On the Stable Birationality of Hilbert schemes of points on surfaces

TL;DR

This paper investigates when Hilbert schemes of points on surfaces are stably birational, showing finiteness of classes for rational surfaces and implications for the rationality of the motivic zeta function.

Contribution

It establishes criteria for stable birationality of Hilbert schemes on surfaces and proves finiteness of classes for rational surfaces, linking to motivic zeta functions.

Findings

Finiteness of stable birational classes among Hilbert schemes on rational surfaces

Criteria for stable birationality on surfaces with irregularity zero

Rationality of the motivic zeta function over characteristic zero fields

Abstract

The aim of this paper is to study the stable birational type of $Hilb^n_X$, the Hilbert scheme of degree $n$ points on a surface $X$. More precisely, it addresses the question for which pairs of positive integers $(n,n')$ the variety $Hilb^n_X$ is stably birational to $Hilb^{n'}_X$, when $X$ is a surface with irregularity $q(X)=0$. After general results for such surfaces, we restrict our attention to geometrically rational surfaces, proving that there are only finitely many stable birational classes among the $Hilb^n_X$'s. As a corollary, we deduce the rationality of the motivic zeta function $\zeta(X,t)$ in $K_0(Var/k)/([\mathbb{A}^1_k])[[t]]$ over fields of characteristic zero.