Lemma 33.23.2. Let $k$ be an algebraically closed field. Let $X$ be a proper $k$-scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Suppose that for every closed subscheme $Z \subset X$ of dimension $0$ and degree $2$ over $k$ the map

$H^0(X, \mathcal{L}) \longrightarrow H^0(Z, \mathcal{L}|_ Z)$

is surjective. Then $\mathcal{L}$ is very ample on $X$ over $k$.

Proof. This is a reformulation of Lemma 33.23.1. Namely, given distinct closed points $x, y \in X$ taking $Z = x \cup y$ (viewed as closed subscheme) we get condition (1) of the lemma. And given a nonzero tangent vector $\theta \in T_{X/k, x}$ the morphism $\theta : \mathop{\mathrm{Spec}}(k[\epsilon ]) \to X$ is a closed immersion. Setting $Z = \mathop{\mathrm{Im}}(\theta )$ we obtain condition (2) of the lemma. $\square$

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