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4:00 pm Thursday, January 24, 2013 Colloquium: Vertical Brauer groups and del Pezzo surfaces of degree 4by Tony Várilly-Alvarado (Rice) in HB 227- Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space. They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface. I will explain how to construct, for every non-trivial, non-constant element A of Br X, 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 cohomological obstruction to the existence of a point on X arising from A, 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'' a Brauer-Manin obstruction. The construction also gives a fast, practical algorithm for computing the Brauer group of X. Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces. This is joint work with Bianca Viray.
Submitted by andyp@rice.edu |