Lemma 38.31.3. Let $\varphi : X \to S$ be a separated morphism of finite type 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 $S' \to S$ such that the strict transform $X'$ of $X$ is isomorphic to an open subscheme of $S'$.
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 $S$. Next, we do another $U$-admissible blowup to get to the situation where $X \to S$ is flat and of finite presentation, see Lemma 38.31.1. In this case the result follows from Lemma 38.11.5. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)