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Intersections de deux quadriques et surfaces de Châtelet. (Intersections of two quadrics and Châtelet surfaces). (French) Zbl 0575.14044
From the summary: ”The Hasse principle and weak approximation hold for the smooth locus of certain intersections of two quadrics in n- dimensional projective space (n$$\geq 5)$$ over a number field. Using descent theory for rational surfaces one can then complete, and extend, the arithmetico-geometric analysis of Châtelet surfaces. In particular, this answers in the affirmative a question posed by Yu. I. Manin in his book ”Cubic forms, algebra, geometry, arithmetic” (Moscow 1972; Zbl 0255.14002; see also the English translation: Amsterdam 1974); chapter VI, end of § 5”.
Moreover the authors give the following proposition. Let k be a field, $$a\in k^*$$, a not a square and P(x)$$\in k[x]$$ an irreducible polynomial of degree three, with discriminant a. Let X be a projective model of the k-surface of affine equation $$y^ 2-az^ 2=P(x)$$ (X is called a generalized Châtelet surface). Then $$X\times_ k{\mathbb{P}}^ 9_ k$$ is k-birational with a $${\mathbb{P}}_ k^{11}$$, but X is not k-birational with a $${\mathbb{P}}^ 2_ k$$. This gives an answer in the negative to a problem of Zariski in the case of a non algebraically closed field. - This result will be developed in collaboration with A. Beauville and the authors will give, elsewhere, the same answer in the negative also in the case of an algebraically closed field. Such a result, with a different language, proves the existence of stable rational varieties which are not rational varieties.
Reviewer: E.Stagnaro

##### MSC:
 14M20 Rational and unirational varieties 14J25 Special surfaces 14M10 Complete intersections