# Degree and the Brauer-Manin obstruction

## TL;DR

This paper investigates whether the Brauer-Manin obstruction to the Hasse principle for certain algebraic varieties is determined by specific primary parts of the Brauer group, with positive results for some classes and counterexamples for others.

## Contribution

It proves that for torsors under abelian varieties, Kummer surfaces, and bielliptic surfaces (conditionally), the obstruction is given by the d-primary subgroup, and provides a counterexample in the general case.

## Key findings

- For torsors under abelian varieties, the obstruction is d-primary.
- Kummer surfaces have a 2-primary Brauer-Manin obstruction.
- A constructed conic bundle over an elliptic curve shows the general answer is no.

## Abstract

Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show that the obstruction is already given by the 2-primary torsion. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.