Exercise 111.46.1. Let $k$ be an algebraically closed field. Let $X$ be a 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 $\mathcal{L}$. Prove there is a natural closed subscheme
\[ Z \subset \mathbf{P}^ n \times X \]
such that the closed point $((\lambda _0 : \ldots : \lambda _ n), x)$ is in $Z$ if and only if the section $\lambda _0 s_0 + \ldots + \lambda _ n s_ n$ vanishes at $x$. (Hint: describe $Z$ affine locally.)
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