Singularities in positive characteristic, stratification and simplification of the singular locus

TL;DR

This paper introduces a new stratification method for hypersurface singularities in positive characteristic, enabling a systematic simplification of singularities through blow-ups, advancing the understanding of singularity resolution.

Contribution

It develops an upper semi-continuous function to stratify the highest multiplicity locus, facilitating singularity simplification in arbitrary characteristic.

Findings

Stratification effectively identifies maximum multiplicity loci.

Blow-ups along the stratification reduce singularities to simpler forms.

Method applies in arbitrary characteristic over perfect fields.

Abstract

We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, also known as a reduction to the monomial case.