Connectivity of the branch locus of moduli space of rational maps

2015/02/10 by Hidalgo, Ruben A., Quispe, Saul
#37F10 #37P05 #37P45 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1502.03001

Abstract

Milnor proved that the moduli space \rm M_d of rational maps of degree d ≥ 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 M₂ with \mathbb C² and, within that identification, that \mathcal B₂ is a cubic curve; so \mathcal B₂ is connected and \mathcal S₂=∅. If d ≥ 3, then \mathcal S_d=\mathcal B_d. We use simple arguments to prove the connectivity of it.

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