2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78709Let g be an integer greater than 1. A uniform version of the Parshin-Arakelov theorem on the finiteness of the set of non-isotrivial curves of genus g over a function field, with fixed degeneracy locus, is proved. This is applied to obtain a uniform version of Manin theorem (the Mordell conjecture on rational points for curves of genus g over function fields). By a "function field" is meant the function field of a complex variety V of any dimension. If V is a curve, the uniform bounds will depend on g, on the genus of V and on the cardinality of the degeneracy locus.Improved exposition, added references, to appear on Comp. MathAlgebraic Geometry14Dxxtext