The Stacks project

Lemma 76.39.6. Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $U \subset W \subset Y$ be open subspaces. Let $f : X \to W$ and let $s : U \to X$ be morphisms such that $f \circ s = \text{id}_ U$. Assume

1. $f$ is proper,

2. $Y$ is quasi-compact and quasi-separated, and

3. $U$ and $W$ are quasi-compact.

Then there exists a $U$-admissible blowup $b : Y' \to Y$ and a morphism $s' : b^{-1}(W) \to X$ extending $s$ with $f \circ s' = b|_{b^{-1}(W)}$.