Varieties with prescribed finite unramified Brauer groups and subgroups precisely obstructing the Hasse principle

On varieties defined over number fields, we consider obstructions to the Hasse principle given by subgroups of their Brauer groups. Given an arbitrary pair of non-zero finite abelian groups $B_0\subset B$, we prove the existence of a variety $X$ such that its unramified Brauer group is isomorphic to $B$ and moreover $B_0$ is the smallest subgroup of $B$ that obstructs the Hasse principle. The concerned varieties are normic bundles over the projective line.