On the Brauer group of affine diagonal quadrics

TL;DR

This paper investigates the Brauer group of affine diagonal quadrics, establishing that, similar to cubic surfaces, these quadrics do not possess uniform generators of their Brauer group.

Contribution

It extends the non-existence results of uniform Brauer group generators from diagonal cubic surfaces to affine diagonal quadrics.

Findings

Affine diagonal quadrics lack uniform Brauer group generators.

General diagonal cubic surfaces do not have uniform generators.

The non-existence result parallels previous findings for cubic surfaces.

Abstract

In a previous work, we introduced the notion of uniform generators of the Brauer group and proved that general diagonal cubic surfaces do not have such generators. In this paper, we prove that a similar non-existence result holds for affine diagonal quadrics.