On the frequency of algebraic Brauer classes on certain log K3 surfaces

TL;DR

This paper investigates the algebraic Brauer group of certain log K3 surfaces defined by quadratic equations, showing that Brauer-Manin obstructions causing failures of the integral Hasse principle are rare among generic systems.

Contribution

It demonstrates that for a family of quadratic systems, the algebraic part of the Brauer group is typically trivial, and provides quantitative bounds on exceptions using resolvent constructions.

Findings

Algebraic Brauer group is trivial for most systems.

Failures of the integral Hasse principle due to algebraic Brauer-Manin obstruction are rare.

Quantitative bounds on the number of exceptions are established.

Abstract

Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer-Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer-Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.