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Intersections of two quadrics and Châtelet surfaces. I. (English) Zbl 0622.14029
The authors generalize, extend and prove many new results about the arithmetic of intersections of two quadrics over a number field k. It is impossible to mention all, even most important, results of this fundamental work. We assume that \(X=Q_ 1\cap Q_ 2\subset {\mathbb{P}}^ n\) is geometrically irreducible, reduced, not a cone and \(n\geq 4\), let \(\bar X\) denote a nonsingular model of X. First of all the authors locate the most exceptional cases. These are the case (E) in which \(n=5\) and X is contained in at least 2 quadrics of rank \(\leq 4\), and the case (E’) in which \(n=4\) or 5, and where X is contained in a pair of quadrics of rank \(\leq 4\) which is Gal\((\bar k/k)\)-invariant. The authors prove that if \(n=5\) and X does not belong to case (E), or \(n=4\) and X is smooth and does not belong to case (E’) then \(Br(\bar X)/Br(k)=0\), and there are no obstructions to the Hasse principle and to the weak approximation on \(\bar X.\) In some cases the authors establish the k-rationality of X. For example, this is the case when X contains a k-rational nonsingular point. Many results concern the R-equivalence relation on X. For example, it is proven that if X is smooth, \(n\geq 7\), \(X(k)\neq \emptyset\), then \(X(k)/R=\oplus \pi_ 0(X(k_ v)),\) where the sum is taken along the set of all real archimedean valuations of k. Among some special results we mention one which asserts that any two quadratic forms in \(n\geq 9\) variables over a totally imaginary field k have a non-trivial common zero.
Reviewer: I.Dolgachev

14J25 Special surfaces
14M10 Complete intersections
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