A REFINEMENT OF STEIN FACTORIZATION AND DEFORMATIONS OF SURJECTIVE MORPHISMS

摘要

This paper is concerned with a refinement of the Stein factorization, and with applications to the study of deformations of morphisms. We show that every surjective morphism f : X -> Y between normal projective varieties factors canonically via a finite cover of Y that is eale in codimension one. This "maximally etale factorization" satisfies a strong functorial property. It turns out that the maximally etale factorization is stable under deformations, and naturally decomposes an etale cover of the Hom-scheme into a torus and into deformations that are relative with respect to the rationally connected quotient of the target Y. In particular, we show that all deformations of f respect the rationally connected quotient of Y.