Motivic Hilbert zeta functions for curve singularities and related invariants

In the present paper, we show that the motivic Hilbert zeta function for a curve singularity yields the generating functions for Euler numbers of punctual Hilbert schemes when any punctual Hilbert scheme admits an affine cell decomposition. This fact allows us to derive the relations among the motivic Hilbert zeta function and other invariants such as the generating function for semi-modules of the semi-group of the singularity, the HOMFLY polynomial and the degrees of Severi strata of the miniversal deformation of the singularity. As an application of the fact above, we also generalize Kawai’s result regarding the generating function for the Euler numbers of a singular curve.