Let $k$ be a number field and let $T$ be a $k$-torus. Consider a fibration in torsors under $T$, i.e. a morphism $f: X \to \mathbb{ P}^1_k$ from a smooth, projective $k$-variety $X$ to $\mathbb{ P}^1_k$ such that the generic fibre $X_\eta \to \eta$ is a smooth compactification of a principal homogeneous space under $T \times_k \eta$. We study the Brauer-Manin obstruction to the Hasse principle and weak approximation for $X$, under Schinzel's hypothesis, thereby generalizing recent work of Wei. Our results are unconditional if $k = \mathbb{ Q}$ and the non-split fibres of $f$ are defined over $\mathbb{ Q}$.
Categories: math.NT math.AG