Stably rational non rational varieties
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. \textit{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 ?
scientific article; zbMATH DE number 3946348