An invitation to equimultiplicity of F-invariants

TL;DR

This paper introduces the concept of equimultiplicity in F-invariants, surveys existing results, and presents new applications related to Hilbert-Kunz multiplicity and F-signature, advancing understanding of singularities in local rings.

Contribution

It provides an accessible introduction to equimultiplicity of F-invariants and offers new results on Hilbert-Kunz multiplicity and F-signature, including applications and joint work with Polstra.

Findings

Updated results on Hilbert-Kunz multiplicity with novel applications

Survey of equimultiplicity of F-signature and Hilbert-Kunz multiplicity

New insights into singularities of local rings

Abstract

This note grew from the lectures I delivered at ICTP during the Summer School in honor of Hochster and Huneke. Its purpose is to provide an introduction to the notion of equimultiplicity (of numerical invariants of singularities/local rings) and survey past work on equimultiplicity of Hilbert-Kunz multiplicity and F-signature. The treatment of F-signature is from joint work with Thomas Polstra and the section on Hilbert-Kunz multiplicity updates my previous paper on the topic and gives novel applications.