Stably rational non rational varieties
type: Item
instance of: scholarly article
title: Stably rational non rational varieties
zbMATH Open document ID: 0589.14042
DOI: 10.2307/1971174
author: Arnaud Beauville
author: Jean-Jacques Sansuc
author: Jean-Louis Colliot-Thélène
author: Peter Swinnerton-Dyer
published in: Annals of Mathematics. Second Series
publication date: 1985-01-01T00:00:00Z
review text: 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 ?
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 3946348
zbMATH Keywords: stably rational function field
zbMATH Keywords: Zariski problem
zbMATH Keywords: function field of a conic bundle
zbMATH Keywords: intermediate jacobians
zbMATH Keywords: Prym varieties
reviewed by: M. Miyanishi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971174
OpenAlex ID: W2327651304
Wikidata QID: Q102005056
description: scientific article; zbMATH DE number 3946348