On moduli of real curves (Q1057950)

Denote the space of isomorphism classes of stable complex curves of genus g by \(\bar M^ g\). This paper is an announcement of results concerning the subspace \(\bar M^ g({\mathbb{R}})\) of \(\bar M^ g\) consisting of curves that can be defined by real polynomials. The moduli space \(\bar M^ g\) admits a canonical antiholomorphic involution that takes the isomorphism class of a complex curve onto that of its complex conjugate. This is a real structure of \(\bar M^ g\) and \(\bar M^ g({\mathbb{R}})\) is the quasiregular real part of that real structure provided that \(g\geq 4\). The main result states that \(\bar M^ g({\mathbb{R}})\) is connected.