Exercise 111.46.4. 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. Let $s_0, \ldots , s_ n \in H^0(X, \mathcal{L})$ be global sections of $\mathcal{L}$. Let $r \geq 1$. Prove there is a natural closed subscheme
\[ Z \subset \mathbf{P}^ n \times X \times \ldots \times X = \mathbf{P}^ n \times X^ r \]
such that the closed point $((\lambda _0 : \ldots : \lambda _ n), x_1, \ldots , x_ r)$ is in $Z$ if and only if the section $s_\lambda = \lambda _0 s_0 + \ldots + \lambda _ n s_ n$ vanishes on the divisor $D = x_1 + \ldots + x_ r$, i.e., the section $s_\lambda $ is in $\mathcal{L}(-D)$. Hint: explain how this follows by combining then results of Exercises 111.46.2 and 111.46.3.
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