Projective varieties covered by isotrivial families

Abstract:

Let $X,Y$ be projective schemes over a discrete valuation ring $R$, where $Y$ is generically smooth and $g: X \to Y$ is 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.