Exercise 111.46.6. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve over $k$. Let $r \geq s$ be integers. Show that there is a natural closed subset
\[ Z \subset X^ r \times X^ s \]
such that a closed point $(x_1, \ldots , x_ r, y_1, \ldots , y_ s)$ of $X^ r \times X^ s$ is in $Z$ if and only if $x_1 + \ldots + x_ r - y_1 - \ldots - y_ s$ is linearly equivalent to an effective divisor. Hint: Choose an auxiliary invertible module $\mathcal{L}$ of very high degree so that $\mathcal{L}(-D)$ has a nonvanshing section for any effective divisor $D$ of degree $r$. Then use the result of Exercise 111.46.5 twice.
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