Algorithm for motivic Hilbert zeta function of some curve singularities

TL;DR

This paper presents algorithms to compute motivic Hilbert zeta functions for certain curve singularities, providing new computational tools and insights into the geometry of Hilbert schemes on singular curves.

Contribution

The authors develop and implement algorithms to compute motivic Hilbert zeta functions for specific classes of curve singularities, extending understanding of their geometric properties.

Findings

Algorithms successfully compute motivic Hilbert zeta functions for selected singularities.

Approximation of infinite semigroup  by finite truncation enables effective computation.

Complexity analysis guides the choice of truncation length for reliable results.

Abstract

We develop algorithms to compute two versions of the motivic Hilbert zeta function for curve singularities: the classical version, applicable to singularities with a monomial valuation semigroup or to singular curves defined by \(y^{k}=x^{n}\) with \(\gcd(k,n)=1\), and a finer version introduced by the first and third authors together with Mounir Hajli, which currently applies to the specific family \(y^{k}=x^{n}\) where \(\gcd(k,n)=1\). It is well known that the Hilbert scheme of points on a smooth curve is isomorphic to the symmetric product of the curve. However, the geometry of the Hilbert scheme of points on singular curves remains much less understood. Our algorithms compute the motivic Hilbert zeta functions \[ Z_{(C,O)}^{\mathrm{Hilb}}(q) \in K_{0}(\mathrm{Var}_{\mathbb{C}})[[q]], \qquad Zm_{(C,O)}^{\mathrm{Hilb}}(a^{2},q^{2}) \in K_{0}(\mathrm{Var}_{\mathbb{C}})[[a^{2}, q^{2}]], \] for such curve singularities, expressed as formal power series with coefficients in the Grothendieck ring of complex varieties. The main computational difficulty arises from the fact that \(\Gamma\) is infinite. To overcome this, we approximate \(\Gamma\) by truncating it to a suitable finite subset, which allows the algorithms to run effectively. We analyze the time complexity of the method and provide an estimate for the effective finite length of \(\Gamma\) required to obtain reliable results. A Python implementation of the algorithms is available at https://github.com/whaozhu/motivic_hilbert.