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On the number of real curves associated to a complex algebraic curve. (English) Zbl 0820.30025
Summary: Using non-Euclidean crystallographic groups we give a short proof of a theorem of Natanzon that a complex algebraic curve of genus \(g\geq 2\) has at most \(2(\sqrt g+ 1)\) real forms. We also describe the topological type of the real curves in the case when this bound is attained. This leads us to solve the following question: how many bordered Riemann surfaces can have a given compact Riemann surface of genus \(g\) as complex double.

MSC:
30F10 Compact Riemann surfaces and uniformization
14H55 Riemann surfaces; Weierstrass points; gap sequences
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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[1] Norman L. Alling and Newcomb Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics, Vol. 219, Springer-Verlag, Berlin-New York, 1971. · Zbl 0225.30001
[2] Emilio Bujalance, José J. Etayo, José M. Gamboa, and Grzegorz Gromadzki, Automorphism groups of compact bordered Klein surfaces, Lecture Notes in Mathematics, vol. 1439, Springer-Verlag, Berlin, 1990. A combinatorial approach. · Zbl 0709.14021
[3] A. H. M. Hoare and D. Singerman, The orientability of subgroups of plane groups, Groups — St. Andrews 1981 (St. Andrews, 1981) London Math. Soc. Lecture Note Ser., vol. 71, Cambridge Univ. Press, Cambridge, 1982, pp. 221 – 227. · Zbl 0489.20036 · doi:10.1017/CBO9780511661884.014 · doi.org
[4] A. M. Macbeath, The classification of non-euclidean plane crystallographic groups, Canad. J. Math. 19 (1967), 1192 – 1205. · Zbl 0183.03402 · doi:10.4153/CJM-1967-108-5 · doi.org
[5] S. M. Natanzon, The order of a finite group of homeomorphisms of a surface onto itself, and the number of real forms of a complex algebraic curve, Dokl. Akad. Nauk SSSR 242 (1978), no. 4, 765 – 768 (Russian). · Zbl 0417.57003
[6] David Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29 – 38. · Zbl 0251.20052 · doi:10.1112/jlms/s2-6.1.29 · doi.org
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