Lemma 53.21.4. In Situation 53.6.2 assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $Z \subset X$ be a nonempty $0$-dimensional closed subscheme. If $\deg (\mathcal{L}) \geq 2g - 1 + \deg (Z)$, then $\mathcal{L}$ is globally generated and $H^0(X, \mathcal{L}) \to H^0(X, \mathcal{L}|_ Z)$ is surjective.

**Proof.**
Global generation by Lemma 53.21.3. If $\mathcal{I} \subset \mathcal{O}_ X$ is the ideal sheaf of $Z$, then $H^1(X, \mathcal{I}\mathcal{L}) = 0$ by Lemma 53.21.1. Hence surjectivity.
$\square$

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