Lemma 30.21.4. (For a more general version see More on Morphisms, Lemma 37.50.3.) Let $f : X \to Y$ be a proper morphism of schemes with $Y$ Noetherian. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_ y$ is ample on $X_ y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.
Proof. Pick $d_0$ as in Lemma 30.21.3 for $\mathcal{F} = \mathcal{O}_ X$. Pick $d \geq d_0$ so that we can find $r \geq 0$ and sections $s_{y, 0}, \ldots , s_{y, r} \in H^0(X_ y, \mathcal{L}_ y^{\otimes d})$ which define a closed immersion
\[ \varphi _ y = \varphi _{\mathcal{L}_ y^{\otimes d}, (s_{y, 0}, \ldots , s_{y, r})} : X_ y \to \mathbf{P}^ r_{\kappa (y)}. \]
This is possible by Morphisms, Lemma 29.39.4 but we also use Morphisms, Lemma 29.41.7 to see that $\varphi _ y$ is a closed immersion and Constructions, Section 27.13 for the description of morphisms into projective space in terms of invertible sheaves and sections. By our choice of $d_0$, after replacing $Y$ by an open neighbourhood of $y$, we can choose $s_0, \ldots , s_ r \in H^0(X, \mathcal{L}^{\otimes d})$ mapping to $s_{y, 0}, \ldots , s_{y, r}$. Let $X_{s_ i} \subset X$ be the open subset where $s_ i$ is a generator of $\mathcal{L}^{\otimes d}$. Since the $s_{y, i}$ generate $\mathcal{L}_ y^{\otimes d}$ we see that $X_ y \subset U = \bigcup X_{s_ i}$. Since $X \to Y$ is closed, we see that there is an open neighbourhood $y \in V \subset Y$ such that $f^{-1}(V) \subset U$. After replacing $Y$ by $V$ we may assume that the $s_ i$ generate $\mathcal{L}^{\otimes d}$. Thus we obtain a morphism
\[ \varphi = \varphi _{\mathcal{L}^{\otimes d}, (s_0, \ldots , s_ r)} : X \longrightarrow \mathbf{P}^ r_ Y \]
with $\mathcal{L}^{\otimes d} \cong \varphi ^*\mathcal{O}_{\mathbf{P}^ r_ Y}(1)$ whose base change to $y$ gives $\varphi _ y$.
We will finish the proof by a sleight of hand; the “correct” proof proceeds by directly showing that $\varphi $ is a closed immersion after base changing to an open neighbourhood of $y$. Namely, by Lemma 30.21.2 we see that $\varphi $ is a finite over an open neighbourhood of the fibre $\mathbf{P}^ r_{\kappa (y)}$ of $\mathbf{P}^ r_ Y \to Y$ above $y$. Using that $\mathbf{P}^ r_ Y \to Y$ is closed, after shrinking $Y$ we may assume that $\varphi $ is finite. Then $\mathcal{L}^{\otimes d} \cong \varphi ^*\mathcal{O}_{\mathbf{P}^ r_ Y}(1)$ is ample by the very general Morphisms, Lemma 29.37.7. $\square$
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