Extremal F-thresholds in regular local rings

TL;DR

This paper classifies ideals in regular local rings of characteristic p that minimize the F-threshold among all ideals with a fixed order of vanishing, contributing to understanding singularity invariants.

Contribution

It provides a classification of ideals with extremal F-thresholds in regular local rings, revealing new insights into their structure and properties.

Findings

Identifies ideals with minimal F-thresholds for fixed order of vanishing.

Establishes criteria for extremal F-thresholds in regular local rings.

Enhances understanding of singularity invariants in positive characteristic.

Abstract

Let $(R, \mathfrak{m})$ be a regular local ring of characteristic $p > 0$. Among all proper ideals $\mathfrak{a}\subseteq R$ with a fixed order of vanishing $\text{ord}_{\mathfrak{m}}(\mathfrak{a})$, we classify the ideals for which the $F$-threshold $\text{ft}^{\mathfrak{m}}(\mathfrak{a})$ is minimal.