Lemma 33.44.13 (0E22)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.44: Degrees on curves
Lemma 33.44.13 (cite)

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Lemma 33.44.13. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and equidimensional of dimension $1$ . Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. If $\deg (\mathcal{L}|_ C) \leq 0$ for all irreducible components $C$ of $X$ , then either $H^0(X, \mathcal{L}) = 0$ or $\mathcal{L} \cong \mathcal{O}_ X$ .

Proof. Let $s \in H^0(X, \mathcal{L})$ be nonzero. Since $X$ is reduced there exists an irreducible component $C$ of $X$ with $s|_ C \not= 0$ . But if $s|_ C$ is nonzero, then $s$ is nonwhere vanishing on $C$ by Lemma 33.44.12. This in turn implies $s$ is nowhere vanishing on every irreducible component of $X$ meeting $C$ . Since $X$ is connected, we conclude that $s$ vanishes nowhere and the lemma follows. $\square$

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