Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Brauer groups of fibrations

Yanshuai Qin

Abstract

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.

On the Brauer groups of fibrations

Abstract

Let be a dominant morphism between smooth irreducible varieties over a finitely generated field such that the generic fiber is smooth, projective and geometrically connected. Assuming that is a curve with function field , we build a relation between the Tate-Shafarevich group for and the geometric Brauer groups for and , generalizing a theorem of Artin and Grothendieck for fibered surfaces to arbitrary relative dimension.

Paper Structure

This paper contains 16 sections, 19 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Sketch of the proof of Theorem \ref{['thm1.2']}
  5. Preliminary results
  6. The Galois invariant parts of geometric Brauer groups
  7. The Tate-Shafarevich group
  8. Cofiniteness of Brauer groups and Tate-Shafarevich groups
  9. Proof of the main theorem
  10. The left exactness in Theorem \ref{['thm1.2']}
  11. The pull-back trick
  12. Proof of Theorem \ref{['thm1.2']}
  13. Applications
  14. Proof of Theorem \ref{['finitebrauer']}
  15. Proof of Theorem \ref{['reduction']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k$ be a finitely generated field of characteristic $p\geq 0$. Let $\mathcal{X}$ be a smooth geometrically connected variety over $k$ and $C$ be a smooth projective geometrically connected curve over $k$ with function field $K$. Let $\pi:\mathcal{X}\longrightarrow C$ be a flat $k$-morphism such where $|C_{k^s}|$ denotes the set of closed points in $C_{k^s}$ and $K_v^{sh}$ denotes the fraction

Theorems & Definitions (37)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Proposition 2.1
  • ...and 27 more