Real forms of a Riemann surface of even genus

Abstract:

Natanzon proved that a Riemann surface $X$ of genus $g \ge 2$ has at most $2(\sqrt g+1)$ conjugacy classes of symmetries, and this bound is attained for infinitely many genera $g$. The aim of this note is to prove that a Riemann surface of even genus $g$ has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even $g$ as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus $g$ has at most four real forms which are not birationally equivalent.