Vertical Brauer groups and degree 4 del Pezzo surfaces.

In this talk, I will show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical, that is, that every Brauer class is obtained by pullback from an element of Br k(P^1) for some rational map f : X ----> P^1. As a consequence, we see that a Brauer class does not obstruct the existence of a rational point if and only if there exists a fiber of f that is locally solvable. The proof is constructive and gives a simple and practical algorithm, distinct from that in [Bright,Bruin,Flynn,Logan (2007)], for computing all nonconstant classes in the Brauer group of X. This is joint work with Anthony V\'arilly-Alvarado.