Unramified Brauer groups of conic bundle threefolds in characteristic two

TL;DR

This paper develops a formula for calculating the unramified Brauer group of tame conic bundle threefolds in characteristic 2, revealing new examples of non-stably rational threefolds over the integers.

Contribution

It introduces a novel formula for the unramified Brauer group in characteristic 2, utilizing Artin-Schreier theory, and provides new examples of threefolds that are not stably rational.

Findings

Derived a formula for the unramified Brauer group in characteristic 2

Identified conditions for non-stable rationality of threefolds

Provided explicit examples over the integers

Abstract

We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic 2. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by Artin-Schreier theory (instead of Kummer theory in characteristic not 2). We use this to give new examples of threefold conic bundles defined over the integers that are not stably rational over the complex numbers.