Partial desingularization

TL;DR

This paper develops methods for partial desingularization of algebraic varieties in characteristic zero, reducing singularities to a finite list of minimal types using blowings-up that preserve normal crossings.

Contribution

It introduces a technique to achieve partial desingularization with a finite set of minimal singularities defined via circulant matrices, extending resolution methods up to dimension four.

Findings

Successfully reduces singularities to minimal types in dimensions up to 4.

Provides explicit local normal forms for singularities using circulant matrices.

Achieves partial desingularization while preserving normal crossings of order at most three.

Abstract

We address the following question of partial desingularization preserving normal crossings. Given an algebraic (or analytic) variety X in characteristic zero, can we find a finite sequence of blowings-up preserving the normal-crossings locus of X, after which the transform X' of X has only singularities from an explicit finite list of minimal singularities, which we define using the determinants of circulant matrices. In the case of surfaces, for example, the pinch point or Whitney umbrella is the only singularity needed in addition to normal crossings. We develop techniques for factorization (splitting) of a monic polynomial with regular (or analytic) coefficients, satisfying a generic normal crossings hypothesis, which we use together with resolution of singularities techniques to find local circulant normal forms of singularities. These techniques in their current state are enough for a positive answer to the question above, for dim X up to 4, or in arbitrary dimension if we preserve normal crossings only of order at most three. In these cases, minimal singularities have smooth normalization.