Lemma 54.5.4 (0C5I)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 54: Resolution of Surfaces
Section 54.5: Dominating by normalized blowups
Lemma 54.5.4 (cite)

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Lemma 54.5.4. Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is Noetherian, Nagata, and has dimension $2$ . Let $Y$ be a proper scheme over $S$ . Given an $S$ -rational map $f : U \to Y$ from $X$ to $Y$ there exists a sequence

$X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 \to X$

and an $S$ -morphism $f_ n : X_ n \to Y$ such that $X_0 \to X$ is the normalization, $X_{i + 1} \to X_ i$ is the normalized blowing up of $X_ i$ at a closed point, and $f_ n$ and $f$ agree.

Proof. Applying Divisors, Lemma 31.36.2 we find a proper morphism $p : X' \to X$ which is an isomorphism over $U$ and a morphism $f' : X' \to Y$ agreeing with $f$ over $U$ . Apply Lemma 54.5.3 to the morphism $p : X' \to X$ . The composition $X_ n \to X' \to Y$ is the desired morphism. $\square$

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