Defect and Local Uniformization

TL;DR

This paper presents an algorithm demonstrating that reducing the multiplicity of hypersurface singularities in positive characteristic is feasible if a defectless finite linear projection exists, highlighting defect as the main obstacle to local uniformization.

Contribution

It introduces a simple algorithm linking defectless projections to the possibility of uniformizing hypersurface singularities in characteristic p>0.

Findings

Reduction of multiplicity is possible with defectless projections.

Defect is identified as the main obstacle in positive characteristic.

The method extends Zariski's algorithm from characteristic 0.

Abstract

We give a simple algorithm showing that the reduction of the multiplicity of a characteristic p>0 hypersurface singularity along a valuation is possible if there is a finite linear projection which is defectless. The method begins with the algorithm of Zariski to reduce multiplicity of hypersuface singularities in characteristic 0 along a valuation. This gives a simple demonstration that the only obstruction to local uniformization in positive characteristic is from defect arising in finite projections of singularities.