Surfaces defined by pairs of polynomials

Abstract

We compute the Brauer group of surfaces defined by equating two bilinear forms of the same degree, assuming these forms are, in an explicit sense, sufficiently general. Our method uses a topological deformation argument and does not require full knowledge of the algebraic or transcendental cycles. We obtain a criterion for the triviality of the transcendental Brauer group of an isotrivial variety, which we use to prove that the Brauer group of the generic diagonal surface of arbitrary degree is trivial.