Bujalance, Emilio; Gromadzki, Grzegorz; Singerman, David On the number of real curves associated to a complex algebraic curve. (English) Zbl 0820.30025 Proc. Am. Math. Soc. 120, No. 2, 507-513 (1994). 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. Cited in 1 ReviewCited in 5 Documents MSC: 30F10 Compact Riemann surfaces and uniformization 14H55 Riemann surfaces; Weierstrass points; gap sequences 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) Keywords:non-Euclidean crystallographic groups; bordered Riemann surfaces PDF BibTeX XML Cite \textit{E. Bujalance} et al., Proc. Am. Math. Soc. 120, No. 2, 507--513 (1994; Zbl 0820.30025) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.