Multiplicities of Jumping Numbers

TL;DR

This paper investigates the multiplicities of jumping numbers in multiplier ideals, proving their multiplicity function is a quasi-polynomial and exploring their properties, especially for monomial ideals and Rees valuations.

Contribution

It establishes that the multiplicity function is a quasi-polynomial and connects its degree to Rees valuations, advancing understanding of jumping numbers in algebraic geometry.

Findings

Multiplicity function is a quasi-polynomial.

Poincaré series of jumping numbers is rational.

Characterization of highest degree components related to Rees valuations.

Abstract

We study multiplicities of jumping numbers of multiplier ideals in a smooth variety of arbitrary dimension. We prove that the multiplicity function is a quasi-polynomial, hence proving that the Poincar\'e series is a rational function. We further study when the various components of the quasi-polynomial have the highest possible degree and relate it to jumping numbers contributed by Rees valuations. Finally, we study the special case of monomial ideals.