On the real nerve of the moduli space of complex algebraic curves of even genus

Abstract

The real locus in the moduli space of complex algebraic curves of given genus consists of curves having real forms, that is, Riemann surfaces admitting a symmetry (anticonformal involution). The real locus is covered by subsets, each formed by curves having a given topological type determined by the number of connected components and the separability type of the real models of the curves. In this paper, we study the structure of the nerve corresponding to this covering, called the real nerve of complex algebraic curves, for even genera. We find its geometrical and homological dimension, and we obtain some results concerning its global geometrical properties. In the proofs, we use the equivalent language of Riemann surfaces and their symmetries.