Rationality of Real Conic Bundles With Quartic Discriminant Curve.

The article focuses on the rationality of real conic bundle three-folds over the projective plane \(\mathbb{P}^2\) with smooth quartic discriminant curves, realized as real double covers of \(\mathbb{P}^1 \times \mathbb{P}^2\) branched over a \((2,2)\)-divisor. It classifies rationality behavior across the six real isotopy classes of smooth plane quartics, constructing rational examples in each class and characterizing rationality for four classes using the real locus topology and the intermediate Jacobian torsor (IJT) obstruction—a refinement of the classical intermediate Jacobian obstruction introduced by Hassett–Tschinkel and Benoist–Wittenberg. The study shows that while the IJT obstruction suffices to characterize rationality when the defining polynomial \(Q_1 Q_3 - Q_2^2\) is negative outside the quartic, it fails in other cases, notably when this polynomial is negative inside the quartic, where additional topological obstructions arise. The paper also provides explicit examples illustrating these phenomena, including cases where rationality is obstructed despite the vanishing of the IJT obstruction, and discusses open questions regarding rationality in certain isotopy classes.