Multiplicities of Jumping Numbers

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.