Pullback Method with Applications to Severi--Brauer Fibrations

TL;DR

This paper introduces a pullback construction for varieties with Brauer classes, applies it to Severi--Brauer fibrations, and demonstrates the existence of fibrations with index one that violate the Hasse principle.

Contribution

It develops a general pullback method for varieties with Brauer classes and applies it to study Severi--Brauer fibrations without explicit equations, revealing new counterexamples to the Hasse principle.

Findings

Existence of Severi--Brauer fibrations with index one that fail the Hasse principle.

A new pullback construction for varieties with Brauer classes.

Analysis of Brauer groups of Severi--Brauer fibrations without explicit equations.

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.