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.
[Lemma 3, Jongmin]
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 (53.8.1.1). \square
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