Minimal Hilbert-Kunz multiplicity

TL;DR

This paper investigates the minimal differences in Hilbert-Kunz multiplicities for certain ideals in strongly F-regular rings, introducing a new invariant and computing it for specific algebraic structures.

Contribution

It defines the minimal Hilbert-Kunz multiplicity for strongly F-regular rings and computes this invariant for quotient singularities and Segre embeddings.

Findings

Calculated minimal Hilbert-Kunz multiplicity for quotient singularities

Determined this invariant for the coordinate ring of the Segre embedding

Provided insights into the behavior of Hilbert-Kunz multiplicities in specific algebraic contexts

Abstract

In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal Hilbert-Kunz multiplicity for strongly F-regular rings. Moreover, we calculate this invariant for quotient singularities and for the coordinate ring of the Segre embedding: $P^{r-1} \times P^{s-1} \hookrightarrow P^{rs-1}$, respectively.