On the Stable Birationality of Hilbert schemes of points on surfaces

Abstract

The aim of this paper is to study the stable birational type of $Hil{b}_X^n$, the Hilbert scheme of degree $n$ points on a surface $X$. More precisely, it addresses the question for which pairs of positive integers $(n,n^{\prime })$ the variety $Hil{b}_X^n$ is stably birational to $Hil{b}_X^{n^{\prime }}$, 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 $Hil{b}_X^n$'s. As a corollary, we deduce the rationality of the motivic zeta function $\zeta (X,t)$ in $K_0(Var/k)/([{\mathbb{A}}_k^1])[[t]]$ over fields of characteristic zero.