Pullback Method with Applications to Severi-Brauer Fibrations

TL;DR

The paper develops a general pullback construction that produces varieties carrying a Brauer–Manin obstruction to rational points from a given one, applicable even without explicit equations. It then analyzes Brauer groups of Severi–Brauer fibrations, showing that certain SB fibrations over $\mathbb{P}^1_k$ have injective Br$\,k$ into Br$X$ and possess a $p$-torsion quotient $\overline{\text{Br}}X$, with nontrivial pullbacks from cyclic algebras. Using the pullback method, the authors build SB fibrations whose Brauer–Manin obstructions are captured by a single $p$-torsion class for any odd prime $p$, and they construct index-one varieties that violate the Hasse principle yet have local solubility everywhere. These results provide explicit mechanisms for Brauer–Manin obstructions in SB fibrations and push forward the understanding of which Brauer classes control rational points on rationally connected varieties.

Abstract

Given a variety with a suitable Brauer class, we present a general pullback construction that produces varieties that has Brauer-Manin obstruction to the existence of rational points. We then study Severi-Brauer fibrations and their Brauer groups without relying on explicit defining equations. As a key application, we show that there exist Severi-Brauer fibrations with index one that fails Hasse principle.

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Definition 1.4
• Corollary 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• Theorem 2.3: Amitsur
• Theorem 2.4: Purity theorem