Connectivity of the branch locus of moduli space of rational maps

Ruben A. Hidalgo, Saul Quispe

Published 2015-02-10Version 1

The branch locus ${\mathcal B}_{d}$ in moduli space ${\rm M}_{d}$ of rational maps of degree $d \geq 2$ consits of the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor proved that ${\mathcal B}_{2}$ is a cubic curve; so connected. In this paper we see that ${\mathcal B}_{d}$ is always connected. As, for $d \geq 3$, the singular locus of ${\rm M}_{d}$ equals the branch locus, this also provides the connectivity of that locus.