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

Abstract

Let X be a smooth projective variety over a number field with a fibration into varieties that either satisfy a certain condition. We study the classes in the Brauer group of X 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.