Arithmetic on some singular cubic hypersurfaces. (English) Zbl 0638.14011

Let k be a number field, and let \(X\subset {\mathbb{P}}\) \(n_ k\) (n\(\geq 3)\) be a cubic hypersurface defined over k. If X contains a set of 3 conjugate singular points, and if X has rational points over all completions of k, then X has a rational point over k. Moreover, if X is geometrically integral and is not a cone, and if \(n\neq 4\) then weak approximation holds for \(X_{smooth}\subset X\); in the case \(n=4\), the Brauer-Manin obstruction to weak approximation on \(X_{smooth}\) is the only one.
The case \(n=3\) of the theorem is known (Skolem, Segre, Coray). The proof of the case \(n>3\) is inductive. The main idea is to construct a hyperplane \(H\subset {\mathbb{P}}\) \(n_ k\) such that \(X\cap H\) contains the 3 conjugate singular points and such that \(X\cap H\) has rational points over each completion of k. The nature of this method is such that, if it is to succeed, then it gives results on weak approximation as well. Therefore, since weak approximation may not hold if \(n=4\), one needs additional arguments for some classes of cubic threefolds. It turns out that in all the dubious cases there are descent varieties which are essentially cubic hypersurfaces with 3 conjugate singular points. These descent varieties have a simpler arithmetical structure than the original variety and one may use the hyperplane section method to show that they satisfy the Hasse principle and weak approximation.
Reviewer: P.Salberger


14G05 Rational points
14G25 Global ground fields in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
11D25 Cubic and quartic Diophantine equations
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