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On real forms of a complex algebraic curve

Part of: Special aspects of infinite or finite groups Curves Riemann surfaces Other groups of matrices

Published online by Cambridge University Press:  09 April 2009

E. Bujalance ,
G. Gromadzki  and
M. Izquierdo
E. Bujalance
Affiliation:
Depto de Matemáticas Fund. UNED c/ Senda del Rey s/n 28040 MadridSpain e-mail: eb@mat.uned.es
G. Gromadzki
Affiliation:
Institute of Mathematics UG Wita Stwosza57 80-952 GdańskPoland e-mail: greggrom@ksinet.univ.gda.pl
M. Izquierdo
Affiliation:
Department of Mathematics Mälardalen University721 23 VästeråsSweden e-mail: mio@mdh.se
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Abstract

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Two projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X, that is, those nonisomorphic real algebraic curves whose complexifications are isomorphic to X. In this paper we compute, as a function of genus, the maximum number of such real forms that a complex algebraic curve admits.

MSC classification

Secondary: 14H45: Special curves and curves of low genus 20H10: Fuchsian groups and their generalizations 30F50: Klein surfaces 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 1 , February 2001 , pp. 134 - 142
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Alling, N. L. and Greenleaf, N., Foundations of the theory of Klein surfaces, Lecture Notes in Math. 219 (Springer, Berlin, 1971).Google Scholar
[2]Bujalance, E., Etayo, J. J., Gamboa, J. M. and Gromadzki, G., A combinatorial approach to groups of automorphisms of bordered Klein surfaces, Lecture Notes in Math. 1439 (Springer, Berlin, 1990).CrossRefGoogle Scholar
[3]Bujalance, E., Gromadzki, G. and Singerman, D., ‘On the number of real curves associated to a complex algebraic curve’, Proc. Amer. Math. Soc. 120 (1994), 507–513.Google Scholar
[4]Gromadzki, G. and Izquierdo, M., ‘Real forms of a Riemann surface of even genus’, Proc. Amer. Math. Soc. 126 (1998), 3475–3479.Google Scholar
[5]Macbeath, A. M., ‘The classification of non-Euclidean plane crystallographic groups’, Canad. J. Math. 19 (1967), 1192–1205.Google Scholar
[6]Natanzon, S. M., ‘On the order of a finite group of homeomorphisms of a surface into itself and the real number of real forms of a complex algebraic curve’, Dokl. Akad. Nauk SSSR 242 (1978), 765–768;Google Scholar
English translation: Soviet Math. Dokl. 19 (1978), 1195–1199.Google Scholar
[7]Natanzon, S. M., ‘Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves’, Trans. Moscow Math. Soc. 1989, 1–51.Google Scholar
[8]Singerman, D., ‘On the structure of non-Euclidean crystallographic groups’, Proc. Cambridge Phios. Soc. 76 (1974), 233–240.CrossRefGoogle Scholar