On real forms of a complex algebraic curve (Q2726654)

The authors consider the number of real forms of a complex algebraic curve \(X\), i.e. the number of those non-isomorphic real algebraic curves whose complexifications are isomorphic to \(X\). Let \(g\geq 2\) be an integer and \(\omega(g)\) the maximal number of real forms that a complex algebraic curve of genus \(g\) can have. Using theory of non-Euclidean crystallographic groups, \textit{G. Gromadzki} and \textit{M. Izquierdo} proved that \(\omega(g)=4\) whenever \(g\) is even [Proc. Am. Math. Soc. 126, 765-768 (1998; Zbl 0913.20029)]. In the present paper, the authors extend these methods to arbitrary \(g\). More precisely, they determine \(\omega(g)\) for all \(g\geq 2\).