On Certain Uniformity Properties of Curves over Function Fields

Arguing by induction on the dimension, it is easy to show that the same holds for higher dimensional base varieties, more precisely, let V be a smooth variety over C and let T be a closed subset of V . Define Fg(V, T ) as the set of non-isotrivial families of smooth curves of genus g ≥ 2 over V − T . Then Fg(V, T ) is finite. Partial motivation for the Shafarevich conjecture was that, as Parshin shows in [P], the function field analog of the Mordell conjecture (Theorem M below, which had already been proved by Manin [M]) can be obtained as a corollary of the above Theorem P-A. Moreover, the same implication holds true for the number field analog of the conjectures of Shafarevich and Mordell.