The Stacks project

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

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

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. $\square$