Connectivity of the branch locus of moduli space of rational maps

TL;DR

This paper investigates the structure of the branch locus in the moduli space of rational maps, proving its connectivity for degrees three and higher, and clarifying the relationship between singular and branch loci.

Contribution

It provides a simple proof that the branch locus of the moduli space of rational maps is connected for degrees three and above, extending Milnor's observations.

Findings

The branch locus ${ m B}_d$ is connected for all degrees $d eq 2$.

For $d eq 2$, the singular locus ${ m S}_d$ coincides with the branch locus ${ m B}_d$.

The case $d=2$ features a disconnected branch locus, which is a cubic curve.

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.