On real forms of a complex algebraic curve

Two projective nonsingular complex algebraic curves X and Y defined over th e field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the coun t of the number of real forms of a complex algebraic curve X, that is, thos e nonisomorphic real algebraic curves whose complexifications are isomorphi c to X. In this paper we compute, as a function of genus, the maximum numbe r of such real forms that a complex algebraic curve admits.