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\begin{example*}
  \[ y^2 = (x - \lambda_1)(\lambda_2)(\lambda_3) \]
  Take $P_0 = (0 : 1 : 0)$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/c10f3aee69ff48d4.png}
  \end{center}
\end{example*}

\begin{example*}
  Let $C$ have \gls{genus} $2$. Then $\deg \KC =
  2g - 2 = 2$, $l(\KC) = 2$.

  \textbf{Claim:} $\dbar|\KC|$ is \gls{bpfree},
  hence induces a \gls{gmorph} $f : C \to \P^1$.

  \begin{lemma*}
    Let $C$ be a projective non-\gls{sing}
    \gls{curve}. If there exist $P, Q \in C$, $P
    \neq Q$, $P \sim Q$, then $C \cong \P^1$.
  \end{lemma*}

  \begin{proof}
    Consider the linear system $\dbar|P|$. Since
    $Q \in \dbar|P|$, $\dim\dbar|P| \ge 1$, so
    $l(P) \ge 2$. But we have an upper bound
    $\dim\LD(D) \le \deg D + 1 \le 2$. Thus $l(P)
    = 2$. If $Q, R \in C$ then $\dim\LD(P - Q - R)
    = 0$ since $\deg(P - Q - R) = 1$. Thus
    $\dbar|P|$ induces an embedding of $C$ into
    $\P^1$. So $C \cong \P^1$.
  \end{proof}

  \begin{proof}[Proof of Claim]
    If $\dbar|\KC|$ is not \gls{bpfree}, then
    there exists $P \in C$ such that $l(\KC - P) =
    l(\KC) = 2$. Since $\deg \KC - P = 1$, this
    means there exists $Q, R \in \dbar|\KC - P|$,
    $Q \neq R$, with $Q \sim R$. Hence $C \cong
    \P^1$, contradiction, since $\P^1$ has
    \gls{genus} $0$.
  \end{proof}

  Thus if $g = 2$, we obtain a degree $2$
  \gls{gmorph} $f : C \to \P^1$ induces by
  $\dbar|\KC|$.
\end{example*}

\begin{flashcard}[hyper-elliptic-defn]
\begin{definition*}[Hyperelliptic]
  \glsadjdefn{hypell}{hyperelliptic}{\gls{curve}}
  \cloze{A projective non-\gls{sing} \gls{curve} $C$ is
  \emph{hyperelliptic} if there exists a degree
  $2$ \gls{gmorph} $f : C \to \P^1$.}
\end{definition*}
\end{flashcard}

\vspace{-1em}
Thus all \gls{genus} 2 \glspl{curve} are
\gls{hypell}.

\begin{theorem*}
  Let $C$ be a projective non-\gls{sing}
  \gls{curve} of \gls{genus} $g \ge 3$. Then
  either:
  \begin{enumerate}[(1)]
    \item $C$ is \gls{hypell}, or
    \item $\dbar|\KC|$ induces an embedding $C
      \hookrightarrow \P^{g - 1}$.
  \end{enumerate}
\end{theorem*}

\begin{proof}
  $\dbar|\KC|$ induces an embedding in $\P^{l(\KC)
  - 1} = \P^{g - 1}$ if and only if $\forall P, Q
  \in C$,
  \[ l(\KC - P - Q) = l(\KC) - 2 = g - 2 .\]
  In any event,
  \[ l(P + Q) - l(\KC - P - Q) = \deg(P + Q) + 1 -
  g = 3 - g .\]
  Thus $\dbar|\KC|$ induces an embedding if and
  only if $l(P + Q) = 1$ for all $P, Q \in C$. Now
  suppose $\dbar|\KC|$ does not induce an
  embedding. Then there exist $P, Q \in C$ such
  that $l(P + Q) > 1$. If $l(P + Q) \ge 3$, then
  for $R \in C$, $l(P + Q - R) \ge 2$. So there
  exists $P_1, P_2 \in \dbar|P + Q - R|$ distinct.
  Thus $C \cong \P^1$ by the lemma, a
  contradiction. Thus $l(P + Q) = 2$. Note
  similarly $l(P + Q - R) = 1$ for all $R \in C$.
  Thus $\dbar|P + Q|$ is \gls{bpfree} and induces
  a degree $2$ \gls{gmorph} $f : C \to \P^1$. So
  $C$ is \gls{hypell}.
\end{proof}

\begin{theorem*}[Riemann-Hurwitz formula]
  Let $f : X \to Y$ be a non-constant \gls{gmorph}
  between projective non-\gls{sing} \glspl{curve},
  with $\characteristic \KK = 0$ (or $\K(Y)
  \subseteq \K(X)$ is a separable field
  extension). Then
  \[ 2 - 2g(X) = (\deg f)(2 - 2g(Y)) - \sum_{p \in
  X} (e_p - 1) .\]
  ($e_p = \nu_p(t \cdot f)$ where $t$ is a local
  parameter at $f(p)$).
\end{theorem*}

\begin{proof}
  Omitted.
\end{proof}

\begin{example*}
  $X = C$ \gls{hypell}, $Y = \P^1$, $Y = \P^1$, $f
  : C \to \P^1$ degree 2. Then
  \[ 2 - 2g(C) = \ub{2 \cdot (2 - 2 \cdot 0)}_{4}
  - \sum_{p \in C} (e_p - 1) .\]
  Thus the number number of points $p \in C$ with
  $e_p > 1$ is $\sum_p (e_p - 1) = 2g(C) + 2$, $\deg g =
  \sum_{p \in f^{-1}(q)} e_p$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/98e2dbb8d2f7489e.png}
  \end{center}
\end{example*}
