Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder

Abstract

In a smooth family of projective, complex varieties, stable rationality need not be preserved under generisation. This was proved by Hassett, Pirutka and Tschinkel upon use of the specialisation method. Work of Schreieder produced many more examples and introduced a simplification of the specialisation method (no explicit desingularisation). In this text, I try to describe the combined method from scratch in one of the simplest cases. A small, further simplification consists in using specialisation of R-equivalence in place of Fulton's specialisation for the Chow group of zero-cycles.