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Group-circulant singularities and partial desingularization preserving normal crossings

André Belotto da Silva, Edward Bierstone

TL;DR

This work develops a cohesive framework for partial desingularization that preserves the normal crossings locus in algebraic and analytic varieties. It combines a multivariate Newton–Puiseux–type splitting theorem, a combinatorial treatment of $G$-circulant matrices yielding group-circulant singularities, and a weighted-blowing-up strategy to achieve a controlled desingularization, resulting in a birational model $X'$ whose singularities are restricted to group-circulant forms and their neighbors with explicit residual possibilities described via orbifold structures. The approach extends known results for simple normal crossings and low dimensions to arbitrary dimension, providing constructive procedures (via blow-ups) and explicit invariants (involving ATW’s framework) to track and resolve singularities near the nc locus. The combination of splitting, circulant combinatorics, and weighted blowings-up offers a systematic path to desingularization compatible with group actions, with potential implications for resolution theory and singularity classification in higher dimensions.

Abstract

The subject is partial desingularization preserving the normal crossings singularities of an algebraic or analytic variety X (over the complex field or over an uncountable algebraically closed field of characteristic zero, in the algebraic case). Our approach has three parts involving distinct techniques: (1) a formal splitting theorem for regular or analytic functions which satisfy a generic splitting hypothesis; (2) a study of singularities in the closure of the normal crossings locus, based on the combinatorics of G-circulant matrices, where G is a finite abelian group, leading to a theorem on reduction to group-circulant normal form; (3) a partial desingularization theorem, proved using (1) and (2) together with weighted blowings-up of group-circulant singularities. Previous results were for partial desingularization preserving simple normal crossings, or preserving general normal crossings when dim X < 5.

Group-circulant singularities and partial desingularization preserving normal crossings

TL;DR

This work develops a cohesive framework for partial desingularization that preserves the normal crossings locus in algebraic and analytic varieties. It combines a multivariate Newton–Puiseux–type splitting theorem, a combinatorial treatment of -circulant matrices yielding group-circulant singularities, and a weighted-blowing-up strategy to achieve a controlled desingularization, resulting in a birational model whose singularities are restricted to group-circulant forms and their neighbors with explicit residual possibilities described via orbifold structures. The approach extends known results for simple normal crossings and low dimensions to arbitrary dimension, providing constructive procedures (via blow-ups) and explicit invariants (involving ATW’s framework) to track and resolve singularities near the nc locus. The combination of splitting, circulant combinatorics, and weighted blowings-up offers a systematic path to desingularization compatible with group actions, with potential implications for resolution theory and singularity classification in higher dimensions.

Abstract

The subject is partial desingularization preserving the normal crossings singularities of an algebraic or analytic variety X (over the complex field or over an uncountable algebraically closed field of characteristic zero, in the algebraic case). Our approach has three parts involving distinct techniques: (1) a formal splitting theorem for regular or analytic functions which satisfy a generic splitting hypothesis; (2) a study of singularities in the closure of the normal crossings locus, based on the combinatorics of G-circulant matrices, where G is a finite abelian group, leading to a theorem on reduction to group-circulant normal form; (3) a partial desingularization theorem, proved using (1) and (2) together with weighted blowings-up of group-circulant singularities. Previous results were for partial desingularization preserving simple normal crossings, or preserving general normal crossings when dim X < 5.
Paper Structure (18 sections, 21 theorems, 190 equations)

This paper contains 18 sections, 21 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. The splitting theorem
  4. Group-circulant normal form
  5. Partial desingularization
  6. Splitting theorem
  7. Group-circulant normal form
  8. Circulant combinatorics
  9. Group circulant normal form
  10. Codimension one combinatorics
  11. Codimension one strata
  12. Reduction to group circulant normal form by blowing up
  13. Weighted blowing up of group-circulant singularities, and partial desingularization
  14. Weighted blowing-up of an affine space
  15. Weighted blowing up of group-circulant singularities and their products
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $f(w,u,x,z)$ denote a function eq:weierpoly, where the coefficients $a_i(w,u,x)$ are regular (or analytic) functions, $f$ is in the ideal generated by $x_1,\ldots, x_{k-1}, z$, and $f$ splits formally at every point where $z=x=0,\, w_1\cdots w_r \neq 0$. Assume that ${\mathbb K}$ is uncountable. we can assume that $f$ splits over ${\mathbb K}{[\![} w^{1/p},u, x{]\!]}$, for some positive integ

Theorems & Definitions (71)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Example 1.10
  • Example 1.11
  • ...and 61 more