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

TL;DR

This paper investigates which Brauer classes do not obstruct the Hasse principle on certain algebraic varieties, showing that classes of order prime to a zero-cycle degree are non-obstructive, with applications to specific surfaces.

Contribution

It establishes conditions under which Brauer classes do not obstruct the Hasse principle, especially relating to prime order classes and zero-cycle degrees, with concrete applications.

Findings

Brauer classes of order prime to the zero-cycle degree do not obstruct the Hasse principle.

Odd torsion Brauer classes never obstruct the Hasse principle for specific surfaces.

Results apply to del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.

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.