Arithmetic of conic bundles

Elyes Boughattas (University of Bath)

Wednesday 16th October 16:00-17:00 Maths 311B

Abstract

For a family of conics X->P^1, the behaviour of rational points on X is not very well known, even though a conjecture of Colliot-Thélène and Sansuc seems to give a nice prediction using some cohomological invariants -- the Brauer group of X.

Following a question of Iksovskikh, it is natural to first ask whether rational points are Zariski dense in such surfaces, which is for instance true for unirational varieties, that is, varieties endowed with a dominant rational map from P^n.

In this talk, I will present a new result on unirationality and triviality of R-equivalence of conic bundles over finite fields (or more generally quasi-finite fields). This supplies a partial answer to a conjecture of Yanchevskii.

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