delta} H_\tau^+\), the intersection of
positive half-spaces.
\begin{itemize}
\tightlist
\item
An alternative presentation: picking \(u_1,\cdots, u_t\) generators
of \(\sigma {}^{ \vee }\), one has
\(\sigma = \left\{{v\in N {~\mathrel{\Big\vert}~}{\left\langle {u_1},~{v} \right\rangle} \geq 0, \cdots, {\left\langle {u_t},~{v} \right\rangle}\geq 0}\right\}\).
\end{itemize}
\item
If \(\tau \leq \sigma\) then
\(\sigma {}^{ \vee }\cap\tau {}^{ \vee }\leq \sigma {}^{ \vee }\) and
\(\dim \tau = \operatorname{codim}(\sigma {}^{ \vee }\cap\tau {}^{ \vee })\),
so the faces of \(\sigma, \sigma {}^{ \vee }\) biject contravariantly.
\item
If \(\tau = \sigma \cap u_\tau^\perp\) then
\(S_\tau = S_\sigma + {\mathbb{N}}\left\langle{-u_\tau}\right\rangle\).
\end{itemize}
\end{remark}
\hypertarget{singularities-and-compactness}{%
\section{Singularities and
Compactness}\label{singularities-and-compactness}}
\hypertarget{section}{%
\subsection{2.1}\label{section}}
\begin{remark}
\begin{itemize}
\item
Any cone \(\sigma\in \Sigma\) has a distinguished point \(x_\sigma\)
corresponding to
\(\mathop{\mathrm{Hom}}_{\semigroup}(S_\sigma, {\mathbf{C}})\) where
\(u\mapsto \chi_{u\in \sigma^\perp}\).
\begin{itemize}
\tightlist
\item
Note \(S_\sigma \coloneqq\sigma {}^{ \vee }\cap M\).
\end{itemize}
\item
Define \(A_\sigma \coloneqq{\mathbf{C}}[S_\sigma]\).
\item
Finding singular points:
\begin{itemize}
\tightlist
\item
Easy case: \(\sigma\) spans \(N_{\mathbf{R}}\) so
\(\sigma^\perp = 0\); consider
\({\mathfrak{m}}\in \operatorname{mSpec}A_\sigma\) be the maximal
ideal at \(x_\sigma\), then
\({\mathfrak{m}}= \left\langle{\chi^u {~\mathrel{\Big\vert}~}u\in S_ \sigma}\right\rangle\)
and
\({\mathfrak{m}}^2 = \left\langle{\chi^u {~\mathrel{\Big\vert}~}u \in S_\sigma\setminus\left\{{0}\right\}+ S_\sigma\setminus\left\{{0}\right\}}\right\rangle\),
so
\({\mathbf{T}}_{x_\sigma} {}^{ \vee }U_\sigma = {\mathfrak{m}}/{\mathfrak{m}}^2 = \left\{{\chi^u {~\mathrel{\Big\vert}~}u\not \in S_{\sigma}\setminus\left\{{0}\right\}+ S_\sigma\setminus\left\{{0}\right\}}\right\}\),
i.e.~``primitive'' elements \(u\) which are not the sums of two
other vectors in \(S_\sigma\setminus\left\{{0}\right\}\).
\item
Nonsingular implies \(\dim U_\sigma = n\), so \(\sigma {}^{ \vee }\)
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 \(U_\sigma \cong {\mathbf{A}}^n\).
\end{itemize}
\item
\textbf{Characterization of smoothness}: \(U_\sigma\) is smooth iff
\(\sigma\) is generated by a subset of a lattice basis for \(N\), in
which case
\(U_\sigma \cong {\mathbf{A}}^k \times {\mathbf{G}}_m^{n-k}\).
\item
All toric varieties are normal since each \(A_\sigma\) is integrally
closed.
\begin{itemize}
\tightlist
\item
If \(\sigma = \left\langle{v_1,\cdots, v_r}\right\rangle\) then
\(\sigma {}^{ \vee }= \cap_{i=1}^r \tau_i {}^{ \vee }\) where
\(\tau_i\) is the ray along \(v_i\). Thus
\(A_\sigma = \cap A_{\tau_i}\), each of which is isomorphic to
\({\mathbf{C}}[x_1, x_2^{\pm 1}, \cdots, x_n^{\pm 1}\) which is
integrally closed.
\end{itemize}
\item
All toric varieties are \textbf{Cohen-Macaulay}: each local ring \(R\)
has depth \(n\), i.e.~contains a regular sequence of length
\(n = \dim R\).
\item
All vector bundles on affine toric varieties are trivial, equivalently
all projective modules over \(A_\sigma\) are free.
