Cubic surface bundles and the Brauer group

In this note we define a subgroup $H^i_{nr,\pi}$ of unramified cohomology group $H^i_{nr}$ of a fibration $\pi:X\to S$. 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 $H^2_{nr, \pi}$ systematically for many cubic surface bundles $\pi:X\to S$ 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 $\pi$.