Determining whether a given diophantine equation has a solution is a wide open question in number theory. For some varieties -- e.g. quadrics -- the existence of local points is enough to determine the existence of global points: this is known as the Hasse principle. Nevertheless, the latter does not hold for cubic forms, as shown by Selmer in 1951. Manin introduced in 1970 a set called the Brauer-Manin set, which is expected to describe all obstructions to the Hasse principle, at least for the wide family of rationally connected varieties.

In this talk, I shall present a work in progress which explains how this Brauer-Manin setting is related to fibrations over P^1, whenever the base field is the function field of a curve over a large finite field.