Mathematics – Algebraic Geometry
Scientific paper
2008-09-08
Mathematics
Algebraic Geometry
Lecture notes for a CIME summer school (Cetraro, September 2007), 56 pages, in French
Scientific paper
This survey, which contains very few proofs, addresses the general question: Over a given type of field, is there a natural class of varieties which automatically have a rational point? Fields under consideration here include: finite fields, p-adic fields, function fields in one or two variables over an algebraically closed field. Classical answers are given by the Chevalley-Warning theorem and by Tsen's theorem. More general answers were provided by a theorem of Graber, Harris and Starr and by a theorem of Esnault. The latter results apply to rationally connected varieties. We discuss these varieties from various angles : weak approximation, R-equivalence, Chow group of zero-cycles. Ongoing work on `rationally simply connected' varieties over function fields in two variables is also mentioned. A common thread in this report is the study of the special fibre of a scheme over a discrete valuation ring: if the generic fibre has a simple geometry, what does it imply for the special fibre?
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