Automorphism loci for the moduli space of rational maps

A Note on the Connectedness of the Branch Locus of Rational Maps

“…In fact, our description permits to see explicitly B d (C n ) as a Zariski open subset of Rat r for a suitable r (see the proof of Corollary 2). Consequences of such a description are that B d (C 2 ) is non-empty for every degree d ≥ 2 (Corollary 1) and that B d (C n ) (if non-empty) is connected (Corollary 2); we should say that this was also observed in Proposition 3 of [7].…”

“…We had realized that such a description was previously obtained in [7]. Ours description is more explicit and more adequate for our needs and a proof is provided in Section 2; our arguments are a little different, but follows the same general idea.…”

“…It was then natural to ask for the connectedness problem for the singular locus of moduli spaces of rational maps and this was the genesis of this paper. The techniques we use in this paper are quite similar to those used in [1,5], adapted to the case of rational maps, together with the description of rational maps with extra automorphisms as done in [7].…”

“…In this section we recall a description of those values of d for which G is admissible and the dimensions of B d (G) obtained in [7]. Since our main interest will be in the cyclic and dihedral cases, we present the explicit computations in those cases; in fact, we provide a more complete description as done in Lemmas 2 and 5 of [7] (see Theorems 2 and 3).…”

“…In this way, B 2 is connected. If d ≥ 3, then it was proved in [7] that S d = B d . In [8], Manes proved that the sublocus of S d consisting of those classes having a point of formal period N is geometrically reducible for infinitely many N. In [4], Fujimura proved that the singular locus of the moduli space of polynomial maps of degree three is connected (this being an irreducible algebraic curve of degree three).…”