  Let k be a field, X a smooth, projective k-variety. If X is geometrically
rational, there is an injective map from the quotient of Brauer groups
Br(X)/Br(k) into the first Galois cohomology group of the lattice given by the
geometric Picard group. In this note, where the main attention is on smooth
compactifications of homogeneous spaces of algebraic k-tori, we show how under
some hypotheses the map is onto, and how one may in some special case exhibit
concrete generators in Br(X). This is applied to the analysis of
counterexamples to the local-global principle for norms in biquadratic
extensions of number fields.
