  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.
