Geodesic
A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS
Glasgow mathematical journal, Tome 60 (2018) no. 1, pp. 199-207

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Milnor proved that the moduli space Md 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 Md 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 M2 with C2 and, within that identification, that ${\mathcal B}$ 2 is a cubic curve; so ${\mathcal B}$ 2 is connected and ${\mathcal S}$ 2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$ d = ${\mathcal B}$ d . In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$ d .
DOI : 10.1017/S0017089516000665
Mots-clés : 37F10 
HIDALGO, RUBEN A.; QUISPE, SAÚL. A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS. Glasgow mathematical journal, Tome 60 (2018) no. 1, pp. 199-207. doi: 10.1017/S0017089516000665
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