Multiplier ideals of plane curves

Abstract

Multiplier ideals and their corresponding jumping numbers have been revealed as a fundamental tool to measure the singularity of plane curves. There are approaches to these objects from both Commutative Algebra and Algebraic Geometry. The main goal of this work is, firstly, to introduce the basic concepts of the theory of multiplier ideals and of the singularity theory of plane curves. Secondly, the literature concerning multiplier ideals of plane curves, with special attention to the irreducible ones, has been reviewed, and the results have been translated in the language of Enriques diagrams, which are combinatorial objects encoding the equisingularity classes of plane curves. Finally, a new approach to the problems has been envisaged and has been illustrated through several examples. The computations needed in the examples have been carried out in Macaulay 2 and Maple, and the new routines implemented are included in the Annex.. Una mesura de les singularitats de corbes planes ve donada pels anomenats ideals multiplicadors i els corresponents nombres de salt.El treball consistirà, en primer lloc, en familiaritzar-se amb els conceptes bàsics de la teoria d'ideals multiplicadors i de la teoria de corbes planes. A continuació aprofundiriem en els treballs existents sobre ideals multiplicadors de corbes planes irreductibles. Un dels objectius seria reformular aquests treballs en el llenguatge de diagrames d'Enriques i intentar extendre els resultats al cas de corbes no irreductibles