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Weak embedded desingularization in characteristic zero

Statement

Assume AC (The Axiom of Choice), inherited from the blowup and canonical-resolution suppliers.

Let $K$ be a field of characteristic zero, $X$ a smooth $K$ -scheme of finite type and $Y \subseteq X$ a reduced closed subscheme (Integral schemes, Closed immersions of schemes); put $I_{Y}$ for its ideal sheaf. Then there is a canonical embedded desingularization of $Y$ in $X$ : a sequence $X = X_{0} \leftarrow X_{1} \leftarrow \dots \leftarrow X_{r} = \tilde{X}$ of blowups of regular centers $C_{i - 1} \subseteq X_{i - 1}$ such that (a) the exceptional divisor $E_{i}$ of the composite has only simple normal crossings and each center has SNC with $E_{i - 1}$ ; (b) every center $C_{i}$ is disjoint from the smooth locus $Reg (Y) \subseteq Y_{i}$ , the strict transform of $Y$ in $X_{i}$ ; (c) the strict transform $\tilde{Y} : = Y_{r}$ is smooth and has only simple normal crossings with the exceptional divisor $E_{r}$ ; (d) the construction is canonical and commutes with smooth morphisms and with embeddings of ambient smooth schemes. In particular the induced morphism $\tilde{Y} \to Y$ is proper and birational on every irreducible component which is an isomorphism over the smooth locus of $Y$ .

Facts & Assumptions

Given: A field $K$ of characteristic zero, a smooth finite-type $K$ -scheme $X$ , a reduced closed subscheme $Y \subseteq X$ with ideal sheaf $I_{Y}$ , and the marked ideal $(I_{Y}, \emptyset, 1)$ , whose support is $Y$ .

[F1]

Włodarczyk, Simple Hironaka Resolution, §4.7, Theorem 4.7.1 (pp. 25–26), together with §4.4 for descent to non-algebraically-closed fields, and Hauser, The Hironaka Theorem on Resolution of Singularities, §13 (pp. 386–387): the modified canonical algorithm for a reduced closed subscheme gives conditions (a)–(d), and in fact makes the irreducible strict transforms smooth and disjoint and gives the stronger full-transform factorization. It continues only until a strict-transform component would become the next center; the §4.7 induction shows that component is then regular and transverse to the exceptional divisor, after which the modified procedure ignores it and resolves the remaining components.

[F2]

Since a smooth scheme is regular, its irreducible components are disjoint open-and-closed components. The algorithm is applied componentwise; a component contained in $Y$ is smooth and receives the identity sequence.

[F3]

Strict transform of a closed subscheme, Multiple test blow-ups, controlled transforms and resolutions of marked ideals: the strict transform is obtained by saturating the total pullback along exceptional components, while the controlled transform divides the total pullback by the marking at each blow-up. Residual exceptional factors can remain, so the controlled transform need not vanish exactly on the strict transform; the modified algorithm's stopping rule concerns the strict transform itself.

Proof

1.1F1F2F3

The modified canonical sequence. On each open-and-closed ambient component contained in $Y$ , take the identity sequence by [F2]. On the remaining components, use the modified canonical procedure in [F1], not the full support-clearing resolution of $(I_{Y}, 1)$ . Whenever a strict-transform component would next be chosen as a center, the source induction shows it is already smooth and transverse to the exceptional divisor; the modified rule removes that completed component from the active support and continues on the others. Thus no executed center meets $Reg (Y)$ . By [F3], this stopping rule concerns the strict transform and does not assume that the controlled transform has no exceptional factors.

2.1F1F3step 1.1∎

The resulting sequence and its properties. The modified procedure in [F1] terminates with smooth strict transform having SNC with the exceptional divisor; its centers are regular and SNC with the exceptional boundary, and the construction is canonical and commutes with smooth morphisms and ambient embeddings. Since the centers avoid $Reg (Y)$ , the composite is an isomorphism there. A composition of blowups of regular centers is proper and birational, so its restriction induces the stated proper morphism $\tilde{Y} \to Y$ , birational on each irreducible component. This gives all clauses of the Statement while preserving the source's full-transform strengthening as an additional consequence.

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Sources

Jaroslaw Wlodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005) 779-822; author's arXiv version math/0401401 (28 pp., dated October 25, 2018) (standard reference, not scraped)
Herwig Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc. 40 (2003) 323-403 (standard reference, not scraped)