Lemma 53.6.5. In Situation 53.6.2. Given an exact sequence

$\omega _ X \to \mathcal{F} \to \mathcal{Q} \to 0$

of coherent $\mathcal{O}_ X$-modules with $\dim (\text{Supp}(\mathcal{Q})) = 0$ and $\dim _ k H^0(X, \mathcal{Q}) \geq 2$ and such that there is no nonzero submodule $\mathcal{Q}' \subset \mathcal{F}$ such that $\mathcal{Q}' \to \mathcal{Q}$ is injective. Then the submodule of $\mathcal{F}$ generated by global sections surjects onto $\mathcal{Q}$.

Proof. Let $\mathcal{F}' \subset \mathcal{F}$ be the submodule generated by global sections and the image of $\omega _ X \to \mathcal{F}$. Since $\dim _ k H^0(X, \mathcal{Q}) \geq 2$ and $\dim _ k H^1(X, \omega _ X) = \dim _ k H^0(X, \mathcal{O}_ X) = 1$, we see that $\mathcal{F}' \to \mathcal{Q}$ is not zero and $\omega _ X \to \mathcal{F}'$ is not an isomorphism. Hence $H^1(X, \mathcal{F}') = 0$ by Lemma 53.6.3 and our assumption on $\mathcal{F}$. Consider the short exact sequence

$0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{Q}/\mathop{\mathrm{Im}}(\mathcal{F}' \to \mathcal{Q}) \to 0$

If the quotient on the right is nonzero, then we obtain a contradiction because then $H^0(X, \mathcal{F})$ is bigger than $H^0(X, \mathcal{F}')$. $\square$

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