Desingularization by blowings-up avoiding simple normal crossings

TL;DR

This paper presents a method for resolving most singularities in algebraic varieties over characteristic zero fields using a sequence of blowings-up that avoids creating simple normal crossings, improving the desingularization process.

Contribution

It introduces a novel approach to desingularization that selectively avoids points with simple normal crossings during blowings-up, refining existing resolution techniques.

Findings

Resolves all but simple normal crossings singularities in algebraic varieties.

Uses desingularization invariant and geometric information for local normal forms.

Provides a constructive method for controlled blowings-up avoiding certain singularities.

Abstract

It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points where the transformed variety together with the exceptional divisor has only snc singularities. The proof follows the philosophy of arXiv:1107.5595 that the desingularization invariant can be used together with natural geometric information to compute local normal forms of singularities.