On the FOD/FOM parameter of rational maps

Abstract

Let $\chi$ be a (right) action of ${\rm PSL}_{2}({\mathbb L})$ on the space ${\mathbb L}(z)$ of rational maps defined over an algebraically closed field ${\mathbb L}$. If $R \in {\mathbb L}(z)$ and ${\mathcal M}_{R}^{\chi}$ is its $\chi$-field of moduli, then the parameter ${\rm FOD/FOM}_{\chi}(R)$ is the smallest integer $n \geq 1$ such that there is a $\chi$-field of definition of $R$ being a degree $n$ extension of ${\mathcal M}_{R}^{\chi}$. When ${\mathbb L}$ has characteristic zero and $\chi=\chi_{\infty}$ is the conjugation action, then it is known that ${\rm FOD/FOM}_{\chi_{\infty}}(R) \leq 2$. In this paper, we study the above parameter for general actions and any characteristic.