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Oberseminar WiSe 12/13 Zusammenfassung des Vortrags von Tony Varilly-Alvarado

Vertical Brauer groups and del Pezzo surfaces of degree 4

This is joint work with Bianca Viray. Let X be a locally soluble del Pezzo surface of degree 4 over a number field k. I will explain how to construct, for every non-trivial element A of Br X/ Br k, a rational genus-one fibration X --> P^1 such that A is vertical for this map. This implies, for example, that if there is a Brauer-Manin obstruction to the Hasse principle on X arising from a single Brauer class, then there is a genus-one fibration X --> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of ``seeing'' the Brauer-Manin obstruction.