On the Brauer groups of fibrations

Let X→ C X → C be a flat k -morphism between smooth integral 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 of Pic⁰X/K Pic X / K 0 and the geometric Brauer groups of X X and X , generalizing a theorem of Artin and Grothendieck for fibered surfaces to higher relative dimensions.