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

Bujalance, Emilio; Gromadski, Grzegorz; Singerman, David

doi:10.1090/S0002-9939-1994-1165047-6

On the number of real curves associated to a complex algebraic curve
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by Emilio Bujalance, Grzegorz Gromadski and David Singerman PDF

Proc. Amer. Math. Soc. 120 (1994), 507-513 Request permission

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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