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

  • RUBEN A. HIDALGO (a1) and SAÚL QUISPE (a1)

Abstract

Milnor proved that the moduli space 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 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 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 .

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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