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\begin{document}
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	\large
	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#10\hfill Due: 19 Nov 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
    Let $0 \to V_0 \to \cdots \to V_n \to 0$ be a complex of finite-dimensional vector spaces. Show that
    \begin{equation*}
      \sum_i (-1)^i \dim(V_i) = \sum_q (-1)^q \dim(H^q(V^\bullet)) \, .
    \end{equation*}

  \item
    On the projective line $\P^1$, when are two direct sums $\O(m) \oplus \O(n)$ and $\O(r) \oplus \O(s)$ isomorphic? (\textit{Hint}: Use that twisting is an exact functor.)

  \item
    \begin{enumerate}
      \item
        Let $X = \P^n$ and $K$ be the function field of $X$. Let $\Fcal$ be the constant function field $\O_X$-module, that is $\Fcal(U) = K$ for any nonempty open $U \subseteq X$. Prove that $H^q(X, \Fcal) = 0$ for all $q > 0$.
      \item
        Let $X = \P^n$ and $\{U^i\}_i$ be an open covering of $X$. Suppose that rational functions $f_i$ are given such that $f_i - f_j$ is a regular function on $U^{ij} \colonequals U^i \cap U^j$ for all $i,j$. Let $\Qcal = \Fcal/\O$ and consider the exact sequence
        \begin{equation*}
          0 \to \O \to \Fcal \to \Qcal \to 0 \, .
        \end{equation*}
        Using this exact sequence, solve the \textit{Cousin Problem}: is there a rational function $f$ such that $f - f_i$ is a regular function on $V^i$ for each $i$?
    \end{enumerate}

  \item
    \begin{enumerate}
      \item 
        Let $X = \P^2 \setminus \{(0:0:1)\}$. Use the covering of $X$ by the two standard affine open sets $\U^0, \U^1$ to compute the cohomology $H^q(X, \O_X)$.
      \item
        Suppose $Y$ is a variety covered by two affine open sets, and $\Mcal$ is an $\O_Y$-module. Show that $H^q(Y, \Mcal) = 0$ for all $q > 1$.
    \end{enumerate}

 % \item
 %   Let $f,g \in \C[x_0, x_1, x_2, x_3]$ be homogeneous polynomials of degrees $d, e$, respectively, and let $I = (f,g)$. Assume that their vanishing locus $C \colonequals \V(I) \subseteq \P^3$ is an irreducible curve. Let $\Iscr$ be the ideal sheaf of $C$.
 %   \begin{enumerate}
 %     \item 
 %       Construct an exact sequence
 %       \begin{equation*}
 %         0 \to \O(-d-e) \to \O(-d) \oplus \O(-e) \to \Iscr \to 0 \, .
 %       \end{equation*}
 %       \item
 %         Using the short exact sequence
 %         \begin{equation*}
 %           0 \to \Iscr \to \O \to i_* \O_C \to 0 \, ,
 %         \end{equation*}
 %         where $i : C \hookrightarrow \P^3$ is the inclusion map, and the exact sequence from the previous part, compute the arithmetic genus $\dim(H^1(C, \O_C))$ of $C$.
 %   \end{enumerate}
\end{enumerate}
\end{document}
