## EVENT DETAILS AND ABSTRACT

**Algebra Seminar**
**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.

**Date:** Tuesday, November 16, 1999

**Time:** 4:00pm

**Where:** Lunt 104

**Contact Person:** Prof. Ionut Ciocan-Fontanine

**Contact email:** ciocan@math.nwu.edu

**Contact Phone:** 847-467-1634

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