The Stacks project

Lemma 38.31.5. Let $S$ be a scheme. Let $U \subset W \subset S$ be open subschemes. Let $f : X \to W$ be a morphism and let $s : U \to X$ be a morphism such that $f \circ s = \text{id}_ U$. Assume

  1. $f$ is proper,

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

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

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

Proof. We may and do replace $X$ by the scheme theoretic image of $s$. Then $X \to W$ is an isomorphism over $U$, see Morphisms, Lemma 29.6.8. By Lemma 38.31.4 there exists a $U$-admissible blowup $W' \to W$ and an extension $W' \to X$ of $s$. We finish the proof by applying Divisors, Lemma 31.34.3 to extend $W' \to W$ to a $U$-admissible blowup of $S$. $\square$

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