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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)