Motivic classes of fixed-generators Hilbert schemes of unibranch curve singularities and Igusa zeta functions

Abstract

This paper delves into the study of Hilbert schemes of unibranch plane curves whose points have a fixed number of minimal generators. Building on the work of Oblomkov, Rasmussen and Shende we provide a formula for their motivic classes and investigate the relationship with principal Hilbert schemes of the same given unibranch curve. In addition, the paper specializes this study to the case of $(p,q)$-curves, where we obtain more structured results for the motivic classes of fixed-generators Hilbert schemes: their positivity and topological invariance, and an explicit relationship to one-generator schemes i.e. principal ideals in $\widehat{\mathcal{O}}_{C,0}$. Finally, we focus on a special open component in the one-generator locus, whose motivic class is naturally related to the motivic measure on the arc scheme $\mathbb A^2_\infty$ of the plane introduced by Denef and Loeser as well as to the Igusa zeta function. We also provide an explicit formulation of these motivic classes in terms of an embedded resolution of the singularity, proving their polynomiality as well as making them an interesting topological invariant of the given curve.