A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/ h has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the nonconnectedness of the subspace M g,R,eh of real elliptic-hyperelliptic algebraic curves in the moduli space M g,C of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space M g,R,eh of real elliptic-hyperelliptic surfaces of genus > 5: we show that M g,R,eh is connected if g is odd and has exactly two connected components if g is even; in both cases the closure M g,R,eh of M g,R,eh in the compactified moduli space M g,C is connected.
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