Degree 2 del Pezzo surface bundles and stable rationality

We study the arithmetic of del Pezzo surfaces $Y$ of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\mathrm{Pic}Y\to (\mathrm{Pic}\bar{Y}{)}^G$. Applying this to a fibration $\pi :X\to S$ in del Pezzo surfaces of degree 2 over a rational surface $S$, we construct examples with nontrivial relative unramified cohomology group $H_{nr,\pi }^2(k(X)\mathrm{/}k)$. A specialization argument implies the failure of stable rationality of varieties specializing to $X$.