John Christian Ottem (UiO): The Limits of Rationality

Abstract: Given a smooth family \(f:X\to B\) of varieties, it is natural to ask how the rationality varies in the fibers. In this talk I will survey two very recent breakthroughs in this direction. The first is a construction due to Hassett--Pirutka--Tschinkel (building on work of Voisin), showing that rationality is not a deformation invariant. The second is the recent results of Nicaise--Shinder and Kontsevich--Tschinkel, stating that rationality is a closed condition in smooth families. The proofs here revolve around constructing various specialization morphisms for the Grothendieck ring and Burnside ring of varieties. I will explain the main ingredients of the proofs of these results, and apply the theorems to some examples of specific interest (e.g., cubic fourfolds).