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

arXiv commentSome corrections done. Comments Welcome !!

References in corpus (2)

• Questions on self maps of algebraic varieties
• Complete toric varieties with reductive automorphism group