On the FOD/FOM parameter of rational maps

Abstract

Let $\chi$ be a (right) action of ${\mathrm{PSL}}_2(\mathbb{L})$ 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 ${FOD/FOM}_{\chi }(R)$ is the smallest integer $n\ge 1$ 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 ${FOD/FOM}_{{\chi }_{\infty }}(R)\le 2$. In this paper, we study the above parameter for general actions and any characteristic.