Lemma 75.39.5. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U \subset B$ be an open subspace. Assume

1. $B$ is quasi-compact and quasi-separated,

2. $U$ is quasi-compact,

3. $f : X \to B$ is proper,

4. $f^{-1}(U) \to U$ us 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 75.39.3 we may find a $U$-admissible blowup $B' \to B$ such that the strict transform $X'$ maps isomorphically to $B'$. Then we can use $B' = X' \to X$ as the factorization. $\square$

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