Cubic surface bundles and the Brauer group

In this note we define a subgroup $H_{nr,\pi }^i$ of unramified cohomology group $H_{nr}^i$ 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 $H_{nr,\pi }^2$ 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 $\pi$.