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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

##### MSC:
 14J25 Special surfaces 14M10 Complete intersections
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