Algebraic Geometry Seminar: On the stable birationality of Hilbert schemes of points on surfaces

In this talk, we will address the question for which pairs of integers (n,n') the variety Hilb^n_X is stably birational to Hilb^n'_X, when X is a surface with H^1(X,O_X)=0. In order to do so we will relate the existence of degree n' effective cycles on X with the existence of degree n ones using curves on X.
We will then focus on 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 a generalization of the Hasse-Weil zeta function Z(X, t) in K_0(Var/k)/([A^1_k])[[t]] when char(k) = 0.