On the Brauer group of diagonal cubic surfaces

TL;DR

This paper investigates the structure of the Brauer group of diagonal cubic surfaces, introducing the concept of uniform generators, and shows that such generators exist only for specific classes, highlighting limitations of previous results.

Contribution

It introduces the notion of uniform generators for the Brauer group and demonstrates their existence for certain diagonal cubic surfaces, while proving their non-existence in general cases.

Findings

Uniform generators exist for some classes of diagonal cubic surfaces.

The Brauer group of general diagonal cubic surfaces lacks uniform generators.

Manin's results cannot be universally generalized to all diagonal cubic surfaces.

Abstract

We are concerned with finding explicit generators of the Brauer group of diagonal cubic surfaces in terms of norm residue symbols, which was originally studied by Manin. We introduce the notion of uniform generators and find that the Brauer group of some classes of diagonal cubic surfaces have uniform generators. However, we also prove that the Brauer group of general diagonal cubic surfaces do not have such ones. This reveals that a result of Manin for certain diagonal cubic surfaces cannot be generalized in some sense.