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 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 HilbⁿX's. As a corollary, we deduce the rationality of the motivic zeta function (X, t) in K₀ (Var/k) / (A¹ₖ) [t] over fields of characteristic zero.