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On moduli of real curves. (English) Zbl 0655.14011
Sémin. géométrie algébrique réelle, Paris 1986, Tome 2, Publ. Math. Univ. Paris VII 24, 85-95 (1986).
[For the entire collection see Zbl 0623.00004.]
The moduli space \(\bar M^ g\) of isomorphism classes of stable complex algebraic curves of genus g is known to be a projective variety. The subspace \(\bar M^ g({\mathbb{R}})\subset \bar M^ g\) of those curves definable by real polynomials is called the moduli space of real algebraic curves of genus g. It is shown that the space \(\bar M^ g({\mathbb{R}})\) is connected. This contrasts with the situation for \(M^ g({\mathbb{R}})\), the moduli space of smooth complex algebraic curves of genus g defined over the real numbers. This space is not connected.
Reviewer: N.Schwartz

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