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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