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

Abstract

Using non-Euclidean crystallographic groups we give a short proof of a theorem of Natanzon that a complex algebraic curve of genus g ⩾ 2 g \geqslant 2 has at most 2 ( g + 1 ) 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 g as complex double?