Rationally connected varieties over the maximally unramified extension of p-adic fields

Details Rationally connected varieties over the maximally unramified extension of p-adic fields Rationally connected varieties over the maximally unramified extension of p-adic fields

2009-06-12

arxiv.org/abs/0906.2364v1

Mathematics

Algebraic Geometry

16 pages

Scientific paper

A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here we show that rationally connected varieties over the maximally unramified extension of the p-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result saying that the p-adics are usually $C_{2}$ fields. The method of proof utilizes a construction from mathematical logic called the ultraproduct. The ultraproduct is used to lift the de Jong, Starr result in the equicharacteristic case to the mixed characteristic case.

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