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	\large
	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#12\hfill Due: 3 Dec 2021}
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\subsection*{Description}

  Solutions are to be prepared in typeset form (typically via latex) and submitted electronically as a pdf file via \href{https://www.gradescope.com/courses/281407}{Gradescope} by midnight on the due date.
  Late problem sets will not be accepted -- your lowest score is dropped, so you can afford to skip one problem set without penalty.

  Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing \textbf{Sources consulted:\ none} at the top of your submission.

\medskip

\noindent
\begin{enumerate}
  \item
    \begin{enumerate}
      \item 
        Let $D$ be a divisor of degree $d$ on a smooth projective curve $Y$, and assume that $h^0(D) > 0$. Prove that if $p$ is a generic point of $Y$, then $h^0(\O(D-p)) = h^0(D)-1$.
      \item
        Let $Y$ be a smooth projective curve of genus $g \geq 2$, and let $D = \sum_{i=1}^{g+1} q_i$ be a divisor that is the sum of $g+1$ generic points of $Y$. Prove that $h^0(\O(D)) = 2$.
    \end{enumerate}

  \item
    Let $Y$ be a smooth curve of genus $2$ and suppose $p$ is a point of $Y$ such that $h^1(\O(2p)) = 0$. Show that there exists a basis of global sections of $\O(4p)$ of the form $(1,x,y)$, where $x$ and $y$ have poles of order $3$ and $4$ at $p$, respectively. Prove that this basis defines a morphism $Y \to \P^2$ whose image is a singular curve of degree $4$.

  \item
    Let $Y$ be a smooth curve of genus $g$.
    \begin{enumerate}
      \item 
        Show that if $D$ is a divisor on $Y$ with $\deg(D) < 0$, then $h^0(\O(D)) = 0$.
      \item
        Show that if $D$ is a divisor on $Y$ and $\deg(D) > 2g-2$, then $h^0(\O(D)) = 1 - g + \deg(D)$.
      \item
        Suppose that $g \geq 2$ and let $p$ be a point of $Y$. Note that $h^0(\O(kp)) \leq h^0(\O((k+1)p))$ for all $k \in \Z$. Show that for $k$ in the range $0 \leq k \leq 2g-2$, there are exactly $g$ values of $k$ where $h^0(kp) = h^0(\O((k+1)p))$. (These are called \textsf{Weierstrass gaps}.)
    \end{enumerate}
\end{enumerate}
\end{document}
