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Birational invariants and decomposition of the diagonal. (English) Zbl 1442.14052
Hochenegger, Andreas (ed.) et al., Birational geometry of hypersurfaces. Lectures given at the “School on Birational Geometry of Hypersurfaces”, Gargnano del Garda, Italy, March 19–23, 2018. Cham: Springer. Lect. Notes Unione Mat. Ital. 26, 3-71 (2019).
Summary: We give a rather detailed account of cohomological and Chow-theoretic methods in the study of the stable version of the Lüroth problem, which ask how to distinguish (stably) rational varieties from general unirational varieties. In particular, we study the notion of Chow or cohomological decomposition of the diagonal, which is a necessary criterion for stable rationality. Having better stability properties than the previously known obstructions under specialization with mildly singular central fibers, it has been very useful in the recent study of rationality questions.
For the entire collection see [Zbl 1430.14003].

##### MSC:
 14E08 Rationality questions in algebraic geometry 14C25 Algebraic cycles 14M20 Rational and unirational varieties
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