On elimination of variables in the study of singularities in positive characteristic

TL;DR

This paper explores invariants derived from variable elimination to simplify singularities of algebraic schemes in positive characteristic, refining multiplicity concepts and guiding blowup procedures.

Contribution

It introduces a new approach using inductively defined invariants for simplifying singularities via variable elimination in positive characteristic.

Findings

Invariants lead to a refined understanding of multiplicity.

Singularities can be simplified through blowups guided by these invariants.

Numerical conditions ensure the effectiveness of the simplification process.

Abstract

The objective of this paper is to discuss invariants of singularities of algebraic schemes over fields of positive characteristic, and to show how they yield the simplification of singularities. We focus here on invariants which arise in an inductive manner, namely by successive elimination of variables. When applied to hypersurface singularities they lead us to a refinement of the notion of multiplicity. The main theorem proves that, under some numerical conditions expressed by these invariants, singularities can be simplified by blowups at centers prescribed by this refinement.