Isotriviality and the space of morphisms on projective varieties

TL;DR

This paper investigates conditions under which morphisms on projective varieties over function fields are isotrivial and explores how morphisms between dynamical systems over such fields relate to their definitions over the base field.

Contribution

It establishes criteria linking potential good reduction to isotriviality and shows that morphisms intertwining dynamical systems over function fields are defined over the base field under certain conditions.

Findings

A $K$-morphism of degree at least two is isotrivial iff it has potential good reduction at all places.

Morphisms intertwining dynamical systems over $K$ are defined over $k$ if the system on the source is over $k$.

Provides conditions connecting isotriviality, good reduction, and field of definition for morphisms.

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 $k$.