Multiplicities of jumping points for mixed multiplier ideals

Abstract

In this paper we make a systematic study of the multiplicity of the jumping points associated to the mixed multiplier ideals of a family of ideals in a complex surface with rational singularities. In particular we study the behaviour of the multiplicity by small perturbations of the jumping points. We also introduce a Poincaré series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall.arXiv:1807.09839v1 [math.AG]

Poincaré series of multiplier and test ideals

“…Galindo and Monserrat [14] proved that this rationality property holds for multiplier ideals associated to simple m-primary ideals in a complex smooth surface and provided an explicit formula. These results were extended later on by Alberich-Carramiñana et al [1] (see also [2]) to the case of multiplier ideals associated to any m-primary ideal in a complex surface with rational singularities. The techniques used in both cases rely on the theory of singularities in dimension two and, in particular, the fact that the data coming from the log-resolution of any ideal can be encoded in a combinatorial object such as the dual graph.…”