Conic bundle threefolds differing by a constant Brauer class and connections to rationality

TL;DR

This paper investigates the rationality of certain conic threefolds over various fields, linking their geometric structures, Brauer classes, and Galois descent properties to determine conditions for rationality over real and local fields.

Contribution

It characterizes rationality conditions for conic bundle threefolds over real and local fields, connecting Brauer classes and Galois descent to geometric structures.

Findings

Rationality over algebraically closed fields is established.

Conditions for rationality over $\,\mathbb{R}$ are characterized based on the topology of $\,\Delta(\mathbb{R})$.

The difference between conic bundle structures is measured by a constant Brauer class.

Abstract

A double cover $Y$ of $\mathbb{P}^1 \times \mathbb{P}^2$ ramified over a general $(2,2)$-divisor will have the structure of a geometrically standard conic bundle ramified over a smooth plane quartic $\Delta \subset \mathbb{P}^2$ via the second projection. These threefolds are rational over algebraically closed fields, but over nonclosed fields, including over $\mathbb{R}$, their rationality is an open problem. In this paper, we characterize rationality over $\mathbb{R}$ when $\Delta(\mathbb{R})$ has at least two connected components (extending work of M. Ji and the second author) and over local fields when all odd degree fibers of the first projection have nonsquare discriminant. We obtain these applications by proving general results comparing the conic bundle structure on $Y$ with the conic bundle structure on a well-chosen intersection of two quadrics. The difference between these two conic bundles is encoded by a constant Brauer class, and we prove that this class measures a certain failure of Galois descent for the codimension 2 Chow group of $Y$.