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Complex algebraic curves with real moduli. (English) Zbl 0638.32020
The space \(M^ g\) of isomorphism classes of smooth complex algebraic curves of genus g, \(g>1\), is a quasiprojective variety that can be embedded into a complex projective space \({\mathbb{P}}^ N({\mathbb{C}})\) in such a way that the image of \(M^ g\) is defined by polynomials with rational coefficients. Then the complex conjugation defines an antiholomorphic self-mapping of \(M^ g\). In this paper we study the set \(M^ g({\mathbb{R}})\) of real points of \(M^ g\) in \({\mathbb{P}}^ N({\mathbb{C}}).\)
We show first that the moduli spaces \(M_{{\mathbb{R}}}^{p,n}\), which consists of the isomorphism classes of real algebraic curves of genus p with n distinguished points, is, in general, a real analytic and semi- algebraic space. We proceed then and show that \(M^ g({\mathbb{R}})\) is, in a natural way, the moduli space of genus g coverings of complex algebraic curves defined over \({\mathbb{R}}\). These constructions give also families of examples of complex algebraic curves that has real moduli but are not isomorphic to curves defined by real polynomials.
Reviewer: M.Seppälä

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F10 Compact Riemann surfaces and uniformization
14D20 Algebraic moduli problems, moduli of vector bundles
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