The Stacks project

Proof. Throughout the proof all the algebraic spaces will be separated of finite type over $Y$. This in particular implies these algebraic spaces are quasi-compact and quasi-separated and that the morphisms between them will be quasi-compact and separated. See Morphisms of Spaces, Sections 67.4 and 67.8. We will use that if $U \to W$ is an immersion of such spaces over $Y$, then the scheme theoretic image $Z$ of $U$ in $W$ is a closed subspace of $W$ and $U \to Z$ is an open immersion, $U \subset Z$ is scheme theoretically dense, and $|U| \subset |Z|$ is dense. See Morphisms of Spaces, Lemma 67.17.7.

Let $X_{12} \subset X_1 \times _ Y X_2$ be the scheme theoretic image of $U \to X_1 \times _ Y X_2$. The projections $p_ i : X_{12} \to X_ i$ induce isomorphisms $p_ i^{-1}(U) \to U$ by Morphisms of Spaces, Lemma 67.16.7. Choose a $U$-admissible blowup $X_ i^ i \to X_ i$ such that the strict transform $X_{12}^ i$ of $X_{12}$ is isomorphic to an open subspace of $X_ i^ i$, see Lemma 76.39.4. Let $\mathcal{I}_ i \subset \mathcal{O}_{X_ i}$ be the corresponding finite type quasi-coherent sheaf of ideals. Recall that $X_{12}^ i \to X_{12}$ is the blowup in $p_ i^{-1}\mathcal{I}_ i \mathcal{O}_{X_{12}}$, see Divisors on Spaces, Lemma 71.18.3. Let $X_{12}'$ be the blowup of $X_{12}$ in $p_1^{-1}\mathcal{I}_1 p_2^{-1}\mathcal{I}_2 \mathcal{O}_{X_{12}}$, see Divisors on Spaces, Lemma 71.17.10 for what this entails. We obtain a commutative diagram

where all the morphisms are $U$-admissible blowing ups. Since $X_{12}^ i \subset X_ i^ i$ is an open we may choose a $U$-admissible blowup $X_ i' \to X_ i^ i$ restricting to $X_{12}' \to X_{12}^ i$, see Divisors on Spaces, Lemma 71.19.3. Then $X_{12}' \subset X_ i'$ is an open subspace and the diagram

is commutative with vertical arrows blowing ups and horizontal arrows open immersions. Note that $X'_{12} \to X_1' \times _ Y X_2'$ is an immersion and proper (use that $X'_{12} \to X_{12}$ is proper and $X_{12} \to X_1 \times _ Y X_2$ is closed and $X_1' \times _ Y X_2' \to X_1 \times _ Y X_2$ is separated and apply Morphisms of Spaces, Lemma 67.40.6). Thus $X'_{12} \to X_1' \times _ Y X_2'$ is a closed immersion. If we define $X$ by glueing $X_1'$ and $X_2'$ along the common open subspace $X_{12}'$, then $X \to Y$ is of finite type and separated¹. As compositions of $U$-admissible blowups are $U$-admissible blowups (Divisors on Spaces, Lemma 71.19.2) the lemma is proved. $\square$

Proof. Let $Z \subset \mathbf{P}^ n_ Y$ be the scheme theoretic image of the immersion $U \to \mathbf{P}^ n_ Y$. Since $U \to \mathbf{P}^ n_ Y$ is quasi-compact we see that $U \subset Z$ is a (scheme theoretically) dense open subspace (Morphisms of Spaces, Lemma 67.17.7). Apply Lemma 76.40.1 to find a diagram

with properties as listed in the statement of that lemma. As $X' \to X$ and $Z' \to Z$ are $U$-admissible blowups we find that $U$ is a scheme theoretically dense open of both $X'$ and $Z'$ (see Divisors on Spaces, Lemmas 71.17.4 and 71.6.4). Since $Z' \to Z \to Y$ is proper we see that $Z' \subset \overline{X}'$ is a closed subspace (see Morphisms of Spaces, Lemma 67.40.6). It follows that $X' \subset Z'$ (scheme theoretically), hence $X'$ is an open subspace of $Z'$ (small detail omitted) and the lemma is proved. $\square$

Proof. By Limits of Spaces, Lemma 70.13.3 there exists a dense open subspace $U \subset X$ and an immersion $U \to \mathbf{A}^ n_ Y$ over $Y$. Composing with the open immersion $\mathbf{A}^ n_ Y \to \mathbf{P}^ n_ Y$ we obtain a situation as in Lemma 76.40.2 and the result follows. $\square$

Proof. In this first paragraph of the proof we reduce the lemma to the case where $Y$ is of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. We may and do replace the base scheme $S$ by $\mathop{\mathrm{Spec}}(\mathbf{Z})$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ as a directed limit of separated algebraic spaces of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Limits of Spaces, Proposition 70.8.1 and Lemma 70.5.9. For all $i$ sufficiently large we can find a separated finite type morphism $X_ i \to Y_ i$ such that $X = Y \times _{Y_ i} X_ i$, see Limits of Spaces, Lemmas 70.7.1 and 70.6.9. Let $\eta _1, \ldots , \eta _ n$ be the generic points of the irreducible components of $|X|$ ($X$ is Noetherian as a finite type separated algebraic space over the Noetherian algebraic space $Y$ and therefore $|X|$ is a Noetherian topological space). By Limits of Spaces, Lemma 70.5.2 we find that the images of $\eta _1, \ldots , \eta _ n$ in $|X_ i|$ are distinct for $i$ large enough. We may replace $X_ i$ by the scheme theoretic image of the (quasi-compact, in fact affine) morphism $X \to X_ i$. After this replacement we see that the images of $\eta _1, \ldots , \eta _ n$ in $|X_ i|$ are the generic points of the irreducible components of $|X_ i|$, see Morphisms of Spaces, Lemma 67.16.3. Having said this, suppose we can find a diagram

where $X_ i' \to X_ i$ is a $U_ i$-admissible blowup for some dense open $U_ i \subset X_ i$ and the morphism $X_ i' \to \mathbf{P}^ n_{Y_ i}$ is an immersion. Then the strict transform $X' \to X$ of $X$ relative to $X_ i' \to X_ i$ is a $U$-admissible blowing up where $U \subset X$ is the inverse image of $U_ i$ in $X$. Because of our carefully chosen index $i$ it follows that $\eta _1, \ldots , \eta _ n \in |U|$ and $U \subset X$ is dense. Moreover, $X' \to \mathbf{P}^ n_ Y$ is an immersion as $X'$ is closed in $X_ i' \times _{X_ i} X = X_ i' \times _{Y_ i} Y$ which comes with an immersion into $\mathbf{P}^ n_ Y$. Thus we have reduced to the situation of the following paragraph.

Assume that $Y$ is separated of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Then $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated of finite type as well. We apply Lemma 76.40.3 to $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ to find a dense open subspace $U \subset X$ and a commutative diagram

with all the properties listed in the lemma. Note that $Z$ has an ample invertible sheaf, namely $\mathcal{O}_{\mathbf{P}^ n}(1)|_ Z$. Hence $Z' \to Z$ is a H-projective morphism by Morphisms, Lemma 29.43.16. It follows that $Z' \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is H-projective by Morphisms, Lemma 29.43.7. Thus there exists a closed immersion $Z' \to \mathbf{P}^ m_{\mathop{\mathrm{Spec}}(\mathbf{Z})}$ for some $m \geq 0$. It follows that the diagonal morphism

is an immersion (because the composition with the projection to $\mathbf{P}^ m_\mathbf {Z}$ is an immersion) and we win. $\square$