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Real forms of a Riemann surface of even genus

Author(s): G. Gromadzki; M. Izquierdo
Journal: Proc. Amer. Math. Soc. 126 (1998), 3475-3479.
MSC (1991): Primary 20F10, 30F10; Secondary 30F35, 51M10, 14H99
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Abstract: Natanzon proved that a Riemann surface $X$ of genus $g \ge 2$ has at most $2(\sqrt g+1)$ conjugacy classes of symmetries, and this bound is attained for infinitely many genera $g$. The aim of this note is to prove that a Riemann surface of even genus $g$ has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even $g$ as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus $g$ has at most four real forms which are not birationally equivalent.

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S. M. Natanzon, 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. (Soviet Math. Dokl. 19 (5), (1978), 1195-1199.) MR 82b:14019

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Additional Information:

G. Gromadzki
Affiliation: Institute of Mathematics University of Gdansk, ul. Wita Stowsza 57, 80-952 Gdansk, Poland

M. Izquierdo
Affiliation: Department of Mathematics, Mälardalen University, 721 23 Västerås, Sweden
Email: mio@mdh.se

DOI: 10.1090/S0002-9939-98-04735-2
PII: S 0002-9939(98)04735-2
Received by editor(s): April 14, 1997
Additional Notes: The second author was partially supported by The Swedish Natural Science Research Council (NFR)
Communicated by: Linda Keen
Copyright of article: Copyright 1998, American Mathematical Society

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