Descent on fibrations over the projective line-猫眼课题宝

投影线上的纤维下降

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1996

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A. Skorobogatov

我们给出了j - l下降法的一般设置。Colliot-Thelene和J. J. Sansuc应用于数场投影线上的纤维品种。这就有可能证明，只要“简并”纤维很少（在某些情况下通常是两个，三个），并且“足够多”的光滑fc-纤维满足哈斯原理和弱近似，那么对哈斯原理和弱近似的曼宁障碍是唯一的障碍。我们引入了一个应该被称为“简并”纤维的新概念，即所谓的“分裂”纤维，其特性仅取决于一般纤维，而不取决于特定模型的选择。这产生了一种更简单和更通用的方法来处理前面的结果。0. 介绍。由二次型的Hasse定理，我们说Hasse原理适用于数域k9上的一组（几何积分的，光滑的和固有的）变分，如果对于这个族的每一个成员，在k的所有补全上有理点的存在意味着存在一个^-有理点。Hasse原理适用于某些族，这一事实极大地简化了对其算法的研究，因为确定是否存在^-有理点的问题被简化为有限个数的^-有理点

We formulate a general set-up for the descent method of J.-L. Colliot-Thelene and J. J. Sansuc applied to varieties fibred over the projective line over a number field. This makes it possible to prove that the Manin obstruction to the Hasse principle and weak approximation is the only one provided there are only few "degenerate" fibres (usually two, three in some cases), and that "sufficiently many" smooth fc-fibres satisfy the Hasse principle and weak approximation. We introduce a new concept of what should be called a "degenerate" fibre, the so called "split" fibres, the property which depends only on the generic fibre and not on the choice of a particular model. This yields a simpler and more general approach to the previous results of that kind. 0. Introduction. Motivated by Hasse's theorem on quadratic forms one says that the Hasse principle holds for a family of (geometrically integral, smooth and proper) varieties over a number field k9 if for every member of this family the existence of rational points over all the completions of k implies the existence of a ^-rational point. The fact that the Hasse principle holds for some family signif icantly simplifies the study of its arithmetic, because the problem of determining whether or not there is a ^-rational point is then reduced to a finite number of

Descent on fibrations over the projective line

投影线上的纤维下降

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