A smooth complex projective variety is rational if it can be obtained from projective space by algebraic surgeries, i.e. blowups and blowdowns. It is stably rational if it becomes rational after takinga product with some projective space.
Consider a family of such varieties over a connected base. Which members are rational? Stably rational? We focus on recent general results and also outstanding questions that remain. These are illustrated in several key examples, including hypersurfaces of low
degree.
Joint work with Kresch, Pirutka, and Tschinkel.