Lemma 89.4.4. Let S be a scheme. Let X be a Noetherian algebraic space over S. Let T \subset |X| be a finite set of closed points x such that (1) X is regular at x and (2) the local ring of X at x has dimension 2. Let f : Y \to X be a proper morphism of algebraic spaces 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 Morphisms of Spaces, Lemma 76.39.5 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 89.4.3 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 on Spaces, Lemma 71.17.5) we find the desired factorization.
\square
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