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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.

##### MSC:
 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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