\end{itemize}
\end{remark}
\hypertarget{section-1}{%
\subsection{2.2}\label{section-1}}
\begin{remark}
\begin{itemize}
\tightlist
\item
An example: \(\Sigma = { \mathrm{Cone} }(me_1-e_2, e_2)\). Then
\(A_\sigma = {\mathbf{C}}[x, xy, xy^2,\cdots,xy^m] = {\mathbf{C}}[u^m u^{m-1}v,\cdots, uv^{m-1}, v^m]\)
and \(U_\sigma\) is the cone over the rational normal curve of degree
\(m\).
\begin{itemize}
\tightlist
\item
Note \(A_\sigma = {\mathbf{C}}[u,v]^{\mu_m}\) is the ring of
invariants under the diagonal action
\(\zeta.{\left[ {u, v} \right]} = {\left[ {\zeta u, \zeta v} \right]}\).
\end{itemize}
\item
If \(\Sigma\) is simplicial, then \(X_\Sigma\) is at worst an
orbifold.
\end{itemize}
\end{remark}
\hypertarget{section-2}{%
\subsection{2.3}\label{section-2}}
\begin{remark}
\begin{itemize}
\item
\(\mathop{\mathrm{Hom}}_{ \mathsf{Alg}{\mathsf{Grp}}}({\mathbf{G}}_m, {\mathbf{G}}_m) = {\mathbf{Z}}\)
using \(n\mapsto (z\mapsto z^n)\).
\item
Cocharacters:
\begin{itemize}
\tightlist
\item
Pick a basis for \(N\) to get
\(\mathop{\mathrm{Hom}}({\mathbf{G}}_m, T_N) = \mathop{\mathrm{Hom}}({\mathbf{Z}}, N) = N\),
then every cocharacter
\(\lambda \in \mathop{\mathrm{Hom}}({\mathbf{G}}_m, T_N)\) is given
by a unique \(v\in N\), so denote it \(\lambda_v\). Then
\(\lambda_v(z)\in T_N = \mathop{\mathrm{Hom}}(M, {\mathbf{G}}_m)\)
for any \(z\in {{\mathbf{C}}^{\times}}\), so
\begin{align*}
u\in M \implies \lambda_v(z)(u) = \chi^u(\lambda_v(z))= z^{{\left\langle {u},~{v} \right\rangle}}
.\end{align*}
\end{itemize}
\item
Characters:
\(\chi \in \mathop{\mathrm{Hom}}(T_n, {\mathbf{G}}_m) = \mathop{\mathrm{Hom}}(N, {\mathbf{Z}}) = M\)
is given by a unique \(u\in M\) and can be identified with
\(\chii^u \in {\mathbf{C}}[M] = H^0(T_N, {\mathcal{O}}_{T_N}^{\times})\).
\item
\(\lim_{z\to 0}\lambda_v(z) = \lim_{z\to 0} {\left[ {z^{m_1},\cdots, z^{m_n}} \right]} \in U_\sigma \iff m_i \geq 0\)
for all \(i\), and if
\(U_\sigma = {\mathbf{A}}^k\times {\mathbf{G}}_m^{n-k}\), \(m_i = 0\)
for \(i > k\). This happens iff \(v\in \sigma\), and the limit is
\({\left[ {\delta_1,\cdots, \delta_n} \right]}\) 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
\(\sigma\).
\item
Summary: \(v\in {\left\lvert {\Sigma} \right\rvert}\) and
\(v\in \tau^\circ\) then \(\lim_{z\to 0} \lambda_v(z) = x_\tau\), and
the limit does not exist for
\(v\not\in{\left\lvert {\Sigma} \right\rvert}\).
\end{itemize}
\end{remark}
\hypertarget{section-3}{%
\subsection{2.4}\label{section-3}}
\begin{remark}
\begin{itemize}
\item
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.
\item
\(X_\Sigma\) is compact iff
\({\left\lvert {\Sigma} \right\rvert} = N_{\mathbf{R}}\),
i.e.~\(\Sigma\) is complete.
\item
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
\(\phi^{-1}({\left\lvert {\Sigma'} \right\rvert}) = {\left\lvert {\Sigma} \right\rvert}\).
Thus \(X_\Sigma\) is compact iff \(\phi: N\to 0\) is a proper
morphism.
\item
Blowing up at \(x_\sigma\): take a basis \(\left\{{v_i}\right\}\), set
\(v_0\coloneqq\sum v_i\), and replace \(\sigma\) by all subsets of
\(\left\{{v_0,v_1,\cdots, v_n}\right\}\) not containing
\(\left\{{v_1, \cdots, v_n}\right\}\).
\end{itemize}
\end{remark}
\newpage
\printbibliography[title=Bibliography]
\end{document}