$F$-jumping and $F$-Jacobian ideals for hypersurfaces

TL;DR

This paper introduces $F$-jumping and $F$-Jacobian ideals to analyze hypersurface singularities in positive characteristic, providing new characterizations, algorithms, and relations among invariants.

Contribution

It defines new families of ideals, establishes their properties, and connects them to existing invariants, advancing the understanding of hypersurface singularities in positive characteristic.

Findings

Characterization of $F$-jumping numbers for hypersurfaces

An algorithm to determine $F$-jumping numbers

Conditions for $F$-regularity of hypersurfaces

Abstract

We introduce two families of ideals, $F$-jumping ideals and $F$-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the $D$-modules $M_{\alpha}$ that were introduced by Blickle, Musta\c{t}\u{a} and Smith. Using strong connections between $F$-jumping ideals and generalized test ideals, we give a characterization of $F$-jumping numbers for hypersurfaces. Furthermore, we give an algorithm that determines whether certain numbers are $F$-jumping numbers. In addition, we use $F$-Jacobian ideals to study intrinsic properties of the singularities of hypersurfaces. In particular, we give conditions for $F$-regularity. Moreover, $F$-Jacobian ideals behave similarly to Jacobian ideals of polynomials. Using techniques developed to study these two new families of ideals, we provide relations among test ideals, generalized test ideals, and generalized Lyubeznik numbers for hypersurfaces.