Resolution except for minimal singularities II. The case of four variables

TL;DR

This paper characterizes the minimal class of four-variable singularities remaining after resolving all but normal crossings, focusing on limits of triple normal crossings that resist elimination without introducing new normal crossings.

Contribution

It introduces a new classification of four-variable singularities, especially those arising as limits of triple normal crossings, and analyzes their resistance to resolution.

Findings

Identifies the smallest class of singularities in four variables after resolution

Characterizes singularities that are limits of triple normal crossings

Shows these singularities cannot be eliminated without creating new normal crossings

Abstract

In this sequel to Resolution except for minimal singularities I, we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities.