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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The authors are supported by BFM2002-04801.
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Bujalance, J.A., Costa, A.F. & Porto, A.M. On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces. Isr. J. Math. 140, 145–155 (2004). https://doi.org/10.1007/BF02786630
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DOI: https://doi.org/10.1007/BF02786630