Conic bundles with nontrivial unramified Brauer group over threefolds

TL;DR

This paper develops a formula for the unramified Brauer group of certain rationally connected fourfolds, producing new examples with nontrivial groups and demonstrating their non-stable rationality using degeneration techniques.

Contribution

It introduces a formula for the unramified Brauer group of conic bundle fourfolds over threefolds and constructs new examples with nontrivial Brauer groups, advancing understanding of rationality properties.

Findings

Derived a formula for the unramified Brauer group of specific fourfolds.

Constructed new conic bundle examples over P^3 with nontrivial unramified Brauer group.

Proved the existence of universally CH_0-trivial resolutions for these fourfolds.

Abstract

We derive a formula for the unramified Brauer group of a general class of rationally connected fourfolds birational to conic bundles over smooth threefolds. We produce new examples of conic bundles over P^3 where this formula applies and which have nontrivial unramified Brauer group. The construction uses the theory of contact surfaces and, at least implicitly, matrix factorizations and symmetric arithmetic Cohen--Macaulay sheaves, as well as the geometry of special arrangements of rational curves in P^2. We also prove the existence of universally CH_0-trivial resolutions for the general class of conic bundle fourfolds we consider. Using the degeneration method, we thus produce new families of rationally connected fourfolds whose very general member is not stably rational.