EVENT DETAILS AND ABSTRACT

Title: Rational points on symmetric products of surfaces
Speaker: Professor Brendan Hassett
Speaker Info: University of Chicago
Brief Description:
Special Note:
Abstract:

Given a variety over a number field, we say that its rational points are potentially dense if there exists a finite extension over which rational points are Zariski dense. We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. We prove that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an abelian fibration over P^N. It is an interesting geometric problem to find the smallest N with this property. For instance, if S is a generic K3 surface of degree 2m^2 then the symmetric square of S admits an abelian fibration. This is joint work with Yuri Tschinkel.