Degree and the Brauer–Manin obstruction

Abstract

Let $X\subset {\mathbb{P}}_k^n$ be a smooth projective variety of degree $d$ over a number field $k$ 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. In the case of Kummer surfaces we show, more specifically, that the obstruction is already given by the $2$-primary torsion, and indeed that this holds for higher-dimensional Kummer varieties as well. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.