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