Isotriviality and the space of morphisms on projective varieties

Abstract

Let $K=k(C)$ be the function field of a smooth projective curve $C$ over an infinite field $k$, let $X$ be a projective variety over $k$. We prove two results. First, we show with some conditions that a $K$-morphism $\phi: X_K \to X_K$ of degree at least two is isotrivial if and only if $\phi$ has potential good reduction at all places $v$ of $K$. Second, let $(X,\phi), (Y,\psi)$ be dynamical systems where $X,Y$ are defined over $k$ and $g:X_{K} \to Y_{K}$ a dominant $K$-morphism, such that $g \circ \phi = \psi \circ g$. We show under certain conditions that if $\phi$ is defined over $k$, then $\psi$ is defined over $k$.