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.
$\square$
Comments (0)