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Isotriviality and the space of morphisms on projective varieties

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 $ϕ: X_K \to X_K$ of degree at least two is isotrivial if and only if $ϕ$ has potential good reduction at all places $v$ of $K$. Second, let $(X,ϕ), (Y,ψ)$ be dynamical systems where $X,Y$ are defined over $k$ and $g:X_{K} \to Y_{K}$ a dominant $K$-morphism, such that $g \circ ϕ= ψ\circ g$. We show under certain conditions that if $ϕ$ is defined over $k$, then $ψ$ is defined over $k$.