2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154876In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a 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.Replaced Introduction, Includes Minor RevisionsAlgebraic Geometry14B05;14H20;14H20text