The Stacks project

Lemma 73.39.4. Let $S$ be a scheme. Let $\varphi : X \to B$ be a proper morphism of algebraic spaces over $S$. Assume $B$ quasi-compact and quasi-separated. Let $U \subset B$ be a quasi-compact open subspace such that $\varphi ^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $B' \to B$ which dominates $X$, i.e., such that there exists a factorization $B' \to X \to B$ of the blowup morphism.

Proof. By Lemma 73.39.3 we may find a $U$-admissible blowup $B' \to B$ such that the strict transform $X'$ is an open subspace of $B'$ and $U$ is scheme theoretically dense in $B'$. Since $X' \to B'$ is proper we see that $|X'|$ is closed in $|B'|$. As $U \subset B'$ is dense $X' = B'$. $\square$


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