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On real forms of a complex algebraic curve. (English) Zbl 0992.14011
The authors consider the number of real forms of a complex algebraic curve \(X\), i.e. the number of those non-isomorphic real algebraic curves whose complexifications are isomorphic to \(X\). Let \(g\geq 2\) be an integer and \(\omega(g)\) the maximal number of real forms that a complex algebraic curve of genus \(g\) can have. Using theory of non-Euclidean crystallographic groups, G. Gromadzki and M. Izquierdo proved that \(\omega(g)=4\) whenever \(g\) is even [Proc. Am. Math. Soc. 126, 765-768 (1998; Zbl 0913.20029)]. In the present paper, the authors extend these methods to arbitrary \(g\). More precisely, they determine \(\omega(g)\) for all \(g\geq 2\).

14H55 Riemann surfaces; Weierstrass points; gap sequences
30F10 Compact Riemann surfaces and uniformization
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
14H45 Special algebraic curves and curves of low genus
14P25 Topology of real algebraic varieties
Full Text: DOI
[1] DOI: 10.1090/S0002-9939-98-04735-2 · Zbl 0913.20029 · doi:10.1090/S0002-9939-98-04735-2
[2] DOI: 10.2307/2159888 · Zbl 0820.30025 · doi:10.2307/2159888
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[4] Alling, Foundations of the theory of Klein surfaces (1971) · doi:10.1007/BFb0060987
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[6] Natanzon, Trans. Moscow Math. Soc. 1989 pp 1–
[7] Soviet Math. Dokl. 19 pp 1195– (1978)
[8] Natanzon, Dokl. Akad. Nauk SSSR 242 pp 765– (1978)
[9] DOI: 10.1017/S0305004100048891 · doi:10.1017/S0305004100048891
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