Automorphism loci for the moduli space of rational maps
Volume 180 / 2017
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 $\mathcal{M}_d$, $\mathcal{M}_d^s$, and $\mathcal{M}_d^{ss}$.
