Surveys in Differential Geometry
Volume 21 (2016)
Stable birational invariants and the Lüroth problem
Pages: 313 – 342
We describe recent progress on rationality and stable rationality questions. We discuss the cohomological or Chow decomposition of the diagonal, a very strong stably birationally invariant property which controls many of the previously defined stable birational invariants. On the other hand, it behaves very well under specialization and desingularization of mild singularities.