Desingularization of Quasi-excellent Schemes in Characteristic Zero

For a Noetherian scheme X, let Xreg denote the regular locus of X. The scheme X is said to admit a resolution of singularities if there exists a blowup X ′ → X with center disjoint from Xreg and regular X ′. More generally, for a closed subscheme Z ⊂ X, let (X,Z)reg denote the set of points x ∈ Xreg such that, in an open neighborhood of x, Z is defined by an equation of the form t1 1 · · · · · t nd d = 0, where t1, . . . , td is a regular system of parameters at x and n1, . . . , nd ≥ 0. (For example, (X, ∅)reg = Xreg.) A desingularization of a pair (X,Z) is a blow-up f : X ′ → X with center disjoint from (X,Z)reg and (X ′, Z )reg = X ′, where Z ′ = Z ×X X ′. If, in addition, f is a succession of blow-ups with regular centers, it is said to be a successive desingularization. The scheme X ′ is said to admit an embedded (resp. successive embedded) resolution of singularities if, for any closed subscheme Z ⊂ X, the pair (X,Z) admits a desingularization (resp. successive desingularization). In his celebrated paper [Hir] published in 1964, Hironaka proved that any integral scheme of finite type over a local quasi-excellent ring of residue characteristic zero admits a successive embedded resolution of singularities (see Remark 2.2.7 and appendix). Recall that a Noetherian ring A is said to be quasi-excellent if, for any prime ideal ℘ ⊂ A, the canonical homomorphism A℘ → Â℘ is regular, and, for any finitely generated A-algebra B, Spec(B)reg is open in Spec(B). (Excellent rings are those which, in addition to the above two properties, are universally catenary.) The result of Hironaka is extremely important and has many applications, but its proof is very difficult and long. It is therefore very natural that mathematicians are still trying to understand and simplify the proof. Simplified proofs of successive embedded resolution of singularities for integral schemes of finite type over a field of characteristic zero were given by Villamayor [Vil], Bierstone-Milman [BM], and Wlodarczyk [W l]. On the other hand, Grothendieck proved in [EGA4, 7.9.5], that if X is a locally Noetherian scheme such that every integral scheme of finite type over X admits a resolution of singularities, then X is quasi-excellent (i.e., it has a covering by open affine subschemes which are spectra of quasiexcellent rings). Furthermore, in loc.cit. 7.9.6, he conjectured that the converse implication is also true, and claimed that Hironaka’s proof gives also resolution of singularities for arbitrary quasi-excellent schemes with