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). $\square$

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