Brendan Hassett, Recent Progress and Questions on Stable Rationality

A complex variety X is rational if its field of meromorphic functions is isomorphic to C(t₁, …, t_d), the function field of projective space P^d. It is stably rational if X × P^m is rational for some m. Topological and complex invariants give criteria for whether varieties are stably rational but a simple characterization remains elusive. We present results for varieties admitting fibrations in conics, quadric hypersurfaces, and rational surfaces using techniques like decomposition of the diagonal. We close with some outstanding open problems. (joint with Kresch and Tschinkel)