Rationality of Real Conic Bundles With Quartic Discriminant Curve

We study real double covers of P¹ × P² branched over a (2, 2)-divisor, which are conic bundles with smooth quartic discriminant curve by the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space is R-rational. For five of the six isotopy classes, we construct C-rational examples with obstructions to rationality over 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 characterize rationality using the real locus and the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg. These double cover models were introduced by Frei, Sankar, Viray, Vogt, and the first author, who determined explicit descriptions for their intermediate Jacobian torsors.