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Partial desingularization in characteristic 0

Dan Abramovich, Michael Temkin

Abstract

It is shown by Belotto da Silva and Bierstone [arXiv:2602.09114] and Włodarczyk [arxiv:2602.14266] that, if one allows to introduce stack theoretic weighted blowups, any variety $X$ over a field of characteristic 0 admits a normal crossings resolution. We introduce a principle that makes such results possible and inevitable, see Theorem 3.

Partial desingularization in characteristic 0

Abstract

It is shown by Belotto da Silva and Bierstone [arXiv:2602.09114] and Włodarczyk [arxiv:2602.14266] that, if one allows to introduce stack theoretic weighted blowups, any variety over a field of characteristic 0 admits a normal crossings resolution. We introduce a principle that makes such results possible and inevitable, see Theorem 3.
Paper Structure (6 sections, 3 theorems)

This paper contains 6 sections, 3 theorems.

Table of Contents

  1. Wonderful invariants
  2. Adequate classes of singularities.
  3. Reduction to the normal cone.
  4. The case of the normal cone.
  5. Where this comes from.
  6. Where this might go.

Key Result

Theorem 1

Let $X$ be a pure dimensional variety over a field of characteristic 0. There is a functorial stack theoretic normal-crossings resolution $X'\to X$.

Theorems & Definitions (6)

  • Theorem 1: Belotto da Silva--Bierstone BDS-B-nc, Włodarczyk Wlodarczyk-nc
  • Definition 2
  • Theorem 3
  • proof
  • Lemma 4
  • proof