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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
  \(r&gt;0\), 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 &amp;= \left\langle{x_1, x_2,\cdots, x_n}\right\rangle\\
    k[x_1, \cdots, x_{n}]_2 &amp;= \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 &amp;\to \left\{{0, 1}\right\} \\
  p &amp;\mapsto
  \begin{cases}
  0 &amp; f(p) = 0 
  \\
  1 &amp; 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)} &amp;&amp;&amp; {\mathsf{Ab}}{\mathsf{Grp}}\\
    U &amp;&amp; {F(U)} &amp;&amp; {G(U)} \\
    \\
    V &amp;&amp; {F(V)} &amp;&amp; {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 &lt;=15pt, shorten &gt;=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 Ã©talÃ©}: define
  \(\text{Ã‰t}(F) = {\textstyle\coprod}_{p\in X} F_p\) and a projection
  \(\pi: \text{Ã‰t}(F) \to X\) by sending \(s\in F_p\) to \(p\). For each
  \(U \subseteq X\) and \(s\in F(U)\), there is a local section
  \(\overline{s}: U\to \text{Ã‰t}(F)\) where \(p\mapsto s_p\), its germ
  at \(p\); this satisfies
  \(\pi \circ \overline{s} = \operatorname{id}_U\). Give
  \(\text{Ã‰t}(F)\) the strongest topology such that the \(\overline{s}\)
  are all continuous. Then
  \(F^+(U) \coloneqq{\mathsf{Top}}(U, \text{Ã‰t}(F))\) is the set of
  continuous sections of \(\text{Ã‰t}(F)\) 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 &amp; p\in U 
  \\
  0 &amp; \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 &amp; q\in { \operatorname{cl}}_X(\left\{{p}\right\}) 
    \\
    0 &amp; \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 &amp; p\in  Z
  \\
  0 &amp; \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 &amp; p\in U 
    \\
    0 &amp; \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=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}{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
  \(h^0({\mathcal{L}}(D)) &gt;0 \implies \deg(D) \geq 0\), 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) &amp;= h^0({\mathcal{L}}(D)) - h^1({\mathcal{L}}(D)) \\ 
  &amp;= h^0({\mathcal{L}}(D)) - h^0({\mathcal{L}}(K-D)) \\
  &amp;= \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}
  \(\ell(K-D) &gt; 0\), 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 \(\deg D &gt; 2g-2\) 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 &amp; { \, \xrightarrow{\sim}\, }\operatorname{Pic}(X) \\
x &amp;\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
    \(e_p &gt; 1 \implies\) 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}: \(e_p &gt; 1\), 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) &amp;\to \operatorname{Div}(X) \\
  Q &amp;\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 Ã©tale
  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 &amp; f \text{ is tamely ramified at } P 
  \\
   &gt; e_p -1 &amp; 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 ) &amp;= \deg(f^* K_Y + R) \\
\implies \chi_{\mathsf{Top}}(X) &amp;= \deg(f^* K_Y) + \deg(R) \\
\implies 2g(X) - 2 &amp;= \deg(f) \deg(K_Y) + \deg(R) \\
\implies 2g(X) - 2 &amp;= \deg(f) \chi_{\mathsf{Top}}(Y) + \deg(R) \\
\implies 2g(X) - 2 &amp;= \deg(f) (2g(Y) - 2) + \deg(R) \\
\implies 2g(X) - 2 &amp;= \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{Ã‰tale covers}: \(X \xrightarrow{f} Y\) is an Ã©tale cover if
  \(f\) is a finite Ã©tale 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 Ã©tale
  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 Ã©tale, then \(f\) must be separable.
\item
  Show that \(\pi_1^\text{Ã©t}({\mathbf{P}}^1) = 0\).
\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}[LÃ¼roth]

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 \(\deg D &gt; 0\).
  \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 \(\deg D &gt; 0\).
\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 \(\deg D &gt; 0\)
\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} }
    &amp;\rightleftharpoons
    {\mathbf{A}}^1_{/ {k}} \\
    E &amp;\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 &amp; j(E) \neq 0,1728\\
4 &amp; j(E) = 1728, \operatorname{ch}k \neq 3 \\
6 &amp; j(E) = 0,\operatorname{ch}k \neq 3 \\ 
12 &amp; 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} }  } &amp;\to \operatorname{Jac}(X) \\
  (p_1,\cdots, p_n) &amp;\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 \(\iff \sum a_i &gt; 0\).
  \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 \(a,b &gt; 0\).
  \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&lt; \cdots &lt; 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 &amp; \dim \tau &lt; n 
    \\
    1 &amp; \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 &amp; { \, \xrightarrow{\sim}\, }{\mathsf{Grp}}(T, {{\mathbf{C}}^{\times}}) \\
  m = {\left[ {m_1,\cdots, m_n} \right]} &amp;\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 &amp;\mapsto {\mathsf{Grp}}({{\mathbf{C}}^{\times}}, T) \\
  u ={\left[ {u_1,\cdots, u_n} \right]} &amp;\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 &amp;\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}}&amp;\to T \\
    u\otimes t &amp;\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 }&amp;\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 &amp;\coloneqq\left\{{u\in N_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} = 0}\right\} \subseteq N_{\mathbf{R}}\\
  H_m^+ &amp;\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 &amp;\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\} &amp;\rightleftharpoons\left\{{T{\hbox{-}}\text{orbits in } X_\Sigma}\right\} \\
  \sigma &amp;\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 &amp;\to  \operatorname{Cl}(X) \twoheadrightarrow \operatorname{Cl}(X\setminus\cup D_i) \\
  e_1 &amp;\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 &amp;\to \operatorname{Div}_T(X) \twoheadrightarrow \operatorname{Cl}(X) \\
  m &amp;\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} &amp;\coloneqq\left\{{m\in M_{\mathbf{R}}{~\mathrel{\Big\vert}~}{\left\langle {m},~{u} \right\rangle} = b}\right\} \\
  H_{u, b}^+ &amp;\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 &amp;\to {\mathbf{P}}^{s-1} \\
  t &amp;\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
  \(a,b&lt; 0\) 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
  \(\operatorname{det}{ \begin{bmatrix}  {-1} &amp; {a} \\  {b} &amp; {-1} \end{bmatrix} } = 1-ab = \pm 1 \implies ab=0,2\),
  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 PoincarÃ© 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}:
  \(V\qty{2\times 2\text{ minors of } \left[\begin{array}{lll} x_0 &amp; x_1 &amp; y_0 \\ x_1 &amp; x_2 &amp; y_1 \end{array}\right]}\)
  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 &amp;\to {\mathbf{P}}^2 \\
  (s,t) &amp;\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 &amp;\to {\mathbf{P}}^5 \\
    (s,t) &amp;\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 &amp;\hookrightarrow{\mathbf{P}}^n \\
  {\left[ {a_1,\cdots, a_n} \right]} &amp;\mapsto {\left[ {1: a_1 : \cdots : a_n} \right]}
  .\end{align*}
\item
  The torus action is
  \begin{align*}
  ({{\mathbf{C}}^{\times}})^n &amp;\curvearrowright{\mathbf{P}}^n \\
  {\left[ {t_1,\cdots, t_n} \right]} . {\left[ {x_0: x_1:\cdots:x_n} \right]} &amp;= {\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
  \({\left\langle {u},~{v} \right\rangle} &lt; 0\). 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 \(i &gt; k\). 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 \(\delta_i = 0\iff m_i &gt; 0\); 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}



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\printbibliography[title=Bibliography]


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