Frobenius powers

TL;DR

This paper generalizes Frobenius powers of ideals to real exponents, linking them to test and multiplier ideals, and explores their applications in studying singularities and hypersurfaces in prime characteristic.

Contribution

It introduces a new framework for Frobenius powers with real exponents, connecting them to existing concepts and analyzing their role in singularity theory.

Findings

Frobenius powers coincide with test ideals for principal ideals.

Frobenius powers exhibit critical exponents similar to jumping numbers.

Applications demonstrate Frobenius powers' refined measurement of singularities.

Abstract

This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincides with its test ideals, but appear to be a more refined measure of singularities in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.