Rationality of real conic bundles with quartic discriminant curve

TL;DR

This paper investigates the rationality of real conic bundle threefolds with quartic discriminant curves, constructing examples across all isotopy classes and analyzing obstructions to rationality using topology and intermediate Jacobian torsors.

Contribution

It provides the first comprehensive analysis of rationality for real conic bundles with quartic discriminant curves across all isotopy classes, including explicit constructions and obstructions.

Findings

Constructed rational examples in all isotopy classes.

Identified obstructions to rationality over  for five classes.

Characterized rationality using topology and intermediate Jacobian torsors.

Abstract

We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we give characterizations of rationality using the topology of the real locus and the intermediate Jacobian torsor obstruction of Hassett--Tschinkel and Benoist--Wittenberg. The double cover models we consider were introduced and previously studied by S. Frei, S. Sankar, B. Viray, I. Vogt, and the first author.