  Let $\mathcal{X}\rightarrow C$ be a dominant morphism between smooth
irreducible varieties over a finitely generated field $k$ such that the generic
fiber $X$ is smooth, projective and geometrically connected. Assuming that $C$
is a curve with function field $K$, we build a relation between the
Tate-Shafarevich group for $\mathrm{Pic}^0_{X/K}$ and the geometric Brauer
groups for $\mathcal{X}$ and $X$, generalizing a theorem of Artin and
Grothendieck for fibered surfaces to arbitrary relative dimension.
