%\documentclass{amsart}
\documentclass[11pt]{amsart}
\usepackage[hmargin=1 in, vmargin = 1 in]{geometry}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage[normalem]{ulem}
\parindent=0 in
\parskip=0.05 in



\newcommand{\isom}{\cong} %The isomorphism symbol
\newcommand{\union}{\cup}
\newcommand{\intersection}{\cap}
\newcommand{\bigunion}{\bigcup}
\newcommand{\bigintersection}{\bigcap}
\newcommand{\disjointunion}{\sqcup}
\newcommand{\bigdisjointunion}{\bigsqcup}

\newcommand{\join}{\vee}
\newcommand{\meet}{\wedge}
\newcommand{\bigjoin}{\bigvee}
\newcommand{\bigmeet}{\bigwedge}


%This makes alternating tensors look right in displayed equations
\newcommand{\Alt}{\bigwedge\nolimits}


%Blackboard bold letters.
\renewcommand{\AA}{\mathbb{A}}
\newcommand{\BB}{\mathbb{B}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\DD}{\mathbb{D}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\GG}{\mathbb{G}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\II}{\mathbb{I}}
\newcommand{\JJ}{\mathbb{J}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\LL}{\mathbb{L}}
\newcommand{\MM}{\mathbb{M}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\renewcommand{\SS}{\mathbb{S}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\UU}{\mathbb{U}}
\newcommand{\VV}{\mathbb{V}}
\newcommand{\WW}{\mathbb{W}}
\newcommand{\XX}{\mathbb{X}}
\newcommand{\YY}{\mathbb{Y}}
\newcommand{\ZZ}{\mathbb{Z}}

%Calligraphic letters

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}}
\newcommand{\GL}{\text{GL}}
\newcommand{\Mat}{\text{Mat}}

\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator{\CoKer}{CoKer}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Hom}{Hom}


\theoremstyle{definition}
\newtheorem{Problem}{Problem}
\newtheorem{problem}{Problem}





\newcommand{\into}{\hookrightarrow}
\newcommand{\onto}{\twoheadrightarrow} 
\newcommand{\from}{\leftarrow}

\newenvironment{sbm}
    {\left[ \begin{smallmatrix}
    }
    { 
     \end{smallmatrix} \right]
    }
    
\newcommand{\newword}[1]{\textbf{\emph{#1}}}

     
\pagestyle{empty}
\begin{document}

\thispagestyle{empty}
\centerline{\sc Problem Set Ten: Due Thursday, April 18.}
\centerline{\sc Last problem set! Due on the last day of class!}

See the course website for policy on collaboration. 

\textbf{Textbook problems}: \\
\textbf{18.2.H, 18.2 I}: These justify the assertion from class that we can often base change to $\overline{k}$ before thinking about projective embeddings. (Problem 19.2.E is another problem with this theme but I'm not assigning it because it relies on too many earlier problems that I didn't assign.)\\
\textbf{19.2.A}: We'll use this in Problem~\ref{canonical} below.\\
\textbf{19.8.C} This exercise has you think about curves of genus five, in the same way that I plan to cover curves of genus $4$.\\
%\textbf{19.9.E} We don't have to restrict ourselves to canonical embeddings; this exercise studies a genus $1$ curve with respect to a degree $4$ line bundle.\\

\begin{Problem} \label{canonical}
Here are three problems where we can describe the canonical sheaf of a curve directly, without mentioning differentials, by using Problem 19.2.A.
\begin{enumerate}
\item Let $X$ be a smooth projective irreducible curve of genus $g$ and let $\pi : X \to \PP^1$ be a degree $2$ map. Show that $\pi^{\ast}(\cO(g-1)) \cong \omega_X$.
\item Let $Y$ be a smooth projective irreducible degree $d$ curve in $\PP^2$; we write $\iota$ for the inclusion $Y \into \PP^2$. Show that $\iota^{\ast} \cO(d-3) \cong \omega_C$. (Hint: As we so often do, start with $0 \to \cO(-d) \to \cO \to \iota^{\ast} \cO \to 0$. )
\item  Let $f$ and $g$ be homogeneous polynomials of degrees $a$ and $b$ in $k[x_0, x_1, x_2, x_3]$, and let $Z = V_+(f,g)$; suppose that $Z$ is smooth and irreducible. In , Problem 3, Problem Set 9, you hopefully computed that $Z$ has genus $(a^2 b + a b^2 - 4 ab + 2)/2$. Show that $\iota^{\ast} \cO(a+b-4) \cong \omega_Z$.
\end{enumerate}
\end{Problem}
%
%\begin{Problem}
%We return to hyperelliptic curves. Let $k$ be a field, let $C$ be a smooth, projective, irreducible curve of genus $g$ and let $\pi : C \to \PP^1$ be a degree $2$ map. At the start of this problem, we do \textbf{not} assume that $C$ is ``the" hyperelliptic curve that we have met in the past.
%
%Let $\infty$ be the point $[1:0]$ of $\PP^1$; let $U_0 = \pi^{-1}(\PP^1 \setminus \infty)$. Let $x_0$ be the standard coordinate on $\PP^1 \setminus \infty \cong \AA^1$ and abuse notation by also writing $x_0$ to mean $\pi^{\ast} x_0$. Let $D_{\infty}$ be the divisor $\pi^{-1}([\infty])$ considered with multiplicity, so $\deg D_{\infty} = 2$.
%\begin{enumerate}
%\item Show that we can choose a basis for $H^0(C, \cO((g+1) D_{\infty}))$ of the form $\{ 1, x_0, x_0^2, \ldots, x_0^{g+1}, y_0 \}$. 
%\item Show that there are coefficients $a_0$, $a_1$, \dots, $a_{g+1}$, $b_0$, $b_1$, \dots, $b_{2g+2}$ in $k$ such that 
%\[ y_0^2 + \sum_{i=0}^{g+1} a_i x_0^i y + \sum_{j=0}^{2g+2} b_j x_0^j . \]
%\item Show that $U_0$ is the affine scheme with coordinate ring $k[x_0, y_0] / \left\langle  y_0^2 + \sum_{i=0}^{g+1} a_i x_0^i y + \sum_{j=0}^{2g+2} b_j x_0^j  \right\rangle$.
%\item Define $x_{\infty} = x_0^{-1}$ and define $D_0$ and $U_{\infty}$ as above. Put $y_{\infty} = x_0^{-g-1} y_0$. Give a similar description of the coordinate ring of $U_{\infty}$ as $k[x_{\infty}, y_{\infty}]$ 
%\end{enumerate}
%We have shown that $C$ can be covered by two open charts with equations of a certain form, and we have explained how to glue them. We now show how to reduce to the simpler form we saw in earlier problem sets.
%\begin{enumerate}[resume]
%\item Suppose that $k$ does not have characteristic $2$. Show that, by making an appropriate choice of $y_0$, we can arrange that $a_0 = a_1 = \cdots = a_{g+1} = 0$.
%\item Suppose that $k = \overline{k}$. Show that, by making a change of coordinates on $\PP^1$, we can arrange that $b_0 = b_{2g+2} = 0$ and that $D_0$ and $D_{\infty}$ are of the form $2 [\mathfrak{p}_0]$ and $2 [ \mathfrak{p}_{\infty}]$. Hint: Use the fact that the polynomial $\sum b_j x^j=0$ factors in $k$. 
%\end{enumerate}
%\end{Problem}

\end{document}


