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Formal splitting and stack-theoretic normal crossings desingularization

André Belotto da Silva, François Bernard, Edward Bierstone

Abstract

We show that stack-theoretic resolution of singularities preserving normal crossings (partial desingularization) by weighted blowings-up, can be obtained in a simple direct way from a splitting theorem of the first and third authors, using the algorithm of Abramovich, Temkin and Włodarczyk for resolution of singularities by weighted blowings-up.

Formal splitting and stack-theoretic normal crossings desingularization

Abstract

We show that stack-theoretic resolution of singularities preserving normal crossings (partial desingularization) by weighted blowings-up, can be obtained in a simple direct way from a splitting theorem of the first and third authors, using the algorithm of Abramovich, Temkin and Włodarczyk for resolution of singularities by weighted blowings-up.
Paper Structure (4 sections, 7 theorems, 17 equations)

This paper contains 4 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. The splitting theorem
  3. The partial desgingularization theorem
  4. Normal crossings strata

Key Result

Theorem 1.1

Let $X\subset Z$ ($Z$ smooth) denote an embedded variety over an uncountable algebraically closed field ${\mathbb K}$ of characteristic zero. Then, for any positive integer $k$, there is a stack-theoretic resolution of singularities ${\widetilde{\sigma}}: {\widetilde{X}} \to X$ preserving the locus

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Corollary 2.4
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 7 more