Jumping Numbers and Multiplier Ideals on Algebraic Surfaces.

Abstract

This dissertation studies certain invariants of singularities of complex algebraic surfaces. In the first main result, we show that all integrally closed ideals on log terminal surfaces are multiplier ideals by extending an existing proof for smooth surfaces. Next, we turn our attention to the computation of the jumping numbers of an ideal in the local ring of a surface at a rational singularity. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of any plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored. We end by giving a new and insightful computation of the jumping numbers of the germ of a unibranch plane curve.