Schottky Uniformization of Real Algebraic Curves and an Application to Moduli

Abstract

We show that a nonsingular compact connected real algebraic curve can be uniformized by a real Schottky group, i. e., a Schottky group in PGL2(ℂ) which is actually contained in PGL2(ℝ). As an application we show that the set Mrpg/ℝ of isomorphismclasses of nonsingularcompact connected real algebraic curves of genus g having real points, has a structure of a semianalytic variety. We show that this structure coincides with the semianalytic structure on Mrpg/ℝ defined via real Teichmuller spaces.