On the Brauer groups of fibrations

Abstract

Let $\mathcal{X}\to C$ 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 ${\mathrm{Pic}}_{X/K}^0$ and the geometric Brauer groups for $\mathcal{X}$ and $X$, generalizing a theorem of Artin and Grothendieck for fibered surfaces to arbitrary relative dimension.

Final version to appear in Mathematische Zeitschrift

References in corpus (1)

• Boundedness of the $p$-primary torsion of the Brauer group of an abelian variety