A refinement of Stein factorization and deformations of surjective morphisms

Details A refinement of Stein factorization and deformations of surjective morphisms A refinement of Stein factorization and deformations of surjective morphisms

2005-08-16

arxiv.org/abs/math/0508285v1

Mathematics

Algebraic Geometry

Scientific paper

This paper is concerned with a refinement of the Stein factorization, and with applications to the study of deformations of surjective morphisms. We show that every surjective morphism f:X->Y between normal projective varieties factors canonically via a finite cover of Y that is etale in codimension one. This "maximally etale factorization" is characterized in terms of positivity of the push-forward of the structure sheaf and satisfies a 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.

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