Group-circulant singularities and partial desingularization preserving normal crossings

TL;DR

This paper introduces new techniques for partial desingularization of algebraic and analytic varieties that preserve normal crossings, utilizing group-circulant matrices and weighted blowings-up to extend previous results.

Contribution

It develops a formal splitting theorem, studies singularities via G-circulant matrices, and proves a partial desingularization theorem that extends existing methods.

Findings

Reduction to group-circulant normal form

Partial desingularization preserving normal crossings

Extension of desingularization results to 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.