Intermediate Jacobian torsors and applications to rationality for conic bundles
Abstract: Clemens–Griffiths introduced the classical intermediate Jacobian obstruction to rationality for complex threefolds in their proof of the irrationality of the cubic threefold. Recently, over non-closed fields, Hassett–Tschinkel and Benoist–Wittenberg refined this obstruction using torsors over the intermediate Jacobian. In this talk, in the setting of conic bundle threefolds, we study this obstruction and its relation to the arithmetic of the ground field. As an application we construct geometrically rational, irrational examples of conic bundles over non-closed fields, where different obstructions witness irrationality. This work is joint with S. Frei, S. Sankar, B. Viray, and I. Vogt.