Differentiating relatively

Given a sequence of algebraic points f(n) of a variety X over a characteristic 0-function field K of unbounded (normalised) height, we construct a limiting derivative in P(Omega(1)(X/K)). The real (Oesterle, 2002 [4]) "a, b, c" conjecture over function fields is an immediate corollary. In principle, every Mordellic problem over function fields reduces to a hyperbolicity problem on the generic fibre by way of the said construction, but, unfortunately, such a conclusion is delicate in the presence of bad reduction. This - as found in McQuillan (2001) [3, 4.3] - together with an alternative approach to the "a, b, c" conjecture by K. Yamanoi (2004) [5] has already been reported in the Seminaire Bourbaki (Gasbarri, 2008 [1])