Automorphism loci for the moduli space of rational maps

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 $\mathrm{R}\mathrm{a}\mathrm{t}$_d$and$$\mathcal{A}$\subset $\mathcal{M}$_d$andthesingularlocus$$\mathcal{S}$\subset$\mathcal{M}$_d$. In particular, we determine which groups occur as subgroups of the automorphism group of some $[\varphi ]\in {\mathcal{M}}_d$ 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}$^{ss}_d$.