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Resolution Except for the Normal-Crossing Locus and Galois actions

Jaroslaw Włodarczyk

Abstract

In characteristic zero we construct a canonical embedded and non-embedded resolution algorithm by weighted blow-ups that preserves the normal crossings (nc) locus and resolves singularities up to normal crossings. The process terminates with a Deligne-Mumford stack whose singularities are precisely normal crossings and are adapted to the exceptional divisor. The construction is governed by a rigidity property of canonical maximal admissible weighted centers together with a Galois-theoretic analysis of splitting forms. We show that normal crossings singularities are étale-locally simple normal crossings and can be viewed as quotients of simple normal crossings by finite Galois groups acting by permutation of branches. This viewpoint explains both the intrinsic role of weighted blow-ups and the natural emergence of stack structures in nc-preserving resolution. Building on the weighted blow-up framework of Abramovich-Temkin-Włodarczyk and its logarithmic refinement developed in my previous work, we obtain in particular a canonical compactification of normal crossings Deligne-Mumford stacks and a functorial nc-preserving resolution for subvarieties and Deligne-Mumford stacks.

Resolution Except for the Normal-Crossing Locus and Galois actions

Abstract

In characteristic zero we construct a canonical embedded and non-embedded resolution algorithm by weighted blow-ups that preserves the normal crossings (nc) locus and resolves singularities up to normal crossings. The process terminates with a Deligne-Mumford stack whose singularities are precisely normal crossings and are adapted to the exceptional divisor. The construction is governed by a rigidity property of canonical maximal admissible weighted centers together with a Galois-theoretic analysis of splitting forms. We show that normal crossings singularities are étale-locally simple normal crossings and can be viewed as quotients of simple normal crossings by finite Galois groups acting by permutation of branches. This viewpoint explains both the intrinsic role of weighted blow-ups and the natural emergence of stack structures in nc-preserving resolution. Building on the weighted blow-up framework of Abramovich-Temkin-Włodarczyk and its logarithmic refinement developed in my previous work, we obtain in particular a canonical compactification of normal crossings Deligne-Mumford stacks and a functorial nc-preserving resolution for subvarieties and Deligne-Mumford stacks.
Paper Structure (45 sections, 33 theorems, 240 equations, 1 figure)

This paper contains 45 sections, 33 theorems, 240 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Center Theorem
  3. Algorithmic Rule
  4. Galois groups, weights, and stacks
  5. Main Theorems
  6. Normal Crossing Schemes
  7. Resolution by Weighted blow-ups
  8. Q-ideals
  9. Canonical invariants and maximal admissible centers
  10. Canonical invariant in the non-logarithmic setup
  11. Rational Rees algebra
  12. Regular centers vs. Rees centers
  13. Canonical invariant for Rees algebras
  14. Weighted, stack-theoretic, and cobordant blow-ups
  15. Cobordant blow-ups at the maximal admissible centers
  16. ...and 30 more sections

Key Result

Theorem 1.1.1

A canonical maximal admissible weighted center intersects the nc locus $X^{\mathrm{nc}}(\mathcal{I})$ only if it is entirely contained in it.

Figures (1)

  • Figure 1: Cobordant blow-up: the role of the vertex $V$ and the exceptional divisor $D$. (Reproduced from W22.)

Theorems & Definitions (81)

  • Theorem 1.1.1: Center Theorem
  • Theorem 2.0.1: NC-preserving principalization
  • Theorem 2.0.2: Embedded NC-preserving resolution
  • Theorem 2.0.3: Non-embedded NC-preserving resolution
  • Theorem 2.0.4: NC compactification
  • Definition 2.1.1
  • Definition 2.1.2: Normal Crossings Divisor and Ideal
  • Definition 2.1.3: NC schemes (non-embedded)
  • Definition 3.0.2: ${\mathbb{Q}}$-ideal
  • Remark 3.0.3
  • ...and 71 more