On the Brauer groups of fibrations

Abstract

Let \({\mathcal {X}}\rightarrow C\) 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 \(\textrm{Pic}^0_{X/K}\) and the geometric Brauer groups of \({\mathcal {X}}\) and X, generalizing a theorem of Artin and Grothendieck for fibered surfaces to higher relative dimensions.