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On moduli of real curves. (English) Zbl 0564.14012
Denote the space of isomorphism classes of stable complex curves of genus g by \(\bar M^ g\). This paper is an announcement of results concerning the subspace \(\bar M^ g({\mathbb{R}})\) of \(\bar M^ g\) consisting of curves that can be defined by real polynomials. The moduli space \(\bar M^ g\) admits a canonical antiholomorphic involution that takes the isomorphism class of a complex curve onto that of its complex conjugate. This is a real structure of \(\bar M^ g\) and \(\bar M^ g({\mathbb{R}})\) is the quasiregular real part of that real structure provided that \(g\geq 4\). The main result states that \(\bar M^ g({\mathbb{R}})\) is connected.

14H15 Families, moduli of curves (analytic)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14Pxx Real algebraic and real-analytic geometry
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