Automorphism loci for the moduli space of rational maps

TL;DR

This paper characterizes automorphism groups and singularities in the moduli space of degree d rational maps, determining possible automorphism groups, their loci, and implications for the space's geometric and algebraic structure.

Contribution

It provides a detailed description of automorphism loci, classifies possible automorphism groups, and analyzes singularities in the moduli space of rational maps.

Findings

Classified subgroups of automorphism groups for rational maps.

Determined dimensions of automorphism loci and singular loci.

Computed Picard and class groups of the moduli spaces.

Abstract

Let $k$ be an algebraically closed field of characteristic $0$ and $\mathcal{M}_d$ the moduli space of rational maps on $\mathbb{P}^1$ of degree $d$ over $k$. This paper describes the automorphism loci $A\subset \mathrm{Rat}_d$ and $\mathcal{A}\subset \mathcal{M}_d$ and the singular locus $\mathcal{S}\subset\mathcal{M}_d$. In particular, we determine which groups occur as subgroups of the automorphism group of some $[\phi]\in\mathcal{M}_d$ for a given $d$ and calculate the dimension of the locus. Next, we prove an analogous theorem to the Rauch-Popp-Oort characterization of singular points on the moduli scheme for curves. The results concerning these distinguished loci are used to compute the Picard and class groups of $\mathcal{M}_d, \mathcal{M}^s_d,$ and $\mathcal{M}^{ss}_d$.