Multiplier ideals in two-dimensional local rings with rational singularities

Abstract

The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is studied extensively.

Monomial generators of complete planar ideals

“…This is an inductive procedure which was already described in the work of Enriques [13, IV.II.17] (see [9, §4.6] for more details). The version that we present here is the one considered in [5].…”

“…In the case of planar ideals, there are methods given by Järviletho [19], Naie [24] and Tucker [29] to compute the set of jumping numbers. The first two authors of this manuscript and Dachs-Cadefau [5] gave an algorithm that computes sequentially the list of jumping numbers of a planar ideal and the antinef divisor associated to the corresponding multiplier ideal.…”

“…· Compute the sequence of jumping numbers {λ j } j∈Z ≥0 and the divisor corresponding to the associated multiplier ideals {J (a λ j )} j∈Z ≥0 , i.e. the antinef closures D λ j of ⌊λ j F −K π ⌋, using the main algorithm of [5].…”

“…The jumping numbers smaller than 1 computed using the algorithm from [5] and generators for the associated multiplier ideals computed using Algorithm 3.5 can be found in Table 1.…”

Multiplicities of jumping points for mixed multiplier ideals

“…, 0) is the origin of the positive orthant R r 0 . Indeed, the mixed multiplier ideal J (a a a 0 ) is described by a divisor D 0 = e 0 j E j which is the antinef closure of −K π that can be computed using the unloading procedure described in [2]. Therefore, the log-canonical wall is supported on hyperplanes of the form e 1,j z 1 + · · · + e r,j z r = k j + 1 + e 0 j , j = 1, .…”