Connectivity of the branch locus of moduli space of rational maps

Abstract

Milnor proved that the moduli space ${\mathrm{M}}_d$ of rational maps of degree $d\ge 2$ has a complex orbifold structure of dimension $2(d-1)$. Let us denote by ${\mathcal{S}}_d$ the singular locus of ${\mathrm{M}}_d$ and by ${\mathcal{B}}_d$ the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify ${\mathrm{M}}_2$ with ${\mathbb{C}}^2$ and, within that identification, that ${\mathcal{B}}_2$ is a cubic curve; so ${\mathcal{B}}_2$ is connected and ${\mathcal{S}}_2=\varnothing$. If $d\ge 3$, then ${\mathcal{S}}_d={\mathcal{B}}_d$. We use simple arguments to prove the connectivity of it.