### 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 ℙ^{N}. It is an interesting geometric problem to find the smallest N with this property.

Original language | English (US) |
---|---|

Pages (from-to) | 1163-1176 |

Number of pages | 14 |

Journal | International Journal of Mathematics |

Volume | 11 |

Issue number | 9 |

State | Published - Dec 2000 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Journal of Mathematics*,

*11*(9), 1163-1176.

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

Research output: Contribution to journal › Article

*International Journal of Mathematics*, vol. 11, no. 9, pp. 1163-1176.

}

TY - JOUR

T1 - Abelian fibrations and rational points on symmetric products

AU - Hassett, Brendan

AU - Tschinkel, Yuri

PY - 2000/12

Y1 - 2000/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034349741&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034349741&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034349741

VL - 11

SP - 1163

EP - 1176

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

IS - 9

ER -