Curve classes on conic bundle threefolds and applications to rationality

We undertake a study of conic bundle threefolds π: X → W over geometrically rational surfaces whose associated discriminant covers (Formula presented) → Δ (Formula presented) W are smooth and geometrically irreducible. We first show that the structure of the Galois module CH² Xk of rational equivalence classes of curves is captured by a group scheme that is a generalization of the Prym variety of (Formula presented) → Δ. This generalizes Beauville’s result that the algebraically trivial curve classes on Xk are parametrized by the Prym variety. We apply our structural result on curve classes to study the refined intermediate Jacobian torsor (IJT) obstruction to rationality introduced by Hassett-Tschinkel and Benoist-Wittenberg. The first case of interest is where W = P² and Δ is a smooth plane quartic. In this case, we show that the IJT obstruction characterizes rationality when the ground field has less arithmetic complexity (precisely, when the 2-torsion in the Brauer group of the ground field is trivial). We also show that a hypothesis of this form is necessary by constructing, over any k (Formula presented) R, a conic bundle threefold with Δ a smooth quartic where the IJT obstruction vanishes, yet X is irrational over k.