Formulas of F-thresholds and F-jumping coefficients on toric rings

TL;DR

This paper derives formulas for F-thresholds on toric rings, explores their relationship with F-jumping coefficients, and applies these results to characterize regularity and rationality in specific cases.

Contribution

It provides a general formula for F-thresholds on toric rings, extending previous examples and establishing inequalities with F-jumping coefficients.

Findings

Derived a formula for F-thresholds on toric rings

Established inequalities between F-jumping coefficients and F-thresholds

Characterized regularity and rationality of F-thresholds in certain toric rings

Abstract

F-thresholds are defined by Mustata, Takagi and Watanabe in [F-thresholds and Bernstein-Sato polynomials], which are invariants of the pair of ideals on rings of characteristic $p$. In their paper, it is proved F-thresholds equal to jumping numbers for the test ideal on regular local rings. In this note, we give an formula of F-thresholds on toric rings. This formula is a generalization of the example in [Huneke, Mustata, Takagi and Watanabe:F-thresholds, tight closure, integral closure, and multiplicity bounds]. We prove that there exists an inequality between F-jumping coefficients and F-thresholds. In particular, we observe a comparison between F-pure thresholds and F-thresholds. As applications, we prove the characterization of regularity for toric rings defined by a simplicial cone, and the rationality of F-thresholds in some cases.