Multiplier Ideals in two-dimensional local rings with rational singularities

The aim of this memoir is to study multiplier ideals in two-dimensional local rings having at worst rational singularities. We also want to extend this study to the case of mixed multiplier ideals. The main achievements in the memoir are the following. We introduce a new method to compute the antinef closure of any given divisor, generalizing previous versions of Casas-Alvero and Reguera. 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 unique minimal and maximal jumping divisors that are studied extensively. We study the multiplicities of jumping numbers of M-primary ideals. 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 any ideal, proving, that it is a rational function. The results obtained above are generalized to the case of mixed multiplier ideals. More precisely, we present a method to compute the jumping walls and the different mixed multiplier ideals in any compact of R^r. This method is implemented as an algorithm that computes the jumping walls for a given family of ideals. We also generalize the notion of jumping divisor and we endow the jumping walls with a notion of multiplicity.Introduction 1 Preliminaries 2 An algorithm to compute jumping numbers and multiplier ideals 3 Jumping Divisors 4 Multiplicities and Poincaré series 5 Mixed multiplier ideals A Examples of jumping walls B Package MultiplierIdealsDim2.m2nrpages: 172status: publishe