F-thresholds and Bernstein-Sato polynomials

TL;DR

This paper introduces F-thresholds as invariants of singularities in positive characteristic, exploring their relation to Bernstein-Sato polynomials and how they connect to invariants in characteristic zero, emphasizing the role of arithmetic properties.

Contribution

It establishes a new link between F-thresholds and Bernstein-Sato polynomial roots, highlighting the arithmetic dependence of invariants across characteristics.

Findings

F-thresholds serve as positive characteristic analogues of multiplier ideal jumping coefficients.

The relation between invariants in characteristic zero and mod p reductions depends on the arithmetic properties of p.

A new connection between mod p invariants and Bernstein-Sato polynomial roots is described.

Abstract

We introduce and study invariants of singularities in positive characteristic called F-thresholds. They give an analogue of the jumping coefficients of multiplier ideals in characteristic zero. We discuss the connection between the invariants of an ideal in characteristic zero and the invariants of the different reduction mod p of this ideal. Our main point is that this relation depends on arithmetic properties of p. We also describe a new connection between invariants mod p and the roots of the Bernstein-Sato polynomial.