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On the number of real curves associated to a complex algebraic curve. (English) Zbl 0820.30025
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.

##### MSC:
 30F10 Compact Riemann surfaces and uniformization 14H55 Riemann surfaces; Weierstrass points; gap sequences 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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##### References:
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