Stably rational non rational varieties. (Variétés stablement rationnelles non rationnelles.) (French) Zbl 0589.14042

Beauville, Arnaud; Colliot-Théléne, Jean-Louis; Sansuc, Jean-Jacques; Swinnerton-Dyer, Peter

Ann. Math. (2) 121, 283-318 (1985).

Let k be a field and let X be a geometrically integral, algebraic variety of dimension d defined over k. If a product of X with the affine m-space \({\mathbb{A}}^ m_ k\) is birationally equivalent to the affine \((m+d)\)- space \({\mathbb{A}}_ k^{m+d}\) for some \(m>0\), X is called stably k- rational. In terms of the function field, if a field extension F/k adjoined with several variables \(F(t_ 1,...,t_ m)\) is k-isomorphic to a rational function field \(k(t_ 1,...,t_{m+d})\), F is called stably rational over k. The problem treated in the present paper is the following: Is a stably rational function field rational over k ? This has been called the ”Zariski problem” [cf. B. Segre [”Sur un problème de M. Zariski”, Colloques internat. Centre Nat. Rech. Sci., No.24, Algèbre et théorie des nombres, Paris 1949, 135-138 (1950; Zbl 0040.082)]. In the case where the dimension \(d=1\), the answer is affirmative for an arbitrary field k by Lüroth’s theorem, and in the case \(d=2\), the answer is affirmative provided k is algebraically closed.
The answer given in the present article is: \((1)\quad There\) is an example of irrational, stably rational function field of dimension 2 over k if k is a suitable, algebraically non-closed field; \((2)\quad there\) is an example of irrational, stably rational function field of dimension 3 over \({\mathbb{C}}.\)
The examples are constructed as the function field of a conic bundle. The irrationality criterion is to use the theory of intermediate jacobians and the Prym varieties. The integer m used in the above definition is 3 in the examples (1) and (2). Can one take m to be 1 ?

Reviewer: M.Miyanishi

MSC:

14M20	Rational and unirational varieties
14J30	\(3\)-folds
14J25	Special surfaces

Keywords:

stably rational function field; Zariski problem; function field of a conic bundle; intermediate jacobians; Prym varieties

Citations:

Zbl 0040.082