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On the connectivity of the branch and real locus of \({\mathcal M}_{0,[n+1]}\)

  • Yasmina Atarihuana
  • Rubén A. HidalgoEmail author
Original Paper
  • 13 Downloads

Abstract

If \(n \ge 3\), then moduli space \({\mathcal M}_{0,[n+1]}\), of isomorphisms classes of \((n+1)\)-marked spheres, is a complex orbifold of dimension \(n-2\). Its branch locus \({\mathcal B}_{0,[n+1]}\) consists of the isomorphism classes of those \((n+1)\)-marked spheres with non-trivial group of conformal automorphisms. We prove that \({\mathcal B}_{0,[n+1]}\) is connected if either \(n \ge 4\) is even or if \(n \ge 6\) is divisible by 3, and that it has exactly two connected components otherwise. The orbifold \({\mathcal M}_{0,[n+1]}\) also admits a natural real structure, this being induced by the complex conjugation on the Riemann sphere. The locus \({\mathcal M}_{0,[n+1]}({\mathbb R})\) of its fixed points, the real points, consists of the isomorphism classes of those marked spheres admitting an anticonformal automorphism. Inside this locus is the real locus \({\mathcal M}_{0,[n+1]}^{\mathbb R}\), consisting of those classes of marked spheres admitting an anticonformal involution. We prove that \({\mathcal M}_{0,[n+1]}^{\mathbb R}\) is connected for \(n \ge 5\) odd, and that it is disconnected for \(n=2r\) with \(r \ge 5\) being odd.

Keywords

Riemann surfaces Automorphisms Teichmüller and moduli spaces 

Mathematics Subject Classification

30F10 30F60 32G15 

Notes

Acknowledgements

The authors would like to thanks to the anonymous referees for their valuable comments and suggestions which permitted to improve the presentation of this paper.

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Departamento de Matemática y EstadísticaUniversidad de la FronteraTemucoChile

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