Lemma 53.21.5. In Situation 53.6.2, assume $X$ is geometrically integral over $k$ and has genus $g$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $\deg (\mathcal{L}) \geq 2g + 1$, then $\mathcal{L}$ is very ample.
Proof. By Lemma 53.21.3, $\mathcal{L}$ is globally generated, and so it determines a morphism $f : X \to \mathbf{P}^ n_ k$ where $n = h^0(X,\mathcal{L}) - 1$. To show that $\mathcal{L}$ is very ample means to show that $f$ is a closed immersion. It suffices to check that the base change of $f$ to an algebraic closure $\overline{k}$ of $k$ is a closed immersion (Descent, Lemma 35.23.19). So we may assume that $k$ is algebraically closed; $X$ remains integral, by assumption. Lemma 53.21.4 gives that for every $0$-dimensional closed subscheme $Z\subset X$ of degree 2, the restriction map $H^0(X, \mathcal{L}) \to H^0(X, \mathcal{L}|_ Z)$ is surjective. By Varieties, Lemma 33.23.2, $\mathcal{L}$ is very ample. $\square$
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