On certain uniformity properties of curves over function fileds

1999/06/23 by Lucia Caporaso, Caporaso, Lucia · 1 citation
Mathematics · #14Dxx #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:14Dxx

paper · pdf · doi:10.48550/arxiv.math/9906156

Improved exposition, added references, to appear on Comp. Math

openalex publication_date 1999/06/23 · arxiv created 2000/11/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let 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.

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