It follows from a conjecture of J.-L. Colliot-Thélène [Bolyai Soc. Math. Stud. 12, 171–221 (2003; Zbl 1077.14029)] that if a smooth projective geometrically integral variety \(X\) over a number field \(k\) is geometrically rationally connected, with Br \(X/\)Br \(k=0\), then it satisfies the Hasse principle and weak approximation. In contrast, this paper gives a large class of varieties which “almost-always” fail to satisfy weak approximation.
To be specific, let \(k\) be a number field and let \(\pi:X\rightarrow\mathbb{P}^n\) be a flat surjective \(k\)-morphism of finite type, with \(X\) smooth, projective and geometrically integral over \(k\). Then the varieties considered are the fibres \(\pi^{-1}(P)\) for \(P\in \mathbb{P}^n(k)\cap\pi(X(\mathbb{A}_k))\). Subject to a list of 7 conditions, it is shown that for 100variety \(X_P\) is smooth, but fails weak approximation. Indeed a positive proportion asymptotically of these varieties have adèlic points, but the proportion with rational points is 0
As a nice example the paper considers diagonal cubic surfaces, \[ a_0X_0^3+a_1X_1^3+a_2X_2^3+a_3X_3^3=0. \] For these a positive proportion, around 86but it is shown that a vanishingly small proportion have rational points. (In contrast, the conjecture mentioned above would imply that the Hasse principle and weak approximation should hold for almost all general cubic surfaces with an adèlic point.) A similar result is obtained for diagonal quartic surfaces.