Israel Journal of Mathematics

, Volume 140, Issue 1, pp 145–155 | Cite as

On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces

  • José A. Bujalance
  • Antonio F. Costa
  • Ana M. Porto
Article

Abstract

LetX be a Riemann surface of genusg. The surfaceX is called elliptic-hyperelliptic if it admits a conformal involutionh such that the orbit spaceX/〈h〉 has genus one. The involutionh is then called an elliptic-hyperelliptic involution. Ifg>5 then the involutionh is unique, see [A]. We call symmetry to any anticonformal involution ofX. LetAut ±(X) be the group of conformal and anticonformal automorphisms ofX and letσ, τ be two symmetries ofX with fixed points and such that {σ, hσ} and {τ, hτ} are not conjugate inAut ±(X). We describe all the possible topological conjugacy classes of {σ, σh, τ, τh}. As consequence of our study we obtain that, in the moduli space of complex algebraic curves of genusg (g even >5), the subspace whose elements are the elliptic-hyperelliptic real algebraic curves is not connected. This fact contrasts with the result in [Se]: the subspace whose elements are the hyperelliptic real algebraic curves is connected.

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Copyright information

© The Hebrew University Magnes Press 2004

Authors and Affiliations

  • José A. Bujalance
    • 1
  • Antonio F. Costa
    • 1
  • Ana M. Porto
    • 1
  1. 1.Departamento de Matemáticas Fundamentales, Facultad de CienciasUNEDMadridSpain

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