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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
##### Keywords:
algebraic curve; real structure; Riemann surface; Fuchsian group
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##### References:
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