Motivic classes of curvilinear Hilbert schemes and Igusa zeta functions

Abstract: This paper delves into the study of curvilinear Hilbert schemes associated with a singular variety $(X,0)$ and the relationship between their motivic classes and the motivic measure on the arc scheme $X_\infty$ of $X$ introduced by Denef and Loeser. We introduce an Igusa zeta function specifically tailored for curvilinear Hilbert schemes for which we provide an explicit formulation in terms of an embedded resolution of the singularity, and we consequently obtain a recursive formula to compute the motivic classes of curvilinear Hilbert schemes in terms of the resolution. In addition, the paper explores and analyzes the geometry and combinatorics of curvilinear Hilbert schemes in the context of plane curve singularities and their topological invariants.