Resolution except for minimal singularities I

TL;DR

This paper presents a method using the desingularization invariant and geometric data to compute local normal forms of singularities, achieving resolution of singularities except for minimal singularities like simple normal crossings.

Contribution

It introduces a novel approach to resolve singularities while avoiding certain simple normal crossings, and identifies the minimal class of singularities remaining in low dimensions.

Findings

Resolution of singularities except for simple normal crossings.

Identification of minimal singularities in low dimensions.

Method based on desingularization invariant and geometric information.

Abstract

The philosophy of the article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollar.