AAG Seminar: On rationality criteria for threefolds over non-closed fields

Abstract: Let X be an algebraic variety over a non-closed field k. If X is rational over k, then it is geometrically rational, i.e. it is birational to projective space over the algebraic closure of k. However, in general, the converse need not hold. When the dimension of X is at most 2, then rationality over k is well-understood, but the picture is less clear starting in dimension 3. In this talk, we study k-rationality obstructions for geometrically rational threefolds. Recently, Hassett–Tschinkel and Benoist–Wittenberg refined the Clemens–Griffiths rationality obstruction by introducing torsors over the intermediate Jacobian. Their results, together with work of Kuznetsov–Prokhorov, showed that this refined obstruction can be used to characterize k-rationality for Fano threefolds of Picard rank 1. In joint work with S. Frei–S. Sankar–B. Viray–I. Vogt and joint work with M. Ji, we study the intermediate Jacobian torsor obstruction for threefolds of higher Picard rank, and we explain why this obstruction does not always characterize k-rationality in this setting.

Host: Roya Beheshti Zavareh