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A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

Published online by Cambridge University Press:  20 March 2017

RUBEN A. HIDALGO  and
SAÚL QUISPE
RUBEN A. HIDALGO
Affiliation:
Departamento de Matemática y Estadística, Universidad de La Frontera. Casilla 54-D, 4780000 Temuco, Chile e-mails: ruben.hidalgo@ufrontera.cl, saul.quispe@ufrontera.cl
SAÚL QUISPE
Affiliation:
Departamento de Matemática y Estadística, Universidad de La Frontera. Casilla 54-D, 4780000 Temuco, Chile e-mails: ruben.hidalgo@ufrontera.cl, saul.quispe@ufrontera.cl
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Abstract

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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.

Keywords

37F10
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 199 - 207
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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