Index of fibrations and Brauer classes that never obstruct the Hasse principle

Abstract

Let X be a smooth projective variety with a fibration into varieties that either satisfy a condition on representability of zero-cycles or that are torsors under an abelian variety. We study the classes in the Brauer group that never obstruct the Hasse principle for X. We prove that if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer-Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.Date: September 24, 2018.

Brauer–Manin obstructions requiring arbitrarily many Brauer classes

“…One can also ask about constraints on the orders of the elements in $$B$$. This and related questions were considered in work of Skorobogatov and Zarhin [19], Creutz and Viray [7], Creutz, Viray, and Voloch [8], and Nakahara [13]. See also [20] for a survey of these questions. (d)In this paper, we are interested in varieties with empty Brauer–Manin set.…”