Abelian fibrations and rational points on symmetric products

Brendan Hassett, Yuri Tschinkel

Research output: Contribution to journalArticle

Abstract

Given a variety over a number field, are its rational points potentially dense, i.e. does there exist 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. Our main result is 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 ℙN. It is an interesting geometric problem to find the smallest N with this property.

Original languageEnglish (US)
Pages (from-to)1163-1176
Number of pages14
JournalInternational Journal of Mathematics
Volume11
Issue number9
StatePublished - Dec 2000

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Symmetric Product
Rational Points
Fibration
K3 Surfaces
Number field
Curve

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Abelian fibrations and rational points on symmetric products. / Hassett, Brendan; Tschinkel, Yuri.

In: International Journal of Mathematics, Vol. 11, No. 9, 12.2000, p. 1163-1176.

Research output: Contribution to journalArticle

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