The Stacks project

[Lemma 3, Jongmin]

Lemma 53.21.3. In Situation 53.6.2 assume $X$ is integral and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $\deg (\mathcal{L}) \geq 2g$, then $\mathcal{L}$ is globally generated.

Proof. Let $Z \subset X$ be the closed subscheme cut out by the global sections of $\mathcal{L}$. By Lemma 53.21.2 we see that $Z \not= X$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal sheaf cutting out $Z$. Consider the short exact sequence

\[ 0 \to \mathcal{I}\mathcal{L} \to \mathcal{L} \to \mathcal{O}_ Z \to 0 \]

If $Z \not= \emptyset $, then $H^1(X, \mathcal{I}\mathcal{L})$ is nonzero as follows from the long exact sequence of cohomology. By Lemma 53.4.2 this gives a nonzero and hence injective map

\[ \mathcal{I}\mathcal{L} \longrightarrow \omega _ X \]

In particular, we find an injective map $H^0(X, \mathcal{L}) = H^0(X, \mathcal{I}\mathcal{L}) \to H^0(X, \omega _ X)$. This is impossible as

\[ \dim _ k H^0(X, \mathcal{L}) = \dim _ k H^1(X, \mathcal{L}) + \deg (\mathcal{L}) + 1 - g \geq g + 1 \]

and $\dim H^0(X, \omega _ X) = g$ by ( $\square$

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