On the Stable Birationality of Hilbert schemes of points on surfaces

The aim of this paper is to study the stable birational type of HilbⁿX, 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 HilbⁿX is stably birational to Hilbn'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 HilbⁿX's. As a corollary, we deduce the rationality of the motivic zeta function ζ(X,t) in K₀(Var/k)/([A¹ₖ])[[t]] over fields of characteristic zero.