Variétés stablement rationnelles non rationnelles. [D’après Beauville, Colliot-Thélène, Sansuc et Swinnerton-Dyer]. (Stably rational non-rational varieties. After Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer.) (French) Zbl 0605.14044

Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. 643, Astérisque 133/134, 223-236 (1986).
[For the entire collection see Zbl 0577.00004.]
Let \(k\) be a field and let \(X\) be an algebraic variety of dimension \(n\) defined over \(k\). \(X\) is called stably \(k\)-rational if \(X\times {\mathbb{P}}^ m_ k\) is k-birationally equivalent to \({\mathbb{P}}_ k^{m+n}\). A problem of Zariski asks if a stably \(k\)-rational variety is \(k\)-rational. This problem was recently solved negatively in the case \(n=3\) over \(k={\mathbb{C}}\) by A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc and D. Swinnerton-Dyer [Ann. Math. (2) 121, 283–318 (1985; Zbl 0589.14042)].
The present article gives a concise and very transparent survey on this problem and related topics. The existence of counterexamples does not imply the end of the problem, but only implies the richness and the complexity of the rationality problem.
Reviewer: M.Miyanishi


14M20 Rational and unirational varieties
14J30 \(3\)-folds
14G05 Rational points
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