Finiteness of the Family of Rational and Meromorphic Mappings Into Algebraic Varieties

Abstract

Introduction. In this paper we deal with the higher dimensional version of de Franchis' theorem which says that there are only a finite number of nonconstant separable rational mappings of an algebraic curve into a complete algebraic curve with genus greater than 1. KobayashiOchiai [11] proved that there are only a finite number of dominant meromorphic mappings onto a complex space of general type, and its algebraic proof which covers the positive characteristic case was given by [14] (cf. also [23]). Here we shall consider the case where the mappings are not necessarily dominant. Let k be an algebraically closed field with arbitrary characteristic. All algebraic varieties and mappings (morphisms) in the following will be defined over k. Let R be an algebraic variety, V a smooth complete algebraic variety, and T(V) denote the tangent bundle over V. Set rank f = sup {dim R - dimtf-1(f (t)); t E R } for rational mappings