4.1. Snc models and simple blowups [015S]

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4.1. Snc models and simple blowups

Given any two models ${\mathcal{X}}$ , ${\mathcal{X}}^{\prime}$ of $X$ , there is a canonical bimeromorphic map ${\mathcal{X}}^{\prime}\dashrightarrow{\mathcal{X}}$ , and we say that ${\mathcal{X}}^{\prime}$ dominates ${\mathcal{X}}$ if this map is a morphism. Any two models ${\mathcal{X}}$ , ${\mathcal{X}}^{\prime}$ is dominated by a third, for instance the normalization of the graph of ${\mathcal{X}}\dashrightarrow{\mathcal{X}}^{\prime}$ . By Hironaka’s theorem, any model is dominated by an snc model. Thus the set of models forms a directed set, in which snc models are cofinal.

Suppose ${\mathcal{X}}$ is an snc model and that ${\mathcal{X}}^{\prime}$ is another model that dominates ${\mathcal{X}}$ via $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . As in [KS06, Definition 22] we say that $\rho$ is a simple blowup if it is a blowup along a smooth, connected complex subspace $W$ of ${\mathcal{X}}_{0}$ meeting transversely (or not at all) every irreducible component of ${\mathcal{X}}_{0}$ that does not contain it. In this case, ${\mathcal{X}}^{\prime}$ is also an snc model.

Lemma 4.1.

Suppose ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ are snc models and that ${\mathcal{X}}^{\prime}$ dominates ${\mathcal{X}}$ via $\rho\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ . Then there exists a third snc model ${\mathcal{X}}^{\prime\prime}$ dominating ${\mathcal{X}}^{\prime}$ , such that the induced map ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is a composition of simple blowups.

We are grateful to Bernard Teissier for help with the following argument.

Proof.

By Hironaka’s version of the Chow theorem (in turn a consequence of the flattening theorem), see [Hir75, Corollary 2], there exists a complex manifold ${\mathcal{X}}^{\prime\prime}$ and a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ such that ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$ . Since ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is an isomorphism above $X$ , the construction in [Hir75] further guarantees that ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is an isomorphism above $X$ . Indeed, the proof proceeds by blowing up well-chosen smooth centers contained in the non-flat locus of ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ , see Définition 4.4.3 (2) in loc. cit.

We may therefore assume that ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ itself is projective, and more precisely the blowup of an ideal $I$ cosupported on ${\mathcal{X}}_{0}$ . By the principalization theorem for ideals, there exists a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ that is a composition of simple blowups, such that the pullback of $I$ to ${\mathcal{X}}^{\prime\prime}$ is a principal ideal, see [Kol07, Theorem 3.45] or [Wło09, Theorem 2.0.3]. In particular, ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$ . ∎