On the order three Brauer classes for cubic surfaces

TL;DR

This paper presents a method to compute the Brauer-Manin obstruction for certain cubic surfaces over rationals, linking Brauer classes to Galois invariant Steiner trihedra and providing explicit examples.

Contribution

It introduces a novel approach to associate order three Brauer classes with Galois invariant triplets of Steiner trihedra on cubic surfaces.

Findings

All order three Brauer classes can be obtained via the proposed method.

Explicit examples demonstrate the effect of the Brauer-Manin obstruction.

The approach simplifies the computation of Brauer classes for specific cubic surfaces.

Abstract

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.