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The Stacks project

Lemma 54.4.2. Let X be a Noetherian scheme. Let T \subset X be a finite set of closed points x such that \mathcal{O}_{X, x} is a regular local ring of dimension 2. Let f : Y \to X be a proper morphism of schemes which is an isomorphism over U = X \setminus T. Then there exists a sequence

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

where X_{i + 1} \to X_ i is the blowing up of X_ i at a closed point x_ i lying above a point of T and a factorization X_ n \to Y \to X of the composition.

Proof. By More on Flatness, Lemma 38.31.4 there exists a U-admissible blowup X' \to X which dominates Y \to X. Hence we may assume there exists an ideal sheaf \mathcal{I} \subset \mathcal{O}_ X such that \mathcal{O}_ X/\mathcal{I} is supported on T and such that Y is the blowing up of X in \mathcal{I}. By Lemma 54.4.1 there exists a sequence

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

where X_{i + 1} \to X_ i is the blowing up of X_ i at a closed point x_ i lying above a point of T such that \mathcal{I}\mathcal{O}_{X_ n} is an invertible ideal sheaf. By the universal property of blowing up (Divisors, Lemma 31.32.5) we find the desired factorization. \square


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