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\begin{document}


\begin{center}\begin{large}\textbf{Math 524 - Problem Set 5\\
Due Wednesday, April 27}\end{large}\end{center}



\pagestyle{empty}

\begin{enumerate}

\item Let $k$ be an algebraically closed field and $X$ a closed subvariety of $\P^n_k$ which is nonsingular in codimension one (and thus satisfies the conditions need to define Weil divisors). For a divisor $D = \sum n_i Y_i$ on $X$, define the \textit{\textbf{degree}} of $D$ to be $\deg D = \sum n_i \deg Y_i$. Let $V$ be an irreducible hypersurface in $\P^n$ which does not contain $X$ and $Y_i$ the irreducible components of $X \cap V$, all of which will have codimension one in $X$. For each $Y_i$, find an open affine $U_i$ such that $Y_i \cap U_i$ is nonempty. Then $Y_i$ is a principal divisor given by some $f_i$ in the function field $K$. Set $n_i = v_{Y_i}(f_i)$ in $U_i$, and define $V.X$ to be $n_iY_i \in \Div X$. 

\begin{enumerate}
\item Show that $V \mapsto V.X$ determines a well-defined homomorphism from the subgroup $H$ of $\Div \P^n$ consisting of divisors, none of whose components contain $X$, to $\Div X$.

\item If $D$ is a principal divisor on $\P^n$, show that $D.X$ is principal on $X$. Thus, we get a homomorphism $\Cl \P^n \to \Cl X$.
\end{enumerate}


\item Let $X = \Spec k[s^4, s^3t, st^3, t^4]$, $k$ a field.
\begin{enumerate}
\item Show that $X$ is Noetherian, separated, integral, and regular in codimension one.
\item Show that $X$ is not normal.
\item Show that the principal divisor $D = (s^2t^2)$ is effective, but $s^2t^2$ is not regular on $X$.
\end{enumerate}

\item Let $X$ be a normal, separated, integral scheme with function field $K$. If $D$ is a Weil divisor on $X$, we defined the associated sheaf $\str_X(D)$ by $$U \mapsto \{f \in K^* \mid (f) + D_U \geq 0\}.$$

\begin{enumerate}
\item Check that $\str(D)$ is indeed a sheaf.
\item As mentioned in class, $\str_X(D)$ is invertible if and only if $D$ is Cartier. As an example, let $R = \C[x,y,z]/(xy-z^2)$, $X = \Spec R$, and $Y \subset X$ the closed subscheme determined by the height one prime $(y,z)$. We showed that the divisor $Y$ is not Cartier. Verify that $\str_X(Y)$ is not invertible.
\end{enumerate} 

\item Let $X$ be a scheme and $f : \mathcal{L} \to \mathcal{M}$ a surjective map of invertible sheaves on $X$. Show that $f$ is an isomorphism.\footnote{Hint: Stalks.}

\item Let $X$ be a scheme over a field $k$. Let $\mathcal{L}$ be an invertible sheaf on $X$. Suppose that $\{s_0, \ldots, s_n\}$ and $\{t_0, \ldots, t_m\}$ are two sets of global sections of $\mathcal{L}$ that span the same subspace $V \subseteq \Gamma(X, \mathcal{L})$ and which each generate each stalk of $\mathcal{L}$. Assume $n \leq m$. Show that the corresponding morphisms $\phi: X \to \P^n$ and $\psi: X \to \P^m$ differ by some linear projection $\P^m - L \to \P^n$ and an automorphism of $\P^n$, where $L$ is some linear subvariety of $\P^m$ of dimension $m-n-1$.
\end{enumerate}

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