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On the number of real curves associated to a complex algebraic curve

Authors: Emilio Bujalance, Grzegorz Gromadski and David Singerman
Journal: Proc. Amer. Math. Soc. 120 (1994), 507-513
MSC: Primary 20H10; Secondary 20H15, 30F50
MathSciNet review: 1165047
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Abstract: Using non-Euclidean crystallographic groups we give a short proof of a theorem of Natanzon that a complex algebraic curve of genus $ g \geqslant 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?

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