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 ∆ → ∆ ⊂ W are smooth and geometrically irreducible.We first show that the structure of the Galois module CH 2 X k of rational equivalence classes of curves is captured by a group scheme that is a generalization of the Prym variety of ∆ → ∆.This generalizes Beauville's result that the algebraically trivial curve classes on X k 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 = P 2 and ∆ 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 ⊂ R, a conic bundle threefold with ∆ a smooth quartic where the IJT obstruction vanishes, yet X is irrational over k.