  Let X,Y be projective schemes over a discrete valuation ring R, where Y is
generically smooth and g:X \to Y a surjective R-morphism such that
g_*\mathcal{O}_X = \mathcal{O}_Y. We show that if the family X \to Spec(R) is
isotrivial, then the generic fiber of the family Y \to Spec(R) is isotrivial.
