# zbMATH — the first resource for mathematics

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

##### MSC:
 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
Full Text: