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%% \begin{frame}
%% \title[Calculus III]{Calculus III -- Recitation 1}
%% \author{Prof. Cueto}
%% \institute{The Ohio State University}
%% \date{January 14, 2016}
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%% \title[Calculus III]{Calculus III -- Recitation 1}
%% \author{Prof. Cueto - The Ohio State University}
%% \date{January 14, 2016}
%% \maketitle
\begin{center}
  \Large{\textsf{\textbf{MATH 8140 - Topics in Algebraic Geometry (Riemann surfaces)}}}\\
\vspace{1ex}
  \Large{\textsf{\textbf{Homework 6}}}\\
\vspace{2ex}

\Large{\textbf{\textsf{Divisors, the Riemann-Roch Theorem, linear systems and maps to projective spaces}}}\normalsize

\vspace{1ex}

\end{center}
\normalsize

Please indicate any source in the literature used in finding the solution to a given problem. You are encouraged to work in teams. If you do so, please indicate the name of your collaborator(s) on your submission.
\vspace{1ex}

Solutions to each problem can be uploaded (on Carmen) by at most one student. There is no deadline, so work at your own pace.
\vspace{1ex}

Please use the following name for the file you upload \texttt{HW\#\_Problem\#.pdf}.
\vspace{1ex}


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%% \noindent
%% {Topics: {the Tropical semiring, valuations, polyhedra, tropical hypersurfaces.}}\normalsize


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\begin{problem} Fix a compact Riemann surface $X$ and a divisor $D$ on it. If $\deg(D)<0$, show that the 0th \v{C}ech cohomology group $H^0(X, \mathcal{O}_D)$ vanishes.
\end{problem}
\vspace{2ex}

\begin{problem}  [Forster $\S 16.4$ (a)]
  Let $X$ be a compact Riemann surface and define the presheaf $\mathcal{D}$ of divisors on $X$ as follows. For each open $U\subset X$ we define $\mathcal{D}(U)$ as the maps $D\colon U\to \ZZ$ such that for every compact subset $K\subset U$ the Support of $D$ restricted to $K$ is finite. The restriction maps are defined in the natural way.
  \begin{enumerate}[(i)]
  \item Show that $\mathcal{D}$ defines a sheaf.
  \item Show that $H^1(X,\mathcal{D})=0$.
    (\emph{Hint:} Use a discontinuos integer value partition of unity and the techniques used to prove that $H^1(X,\mathcal{E})=0$, where $\mathcal{E}$ is the sheaf of differentiable functions on $X$.)
  \end{enumerate}
\end{problem}
  \vspace{2ex}
  \begin{problem}  [Forster $\S 16.4$ (b)] Fix a compact Riemann surface $X$ and the sheaf of divisors $\mathcal{D}$ on $X$ defined in Problem 2. We define a sequence of maps of sheaves:
    \begin{equation}\label{eq:ses} \xymatrix{0 \ar[r] & \mathcal{O}^* \ar[r]^{\alpha} & \mathcal{M}^*\ar[r]^{\beta} & \mathcal{D} \ar[r] & 0,}
    \end{equation}
    where $\alpha\colon \mathcal{O}^*\to \mathcal{M}^*$ is the natural inclusion map and $\beta\colon \mathcal{M}^* \to \mathcal{D}$ satisfies $\beta_U(f) = (f)\in \mathcal{D}(U)$ for each $f\in \mathcal{M}^*(U)$ and each open $U\subset X$.
    \begin{enumerate}[(i)]
    \item Show that~\eqref{eq:ses} is an exact sequence.
      \item Compute the connection map $\delta^*\colon Div(X) \to H^1(X,\mathcal{O}^*)$.
      \end{enumerate}
\end{problem}
  \vspace{2ex}

