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Let $C$ be a projective non-\gls{sing} curve.

\begin{flashcard}[div-zero-pole-defn]
\begin{definition*}[Divisor of zeroes and poles]
  \glssymboldefn{divzp}{$(f)$}{$(f)$}
  \glsnoundefn{divzp}{divisor of zeroes and
  poles}{N/A}
  \cloze{For $f \in \K(C) \setminus \{0\}$, the
  \emph{divisor of zeroes and poles} of $f$ is
  \[ (f) = \sum_{p \in C} \nuf_p(f) \cdot p .\]}
\end{definition*}
\end{flashcard}

\begin{remark*}
  Note $f$ is represented on some open subset $U
  \subseteq C$ by $\frac{g}{h}$, $g, h$
  homogeneous polynomials. We shrink $U$ by
  removing $\Z(g), \Z(h)$. Now, if $p \in U$, $f =
  \frac{g}{h} \in \OXp_{C, p}$ is a regular
  function with $f(p) \neq 0$, so $\nuf_p(f) = 0$.
  Thus the sum defining $\divzp(f)$ is a sum over points
  of $C \setminus U$, which is a finite set.

  (Here we use $\dim C = 1$, so that irreducible
  sets are $C$ and singleton sets).
\end{remark*}

\begin{flashcard}[group-prin-div-defn]
\begin{definition*}[Group of principal divisors]
  \glsnoundefn{gppdiv}{group of principal
  divisors}{N/A}
  \cloze{The group of \emph{principal divisors} on $C$ is
  \[ \Prin C = \{\divzp(f) \mid f \in \K(C) \setminus
  \{0\}\} .\]}
\end{definition*}
\end{flashcard}

\vspace{-1em}
This is a subgroup since:
\begin{itemize}
  \item $\divzp(f \cdot g) = \divzp(f) + \divzp(g)$
  \item $\divzp(f^{-1}) = -\divzp(f)$.
\end{itemize}

\begin{flashcard}[div-class-gp-defn]
\begin{definition*}[Divisor class group]
  \cloze{The \emph{(divisor) class group} is
  \[ \Cl C \defeq \frac{\Div C}{\Prin C} .\]}
\end{definition*}
\end{flashcard}

\begin{flashcard}[linearly-equiv-defn]
\begin{definition*}[Linearly equivalent]
  \glsadjdefn{linequiv}{linearly
  equivalent}{linearly equivalent}
  \cloze{If $D, D' \in \Div C$ satisfy $D - D' =
    \divzp(f)$ for
  some $f \in \K(C)^\times$, then we say $D$ is
  \emph{linearly equivalent} to $D'$, and write
  \[ D \sim D' .\]}
\end{definition*}
\end{flashcard}

\vspace{-1em}
\textbf{Digression:} Extending \glspl{gmorph} to
projective space. $C$ a projective non-\gls{sing}
curve, $\emptyset \neq U \subseteq C$ an open
subset. $f_0, \ldots, f_n$ \gls{gregf} functions on $U$
without a common zero.

Then we obtian a \gls{gmorph}
\begin{align*}
  f : U
  &\to \P^n \\
  p
  &\mapsto (f_0(p) : \cdots : f_n(p))
\end{align*}

\begin{flashcard}[curve-morph-extension-thm]
\begin{theorem*}
  $f : U \to \P^n$ extends to a \gls{gmorph} $f :
  C \to \P^n$.
\end{theorem*}

\begin{proof}
  \cloze{Suppose either $f_i$ has a pole at $p \in C$
  (i.e. $\nuf_p(f_i) < 0$) or all $f_i$ are zero
  at $p$. Let
  \[ m = \min\{\nuf_p(f_i) \st 0 \le i \le n\} .\]
  Let $t$ be a local parameter at $p$, i.e. a
  generator of $m_p \subseteq \OXp_{C, p}$. So
  $\nuf_p(t) = 1$. Then $\nuf_p(t^{-m} f_i) =
  \nuf_p(f_i) - m$. Thus $\nuf_p(t^{-m} f_i) = 0$
  for some $i$ and $\nuf_p(t^{-m} f_j) \ge 0$.
  Thus $t^{-m}  f_0, \ldots, t^{-m} f_n \in
  \OXp_{C, p}$, and these functions don't
  simultaneously vanish at $p$. Hence in some
  neighbourhood $V$ of $p$, we obtain a
  \gls{gmorph} $f_p : V \to \P^n$ given by $q
  \mapsto ((t^{-m} f_0)(p) : \cdots : (t^{-m}
  f_n)(p))$. This agrees with $f$ on $U \cap V$ by
  rescaling.}
\end{proof}
\end{flashcard}

