Which ideals are multiplier ideals?
Multiplier ideals are always complete (integrally closed), but the converse is not true in general. While Lazarsfeld-Lee have found counterexamples in smooth threefolds, the picture is completely different for surfaces: Lipman-Watanabe and Favre-Jonsson proved that every complete ideal at a smooth surface point is a multiplier ideal. This result was extended by Tucker to log-terminal singularities, but his proof cannot be generalized to worse singularities (not even log-canonical ones), where the question is completely open despite the efforts of the preceding authors. In this talk I will discuss some results in a joint work in progress with M. Alberich-Carramiñana and J. Àlvarez Montaner, introducing a new approach to the problem that applies to any (normal) surface singularity.