\begin{problem} [Forster $\S 16.1$]
  Fix a divisor $D$ on $\pr^1$. Show:
  \begin{enumerate}[(i)]
  \item $\dim H^0(\pr^1, \mathcal{O}_D)=\max\{0, 1+\deg(D)\}$,
    \item $\dim H^1(\pr^1, \mathcal{O}_D)=\max\{0, -1-\deg(D)\}$.
  \end{enumerate}
\end{problem}
\vspace{2ex}


\begin{problem} [Forster $\S 17.4$]
  Fix a compact Riemann surface $X$ of genus $g$ and a divisor $D$ on it. Show:
  \begin{enumerate}[(i)]
  \item $0\leq \dim H^0(X,\mathcal{O}_D) \leq 1+\deg(D)$\quad for $-1\leq \deg(D)\leq g-1$,
  \item $1-g+\deg(D) \leq \dim H^0(X,\mathcal{O}_D) \leq g$ \quad for $g-1\leq deg(D) \leq 2g-1$,
  \item $\dim H^0(X,\mathcal{O}_D) = 1-g +\deg(D)$ \quad if $\deg(D)\geq 2g-1$.
  \end{enumerate}
\end{problem}

  \vspace{2ex}

  \begin{problem} [Forster $\S 17.3$] Fix a compact Riemann surface $X$ of genus $g>0$. Consider the canonical divisor $K$ on $X$ and a divisor $D\geq K$ with $\deg(D)=\deg(K)+1$. Show that the sheaf $\mathcal{O}_K$ is globally generated by $\mathcal{O}_D$ is not.
  \end{problem}

  \vspace{2ex}

  \begin{problem} [Forster $\S 17.7$]
    Let $X$ be a compact Riemann surface of genus two. Consider a basis  $\{\omega_1,\omega_2\}$ of $H^0(X,\Omega)$, and define $f\in \mathcal{M}(X)$ via $\omega_{1}=f\,\omega_2$ Show that $f\colon X\to \pr^1$ is a degree two holomorphic map.
  \end{problem}
  \vspace{2ex}

  \begin{problem} [Miranda $\S$VII.1.B] Fix a compact Riemann surface of genus $g\geq 2$ an a divisor $D$ on it with $\deg(D)>0$. Show:
    \begin{enumerate}[(i)]
    \item if $\deg(D)\leq 2g-3$ then $\dim H^0(X,\mathcal{O}_D)\leq g-1$,
    \item if $\deg(D)=2g-2$ then $\dim H^0(X,\mathcal{O}_D)\leq g$.
Deduce from (ii) that among the divisors of degree $2g-2$, the sheaf $\mathcal{O}_D$ with most sections is $\mathcal{O}_K$.
    \end{enumerate}
  \end{problem}

    \vspace{2ex}

\noindent Recall that a divisor $D$ on a compact Riemann surface $X$ is \emph{very ample} if the complete linear system $|D|$ is base point free and $\phi_D\colon X\to \pr^{N}$ is am embedding, where $\dim(H^0(X,\mathcal{O}_D)=N+1$.
    \vspace{1ex}

    \begin{problem} [Miranda $\S$VII.1.C]
      Fix a compact Riemann surface of genus $g\geq 2$.
      \begin{enumerate}[(i)]
      \item Show if $g\geq 3$, then  $mK$ is very ample for every $m\geq 2$.
      \item Show that if $g=2$, then $mK$ is very ample for every $m\geq 3$.
      \item Show that if $g=2$, then $\phi_{2K}\colon X\to \pr^2$ is a degree two holomorphic map. (\emph{Hint:} $\dim(H^0(X,\mathcal{O}_{2K})=3$ by Problem 5(iii))
      \item Show that if $g=2$ the image of $\phi_{2K}$ is a smooth plane conic curve. 
    \end{problem}


    
%% \begin{problem}  [Forster $\S 16.2$] Fix a rank-2 lattice $\Lambda$ on $\CC$ and consider the elliptic curve $X=C/\Lambda$. Fix a point $x_0\in X$ and the associated divisor $D=[x_0]$. 
%% \end{problem}

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