Multiplicities of jumping points for mixed multiplier ideals

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 Poincar\'e series for mixed multiplier ideals and prove its rationality. Finally, we study the set of divisors that contribute to the log-canonical wall.