Abstract

Given a family of rationally connected varieties over the projective line, the fibration method aims at constructing rational points on the total space.  We revisit this method and make it work fully when the locus of non-split fibres has degree at most 2, as well as, under Schinzel's hypothesis, when the bad fibres are split by a cyclic extension.  We do not make any arithmetic assumption on the smooth fibres, thus solving a problem left open since the 1990's.  This is a joint work with Yonatan Harpaz and Dasheng Wei.