The Stacks project

Lemma 29.39.8. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{N}$, $\mathcal{L}$ be invertible $\mathcal{O}_ X$-modules. Assume $S$ is quasi-compact, $f$ is of finite type, and $\mathcal{L}$ is $f$-ample. Then $\mathcal{N} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$.

Proof. By Lemma 29.39.6 we reduce to the case $S$ is affine. Combining Lemma 29.39.4 and Properties, Proposition 28.26.13 we can find an integer $d_0$ such that $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ is globally generated. Choose global sections $s_0, \ldots , s_ n$ of $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ which generate it. This determines a morphism $j : X \to \mathbf{P}^ n_ S$ over $S$. By Lemma 29.39.4 we can also pick an integer $d_1$ such that for all $d \geq d_1$ there exist sections $t_{d, 0}, \ldots , t_{d, n(d)}$ of $\mathcal{L}^{\otimes d}$ which generate it and define an immersion

\[ j_ d = \varphi _{\mathcal{L}^{\otimes d}, t_{d, 0}, \ldots , t_{d, n(d)}} : X \longrightarrow \mathbf{P}^{n(d)}_ S \]

over $S$. Then for $d \geq d_0 + d_1$ we can consider the morphism

\[ \varphi _{\mathcal{N} \otimes \mathcal{L}^{\otimes d}, s_ j \otimes t_{d - d_0, i}} : X \longrightarrow \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_ S \]

This morphism is an immersion as it is the composition

\[ S \to \mathbf{P}^ n_ S \times _ S \mathbf{P}^{n(d - d_0)}_ S \to \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_ S \]

where the first morphism is $(j, j_{d - d_0})$ and the second is the Segre embedding (Constructions, Lemma 27.13.6). Since $j$ is an immersion, so is $(j, j_{d - d_0})$ (apply Lemma 29.3.1). We have a composition of immersions and hence an immersion (Schemes, Lemma 26.24.3). $\square$


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Your email address will not be published.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 0FVC. Beware of the difference between the letter 'O' and the digit '0'.