Logarithmic resolution via multi-weighted blow-ups

Abstract

We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka.Specifically, for a singular, reduced closed subscheme X of a smooth scheme Y over a field of characteristic zero, we resolve the singularities of X by taking proper transforms Xi ⊂ Yi along a sequence of multiweighted blow-ups YN → YN−1 → • • • → Y0 = Y which satisfies the following properties:(i) the Yi are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of Xi, and witness on Xi+1 an immediate improvement in singularities; (iii) and finally, the singular locus of X is transformed into a simple normal crossing divisor on XN .

Desingularization of binomial varieties using toric Artin stacks

“…Motivated by recent works with focus on stacks in the service of resolution of singularities over fields of characteristic zero [AQ,ATW1,ATW2,ATW3,M,Q], we investigate the case of binomials defined over algebraically closed fields of arbitrary characteristic from the perspective of stacks, and with a view towards efficient computations of resolutions and of p-adic and motivic integrals. Classical resolution approaches are discussed in the recent [GaSc].…”