Specialization of stable rationality

In this talk based on joint work with J. Nicaise I will explain how stable rationality behaves in families. The main result is that in characteristic zero stable rationality is preserved under specialization in smooth and nodal families. The proof is based on the Grothendieck ring of varieities and various forms of resolution of singularities. The method has been recently used to by Kontsevich and Tschinkel to prove similar results for specialization of rationality.