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On the connectedness of the locus of real Riemann surfaces. (English) Zbl 1030.14023
This paper contains a lot of interesting information concerning the connectedness of moduli spaces of compact Riemann surfaces with real models. A central role is played by the subset of those Riemann surfaces of genus \(g\) admitting a non-separating symmetry with precisely one oval (the locus of fixed points). This subset turns out to be the (only!) “spine” of the moduli space in question, i.e., it cuts the locus of the Riemann surfaces with real models of all other types. This has particularly nice consequences on connectedness properties of moduli spaces of real hyperelliptic curves. However, these properties do not extend to \(p\)-gonal curves for \(p>2\).
Almost all proofs rely on sophisticated constructions of suitable Fuchsian and non-Euclidean cristallographic groups.

14P25 Topology of real algebraic varieties
14H15 Families, moduli of curves (analytic)
30F10 Compact Riemann surfaces and uniformization
20H15 Other geometric groups, including crystallographic groups
14F45 Topological properties in algebraic geometry
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H10 Families, moduli of curves (algebraic)
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