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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.

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