Brauer groups of surfaces defined by pairs of polynomials

It is known that the Brauer group of a smooth, projective surface
defined by an equality of two homogeneous polynomials in characteristic 0, is
finite up to constants. I will report on new methods to determine these Brauer
groups, or at least their algebraic parts, as long as the coefficients are in a
certain sense generic. This generalises previous results obtained over the
years by Colliot-Thélène--Kanevsky--Sansuc, Bright, Uematsu and Santens.
(Joint work with A. N. Skorobogatov.)