Connectivity of the branch locus of moduli space of rational maps

Abstract: Milnor proved that the moduli space ${\rm M}{d}$ of rational maps of degree $d \geq 2$ has a complex orbifold structure of dimension $2(d-1)$. Let us denote by ${\mathcal S}{d}$ the singular locus of ${\rm M}{d}$ 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 ${\rm M}2$ with ${\mathbb C}^2$ and, within that identification, that ${\mathcal B}{2}$ is a cubic curve; so ${\mathcal B}{2}$ is connected and ${\mathcal S}{2}=\emptyset$. If $d \geq 3$, then ${\mathcal S}{d}={\mathcal B}{d}$. We use simple arguments to prove the connectivity of it.