\begin{flashcard}[non-const-morph-curve-facts]
\begin{proposition*}
  Let $f : X \to Y$ be a non-constant \gls{gmorph}
  between projective non-\gls{sing} \glspl{curve}.
  Then
  \begin{enumerate}[(1)]
    \item \cloze{$f^{-1}(q)$ is a finite set for all $q
      \in Y$}
    \item \cloze{$f$ induces an inclusion $\K(Y)
      \hookrightarrow \K(X)$ such that $[\K(X) :
      \K(Y)]$ is finite. We call $[\K(X) : \K(Y)]$
      the \emph{degree} of $f$.}
  \end{enumerate}
\end{proposition*}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(1)]
    \item \cloze{$f^{-1}(q) \subseteq X$ is closed, and
      since $\dim X = 1$, either $f^{-1}(q)$ is
      finite, or $f^{-1}(q) = X$. The latter
      contradicts $f$ being non-constant.}
    \item \cloze{If $\varphi \in \K(Y)$, then $\varphi$
      defines a \gls{gregf} function on some open
      $U \subseteq Y$. $\varphi : U \to \KK$.
      $\varphi \circ f$ makes sense provided $f(X)
      \not\subseteq Y \setminus U$. But $f(X)$ is
      irreducible (point set topology exercise),
      so $f$ is constant if $f(X) \subseteq Y
      \setminus U$. Thus $\varphi \circ f$ makes
      sense as a rational function on $X$. Thus
      $\K(Y) \to \K(X)$ exists and is
      automatically an injection since both are
      fields. Omit proof of finiteness.} \qedhere
  \end{enumerate}
\end{proof}
\end{flashcard}

\begin{flashcard}[deg-ram-defn]
\begin{definition*}[Degree of ramification]
  \glsnoundefn{degram}{degree of ramification}{N/A}
  \cloze{Suppose $f : X \to Y$ is a non-constant
  \gls{gmorph} between projective non-\gls{sing}
  \glspl{curve}. Let $p \in Y$, $m_p = (t)
  \subseteq \OXp_{Y, p}$, $t$ a local parameter.
  Let $q \in f^{-1}(p)$. Then $t \circ f \in
  \OXp_{X, q}$. Define
  \[ e_q \defeq \nu_q(t \circ f) ,\]
  the \emph{degree of ramification} of $f$ at
  $q$.}
\end{definition*}
\end{flashcard}

\begin{flashcard}[degram-sum-degf-thm]
\begin{theorem*}
  Let $f : X \to Y$ a non-constant \gls{gmorph}
  between projective non-\gls{sing} \glspl{curve}.
  Then for $p \in Y$,
  \[ \cloze{\sum_{q \in f^{-1}(p)} e_q = \deg f} \]
  \cloze{is the degree of $f$.}
\end{theorem*}

\begin{proof}
  \cloze{Omitted, but the theorem statement is
  crucial.}
\end{proof}
\end{flashcard}

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $\characteristic \KK \neq 2$, $f : \P^1 \to \P^1$,
      $(u, v) \mapsto (u^2 : v^2)$. Setting $v =
      1$, this gives a \gls{gmorph} $\AA^1 \to
      \AA^1$ given by $u \mapsto u^2$. If $p \in
      \AA^1$, $t = u - p$ is a local parameter at
      $p$. $t \circ f = u^2 - p = (u - q)(u + q)$
      where $q^2 = p$. Then $e_q = e_{-q} = 1$.
      We have $\deg f = e_q + e_{-q} = 2$.
    \item If $p = 0$, $f^{-1}(p) = \{0\}$, $e_0 =
      \nu_0(u^2) = 2$. Function fields, $\K(\P^1)
      = \KK(u)$, $\KK(u) \to \KK(y)$, $u \mapsto
      u^2$ degree $2$.
    \item $\characteristic \KK = p$, $f : \P^1 \to
      \P^1$, $(u : v) \mapsto (u^p : v^p)$. Set $v
      = 1$, $u \mapsto u^p$. $f^{-1}(q) = \{r\}$
      with $r^p = q$, $q \in \AA^1$. Then $t = u -
      q$. $t \circ f = u^p - q = (u - r)^p$.
  \end{enumerate}
\end{example*}

\vspace{-1em}
\textbf{Application:} Let $X$ be a projective
non-\gls{sing} \gls{curve}, $f \in \K(X)^\times$.
This gives a \gls{gmorph} $U \to \P^1$ where $U$
is the open set in which $f$ is singular. This
extends to $f : C \to \P^1$, non-constant as long
as $f \notin \KK$.