# Curve classes on conic bundles threefolds and applications to rationality

Curve classes on conic bundles threefolds and applications to rationality

Abstract: In this talk I'll discuss joint work with Sarah Frei, Lena Ji, Soumya Sankar and Bianca Viray on the problem of determining when a geometrically rational variety is birational to projective space over its field of definition. Hassett--Tschinkel and Benoist--Wittenberg recently refined the classical intermediate Jacobian obstruction of Clemens--Griffiths by considering torsors under the intermediate Jacobian of a geometrically rational threefold. By work of Hassett--Tschinkel, Benoist--Wittenberg and Kuznetsov--Prokhorov, this obstruction is strong enough to characterize rationality of geometrically rational Fano threefolds of geometric Picard rank 1. Moving into higher Picard rank, we compute this obstruction for conic bundles over P^2. As a consequence of our work, when the ground field is the real numbers, we show that neither the topological obstruction nor the refined intermediate Jacobian obstruction is sufficient to determine rationality.