Multiplier ideals in two-dimensional local rings with rational singularities

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

“…In [1] it was proved that, given a jumping number λ, every jumping divisor G satisfies G λ G H λ for some special jumping divisors G λ and H λ . These divisors are called respectively minimal and maximal jumping divisor, and the former is extensively studied in [1].…”

“…In [1] it was proved that, given a jumping number λ, every jumping divisor G satisfies G λ G H λ for some special jumping divisors G λ and H λ . These divisors are called respectively minimal and maximal jumping divisor, and the former is extensively studied in [1]. The aim of this section is to study the maximal one, which can be defined for any positive real number c and will play a prominent role in the rest of the paper.…”

“…One of the goals of this paper is to extend their result to the case of any m-primary ideal in a surface with a rational singularity at O. To do so we provide first a systematic study of the multiplicities using the theory of jumping divisors introduced in [1]. Another goal that we achieve is to give a simple numerical criterion (see Theorem 5.2) which characterizes whether any given rational number is a jumping number.…”

“…In Section §3 we review the notion of jumping divisors introduced in [1]. In fact we will be mainly interested in the maximal jumping divisor since it satisfies a nice periodicity property.…”

“…Another consequence of the formulas for the multiplicities is that we can describe those jumping numbers contributed by dicritical divisors. In particular we give in Theorem 5.5 a full description of the jumping numbers in the interval (1,2].…”