The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E8W. Beware of the difference between the letter 'O' and the digit '0'.