Exercise 111.46.5. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Show that there is a natural closed subset
\[ Z \subset X^ r \]
such that a closed point $(x_1, \ldots , x_ r)$ of $X^ r$ is in $Z$ if and only if $\mathcal{L}(-x_1 - \ldots -x_ r)$ has a nonzero global section. Hint: use Exercise 111.46.4.
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