Cubic Surface Bundles and the Brauer Group

In this note we define a subgroup Hⁱ _nr,π of unramified cohomology group Hⁱ_nr of a fibration π : X → S. This subgroup can be used efficiently in refined specialization arguments and allows to detect the failure of stable rationality of a variety specializing to X. We compute H²_nr,π systematically for many cubic surface bundles π : X → S over a smooth projective rational surface over an algebraically closed field: we give a combinatorial formula in terms of components of the discriminant divisor of π.