Lemma 76.40.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $U \subset X$ be an open subspace. Assume
$U$ is quasi-compact,
$Y$ is quasi-compact and quasi-separated,
there exists an immersion $U \to \mathbf{P}^ n_ Y$ over $Y$,
$f$ is of finite type and separated.
Then there exists a commutative diagram
\[ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & Y & \mathbf{P}^ n_ Y \ar[l] } \]
where the arrows with source $U$ are open immersions, $X' \to X$ is a $U$-admissible blowup, $X' \to Z'$ is an open immersion, $Z' \to Y$ is a proper and representable morphism of algebraic spaces. More precisely, $Z' \to Z$ is a $U$-admissible blowup and $Z \to \mathbf{P}^ n_ Y$ is a closed immersion.
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
\[ \xymatrix{ X' \ar[d] \ar[r] & \overline{X}' & Z' \ar[l] \ar[d] \\ X & U \ar[l] \ar[lu] \ar[u] \ar[ru] \ar[r] & Z } \]
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.
$\square$
Comments (2)
Comment #7260 by Laurent Moret-Bailly on
Comment #7332 by Johan on