Lemma 53.7.2. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume

1. $\mathcal{L}$ is globally generated,

2. $H^1(X, \mathcal{L}) = 0$, and

3. $\mathcal{L}$ is ample.

Then $\mathcal{L}^{\otimes 2}$ is very ample on $X$ over $k$.

Proof. Choose basis $s_0, \ldots , s_ n$ of $H^0(X, \mathcal{L}^{\otimes 2})$ over $k$. By property (1) we see that $\mathcal{L}^{\otimes 2}$ is globally generated and we get a morphism

$\varphi _{\mathcal{L}^{\otimes 2}, (s_0, \ldots , s_ n)} : X \longrightarrow \mathbf{P}^ n_ k$

See Constructions, Section 27.13. The lemma asserts that this morphism is a closed immersion. To check this we may replace $k$ by its algebraic closure, see Descent, Lemma 35.23.19. Thus we may assume $k$ is algebraically closed.

Assume $k$ is algebraically closed. For each generic point $\eta _ i \in X$ let $V_ i \subset H^0(X, \mathcal{L})$ be the $k$-subvector space of sections vanishing at $\eta _ i$. Since $\mathcal{L}$ is globally generated, we see that $V_ i \not= H^0(X, \mathcal{L})$. Since $X$ has only a finite number of irreducible components and $k$ is infinite, we can find $s \in H^0(X, \mathcal{L})$ nonvanishing at $\eta _ i$ for all $i$. Then $s$ is a regular section of $\mathcal{L}$ (because $X$ is Cohen-Macaulay by Lemma 53.6.1 and hence $\mathcal{L}$ has no embedded associated points).

In particular, all of the statements given in the proof of Lemma 53.7.1 hold with this $s$. Moreover, as $\mathcal{L}$ is globally generated, we can find a global section $t \in H^0(X, \mathcal{L})$ such that $t|_ Z$ is nonvanishing (argue as above using the finite number of points of $Z$). Then in the proof of Lemma 53.7.1 we can use $t$ to see that additionally the multiplication map

$\mu _ n : H^0(X, \mathcal{L}) \otimes _ k H^0(X, \mathcal{L}^{\otimes 2}) \longrightarrow H^0(X, \mathcal{L}^{\otimes 3})$

is surjective. Thus

$S = \bigoplus \nolimits _{n \geq 0} H^0(X, \mathcal{L}^{\otimes n})$

is generated in degrees $0, 1, 2$ over $k$. Arguing as in the proof of Lemma 53.7.1 we find that $S^{(2)} = \bigoplus _{n} S_{2n}$ is generated in degree $1$. This means that $\mathcal{L}^{\otimes 2}$ is very ample as desired. Some details omitted. $\square$

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