Descent on fibrations over the projective line

Source abstract

We formulate a general set-up for the descent method of J.-L. Colliot-Thélène 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 k -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.