Equimultiplicity in Hilbert-Kunz theory

TL;DR

This paper introduces a theory of equimultiplicity for Hilbert-Kunz multiplicity, exploring its behavior on specific hypersurfaces and revealing that it can take infinitely many values and that equimultiple strata may not be locally closed.

Contribution

It develops a new theory of equimultiplicity for Hilbert-Kunz multiplicity and applies it to analyze its behavior on the Brenner-Monsky hypersurface, showing novel properties.

Findings

Hilbert-Kunz multiplicity attains infinitely many values

Equimultiple strata may not be locally closed

New theory of equimultiplicity for Hilbert-Kunz multiplicity

Abstract

This paper develops a theory of equimultiplicity for Hilbert-Kunz multiplicity and uses it to study the behavior of Hilbert-Kunz multiplicity on the Brenner-Monsky hypersurface. A number of applications follows, in particular we show that Hilbert-Kunz multiplicity attains infinitely many values and that equimultiple strata may not be locally closed.