The Stacks project

Lemma 38.31.4. Let $\varphi : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Let $U \subset S$ be a quasi-compact open such that $\varphi ^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \to S$ which dominates $X$, i.e., such that there exists a factorization $S' \to X \to S$ of the blowup morphism.

Proof. The discussion in Remark 38.30.1 applies. Thus we may do a first $U$-admissible blowup and assume the complement $S \setminus U$ is the support of an effective Cartier divisor $D$. In particular $U$ is scheme theoretically dense in $S$. Choose another $U$-admissible blowup $S' \to S$ such that the strict transform $X'$ of $X$ is an open subscheme of $S'$, see Lemma 38.31.3. Since $X' \to S'$ is proper, and $U \subset S'$ is dense, we see that $X' = S'$. Some details omitted. $\square$

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