Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities

Abstract

Abstract. We study the multiplicity of the jumping numbers of an m-primary ideal a in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincaré series of multiplier ideals associated to a proving, in particular, that it is a rational function.

Multiplier ideals in two-dimensional local rings with rational singularities

“…The multiplier ideals corresponding to λ 0 = 0 and λ 1 = lct(m) = 4 9 are given by the antinef divisors D λ 0 = (2, 1, 1, 1, 1, 1) and D λ 1 = (3, 1, 1, 1, 1, 1). Notice that J (m λ 0 ) = m and, using the techniques of [1], we get that the codimension between these multiplier ideals is 4.…”

Multiplicities of jumping points for mixed multiplier ideals

“…In this section we are going to provide a systematic study of the multiplicity of any point c c c ∈ R r 0 . The results that we present are a natural generalization of the ones we obtained in [3]. Our first goal is to compute explicitly these multiplicities using the theory of jumping divisors in this mixed multiplier ideals setting as considered in [4].…”

“…In Section 2 we recall all the basics on mixed multiplier ideals. In Section 3 we extend the results of [3] to this setting. Namely, we make a systematic study of the multiplicities of points in the positive orthant.…”

Multiplicities of Jumping Numbers

“…This generalizes the work in dimension two of Alberich-Carramiñana, Àlvarez Montaner, Dachs-Cadefau and González-Alonso. They proved the dimension two case in [AADG17] by computing the multiplicities explicitly in terms of the intersection matrix of a log resolution. We use numerical intersection theory of divisors as developed by Kleiman in [Kle66].…”