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\textbf{18.782 Introduction to Arithmetic Geometry\hfill Spring~2023}\\\vspace{4pt}
\textbf{Problem Set \#8\hfill Due: 04/20/2023}
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These problems are related to the material covered in Lectures 24-27.
I have made every effort to proof-read them, but some errors may remain.
The first person to spot each error will receive 1-5 points of extra credit.

The problem set is due at 11:59pm Eastern time on 04/20/2023. It is to be submitted electronically as a pdf-file through \href{https://www.gradescope.com/courses/510542/}{gradescope}.
You can use the latex source for this problem set as a template for writing up your solutions;
be sure to include your name in your solutions, and identify collaborators and any sources not listed in the syllabus.

Recall that we have defined a \emph{curve} as a smooth projective variety of dimension one.

\subsection*{Problem 1. Singularities (20 points)}

Let $X$ be the projective closure of the affine curve $y^2 = x^5$ over an algebraically closed field of characteristic $0$.
\begin{enumerate}[(a)]
\item Find the singularities of $X$.
\item Find a smooth projective curve $Y$ that is birational to $X$.
\end{enumerate}

\subsection*{Problem 2. Smooth projective model of hyperelliptic curve (30 points)}

Let $k$ be an algebraically closed field and $f(x) = \sum a_i x^i$ be a squarefree polynomial of degree $5$ or $6$ in $k[x]$. Let $C$ be the affine curve defined by the equation $y^2 = f(x)$, with function field $F = k(x,\sqrt{f(x)})$. In this exercise, we want to show that $X \subseteq \P^4$ defined by
\begin{align*}
    & Y^2 = a_0 X_0^2 + a_1 X_0X_1 + a_2 X_1^2 + a_3 X_1X_2 + a_4 X_2^2 + a_5 X_2X_3 + a_6 X_3^2\\
    & X_0X_2 - X_1^2 = 0\\
    & X_1X_3 - X_2^2 = 0\\
    & X_0X_3 - X_1X_2 = 0
\end{align*}
is a smooth projective curve with function field isomorphic to $F$.
\begin{enumerate}[(a)]
    \item Show that the affine curve $C$ is isomorphic to the affine part of $X$ in the affine chart given by $X_0 = 1$, thus getting the assertion about function field of $X$.
    \item Show all points of $C$ are smooth, and deduce the same about points of $X$ in this affine chart.
    \item Find all points of $X$ not in this affine chart. By choosing another suitable affine chart, and working out the Jacobian matrix, show that these points are smooth as well.
\end{enumerate}

\subsection*{Problem 3. Divisor identities (50 points)}
Let $k$ be a perfect (but not necessarily algebraically closed) field.
Let $\phi\colon C_1\to C_2$ be a morphism of curves defined over $k$, and let $\phi^*\colon k(C_2)\to k(C_1)$ be the corresponding embedding of function fields.
We also use $\phi^*$ to denote the pullback map $\Div_k C_2\to \Div_k C_1$, and $\phi_*$ denotes the pushforward map $\Div_k C_1\to \Div_k C_2$, as defined in Definition 19.19.


\begin{enumerate}
\item[\textbf{(a)}] Prove that $\phi_*\circ\phi^*$ acts as multiplication by $\deg \phi$ on $\Div_k C_2$.
\item[\textbf{(b)}] Show that $\phi^*\circ\phi_*$ is not multiplication by $\deg\phi$ on $\Div_k C_1$, in general, but that we do have
\[
\deg(\phi^*\phi_* D) = (\deg\phi)(\deg D)
\]
for all $D\in \Div_k C_1$.
\item[\textbf{(c)}] Prove that $\ord_P(\phi^*f) = e_\phi(P)\ord_{\phi(P)}(f)$ for all closed points $P$ of $C_1/k$ and functions $f$ in $ k(C_2)^\times$.  Then use this to show $\phi^*(\div f)=\div(\phi^* f)$ for all $f\in k(C_2)^\times$.
\item[\textbf{(d)}] For any curve $C/k$ and $f\in k(C)^\times$ and $D\in \Div_k(C)$ such that $\div f$ and $D=\sum n_P P$ have disjoint supports, define
\[
f(D) = \prod f(P)^{n_P\deg P},
\]
where $P$ ranges over the closed points of $C/k$.
Prove that $f(\phi_* D)=(\phi^*f)(D)$ for all $f\in k(C_2)^\times$ and $D\in \Div_k(C_1)$ where both sides are defined.
\item[\textbf{(e)}] Prove that the sequence
\[
1\to k^\times\to k(\P^1)^\times\overset{\div}{\longrightarrow} \Div_k\P^1\overset{\deg}{\longrightarrow}\Z\to 0
\]
is exact.
\end{enumerate}

\subsection*{Problem 4. Survey}
Complete the following survey by rating each problem on a scale of 1 to~10 according to how interesting you found the problem (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found the problem (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
\end{tabular}
\end{center}

\noindent
Feel free to record any additional comments you have on the problem sets or lectures; in particular, how you think they might be improved.

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