Lemma 54.16.10. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a separated morphism of finite type with $X$ regular of dimension $2$. Then $X$ is quasi-projective over $S$.
Proof. By Chow's lemma (Cohomology of Schemes, Lemma 30.18.1) there exists a proper morphism $\pi : X' \to X$ which is an isomorphism over a dense open $U \subset X$ such that $X' \to S$ is H-quasi-projective. By Lemma 54.4.3 there exists a sequence of blowups in closed points
\[ X_ n \to \ldots \to X_1 \to X_0 = X \]
and an $S$-morphism $X_ n \to X'$ extending the rational map $U \to X'$. Observe that $X_ n \to X$ is projective by Divisors, Lemma 31.32.13 and Morphisms, Lemma 29.43.14. This implies that $X_ n \to X'$ is projective by Morphisms, Lemma 29.43.15. Hence $X_ n \to S$ is quasi-projective by Morphisms, Lemma 29.40.3 (and the fact that a projective morphism is quasi-projective, see Morphisms, Lemma 29.43.10). By Lemma 54.16.9 (and uniqueness of contractions Lemma 54.16.2) we conclude that $X_{n - 1}, \ldots , X_0 = X$ are quasi-projective over $S$ as desired. $\square$
Comments (0)