×

zbMATH — the first resource for mathematics

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 \(g\geq 2\).

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
[3] Bujalance, A combinatorial approach to groups of automorphisms of bordered Klein surfaces (1990) · Zbl 0709.14021 · doi:10.1007/BFb0084977
[4] Alling, Foundations of the theory of Klein surfaces (1971) · doi:10.1007/BFb0060987
[5] Macbeath, Canad. J. Math. 19 pp 1192– (1967) · Zbl 0183.03402 · doi:10.4153/CJM-1967-108-5
[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
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.