%! TEX root = AG.tex
% vim: tw=50
% 06/03/2024 12PM

\begin{remark*}
  This is known as B\'ezout's Theorem. This is
  usually expressed as follows:

  Let $C, C' \subseteq \P^2$ be \glspl{curve} of
  degrees $d$ and $e$ respectively. Then the
  number of points in $C \cap C'$ (assuming $C
  \neq C'$) ``counted with multiplicities'' is $d
  \cdot e$.

  For example, if $C$ is non-\gls{sing}, $f : C
  \hookrightarrow \P^2$ an embedding, then $d =
  \deg f$ and $\deg \fstar f C' = d \cdot e$. So
  if $p \in C \cap C'$, its multiplicity is the
  coefficient of $p$ in $\fstar f C'$. If $C$ is
  \gls{sing}, need a more subtle definition of
  multiplicity.
\end{remark*}

\vspace{-1em}
In general, given a \gls{div} $D$ on a projective
non-\gls{sing} \gls{curve} $C$, we would like to
understand when $\dbar|D|$ induces an embedding
$C$ in projective space.

In other words, suppose $\dbar|D|$ is
\gls{bpfree}, i.e. $\forall p \in C$, there exists
$D' \in \dbar|D|$ with $p \notin \Supp D'$. Then
by choosing $f_0, \ldots, f_n \in \LD(D)$ spanning
$\LD(D)$, we obtain a \gls{gmorph} $f = (f_ :
\cdots : f_n) : C \to \P^n$. When is this an
embedding? We can alsu use a sub-linear system
$\mathcal{D} = \P(V) \subseteq \dbar|D| =
\P(\LD(D))$ and choose $f_0, \ldots, f_n \in V$ a
spanning set.

\begin{flashcard}[induced-morph-embedding-iff-thm]
\begin{theorem*}
  Suppose a linear system $\mathcal{D} \subseteq
  \dbar|D|$ is \gls{bpfree}. Then the induced
  \gls{gmorph} $f : C \to \P^n$ is an embedding if
  and only if
  \begin{enumerate}[(1)]
    \item \cloze{$\mathcal{D}$ separates points: i.e.
      $\forall P, Q \in C$ distinct, there exists
      a $D' \in \mathcal{D}$ such that $P \in
      \Supp D'$ and $Q \notin \Supp D'$. (This is
      equivalent to injectivity of $f$).}
    \item \cloze{$\mathcal{D}$ separates vectors: i.e.
      $\forall P \in C$, $\exists D' \in
      \mathcal{D}$ such that the coefficient of
      $P$ in $D'$ is $1$.}
  \end{enumerate}
\end{theorem*}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/fc7b6a3f146e4002.png}
\end{center}
\end{flashcard}

\begin{flashcard}[very-ample-defn]
\begin{definition*}[Very ample divisor]
  \glsadjdefn{va}{very ample}{\gls{div}}
  \cloze{We say a \gls{div} $D$ is \emph{very ample} if
  $\dbar|D|$ induces an embedding into some
  projective space.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[very-ample-if-thm]
\begin{theorem*}
  $D$ is \gls{va} if \cloze{$\forall P, Q \in C$, not
  necessarily distinct, we have
  \[ \dim \dbar|D - P - Q| = \dim \dbar|D| - 2
  .\]}
\end{theorem*}
  
\begin{proof}
  \cloze{Recall $\dim\dbar|D| = \dim\LD(D) - 1$. For any
  $P \in C$, we have a map $\LD(D) \to \KK$. This
  is constructed as follows. Suppose the
  coefficient of $P$ in $D$ is $n$. Then if $f \in
  \LD(D)$, then $\nuf_p(t^n \cdot f) = n +
  \nuf_p(f) \ge 0$, where $t$ is a local
  parameter at $p$. So $t^n \cdot f \in \OXp_{C,
  p}$. Thus we define
  \begin{align*}
    ev_p : \LD(D)
    &\to \KK \\
    f
    &\mapsto (t^n \cdot f)(p)
  \end{align*}
  If $f \in \Ker(ev_p)$, we have $\nuf_p(t^n \cdot
  f) \ge 1$, so $\nuf_p(f) > -n$. Hence the
  coefficient of $p$ in $D = \divzp(f)$ is at
  least 1. Thus $(D - p) + \divzp(f)$ is
  \gls{eff}, so $f \in \LD(D - P)$. Conversely, if
  $f \in \LD(D - P)$, $(D - P) + \divzp(f)$ is
  \gls{eff}, so $\nuf_p(f) \ge -n + 1$, so
  $\nuf_p(t^n \cdot f) \ge 1$, so $f \in
  \Ker(ev_p)$. Thus $\LD(D - P) = \Ker ev_p$. If
  $\dbar|D|$ is \gls{bpfree}, then $ev_p : \LD(D)
  \to \KK$ is surjective $\forall p$ and
  conversely. So
  \[ \dim\dbar|D - P| = \dim \LD(D - P) - 1 = \dim
  \LD(D) - 2 = \dim\dbar|D| - 1 \]
  for all $p$ if and only if $\dbar|D|$ is
  \gls{bpfree}. Now $\dbar|D|$ separates points
  and tangent vectors if and only if $\dbar|D -
  P|$ is \gls{bpfree} $\forall p \in C$. Indeed,
  if $D' \in \dbar|D - P|$ does not have $Q$ in
  its support, then $D' + P$ separatest $P$ and
  $Q$ if $Q \neq P$. If $P = Q$, and $P \notin
  \Supp D'$, then $D' + P$ has coefficient 1 for
  $P$. Now
  \[ \dim \dbar|D - P - Q| = \dim \dbar|D - P| = 1 \]
  if and only if $\dbar|D - P|$ is \gls{bpfree} so
  $\dbar|D|$ is \gls{va} and \gls{bpfree} if and
  only if
  \[ \dim\dbar|D - P - Q| = \dim\dbar|D - P| - 1 =
  \dim\dbar|D| - 2 \qquad \forall P, Q . \qedhere
  \]}
\end{proof}
\end{flashcard}

\begin{moral*}
  If we can control $\dim\LD(D)$, then we know a
  lot about embeddings.
\end{moral*}

\newpage
\section{Differentials and the Riemann-Roch
Theorem}

\begin{flashcard}[Omega-B-slash-A-defn]
\begin{definition*}[$\Omega_{B / A}$]
  \glssymboldefn{rOmega}{$\Omega_{B /
  A}$}{$\Omega_{B / A}$}
  \cloze{Let $B$ be a ring and $A \subseteq B$ a subring.
  We define
  \[ \Omega_{B / A} = \frac{\text{free $B$-module
  generated by symbols $\dd b$ for $b \in
  B$}}{\text{submodule $R$ of relations}} \]
  where $R$ is the submodule with generators:
  \begin{align*}
    &\dd(bb') - b\dd b' - b'\dd b
    &&\forall b, b' \in B \\
    &\dd(b + b') - \dd b - \dd b'
    &&\forall b, b' \in B \\
    &\dd a
    &&\forall a \in A
  \end{align*}}
\end{definition*}
\end{flashcard}
