Local-global principle for certain biquadratic normic bundles

Let $X$ be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\sqrt{a},\sqrt{b})\mathrm{/}k}(x)=Q(t_1,\mathrm{.}\mathrm{.}\mathrm{.},t_m{)}^2$ over a number field $k$. We prove that : (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of $X$; (2) the analogue for rational points is also valid assuming Schinzel’s hypothesis.