The University of Arizona
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A relation between Brauer groups and Tate-Shafarevich groups for high dimensional fibrations

Algebra and Number Theory Seminar

A relation between Brauer groups and Tate-Shafarevich groups for high dimensional fibrations
Series: Algebra and Number Theory Seminar
Location: ONLINE
Presenter: Yanshuai Qin, UC Berkeley

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