Motivic classes of curvilinear Hilbert schemes and Igusa zeta functions

TL;DR

This paper investigates the motivic classes of curvilinear Hilbert schemes linked to singular varieties, introduces an Igusa zeta function for these schemes, and provides formulas and analysis connecting their geometry, combinatorics, and singularity resolutions.

Contribution

It introduces an explicit Igusa zeta function for curvilinear Hilbert schemes and derives recursive formulas for their motivic classes using embedded resolutions.

Findings

Explicit formulation of the Igusa zeta function for curvilinear Hilbert schemes

Recursive formulas for motivic classes based on singularity resolutions

Analysis of the geometry and combinatorics of curvilinear Hilbert schemes in plane curve singularities

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.