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