Configurations of saddle connections of quadratic differentials on \(\mathbb {CP}^{1}\) and on hyperelliptic Riemann surfaces (Q843198)

The moduli space of the quadratic differentials on a Riemann surface is naturally stratified by the set of orders of poles and zeros. There are two types of the non-connected strata of quadratic differentials, that is, hyperelliptic strata and exceptional strata. A geodesic segment (or loop) joining two singularities (or a singularity to itself) without any singularities in itself is called a saddle connection. The notion of homologue is defined for saddle connections by considering their lifts to a double cover. A collection \(\gamma\) of homologous saddle connections gives its configuration, which is a combinatorial data determined by the decomposition of the surface by \(\gamma\). In this paper, the author gives the classification of the configurations of homologous saddle connections for the hyperelliptic connected components. It is obtained through the double covering of the configurations for strata of quadratic differentials on \(\mathbb{CP}^1\). The author shows that if a stratum \(Q\) of quadratic differentials on a Riemann surface of genus \(g\geq 5\) admits a hyperelliptic connected component, then \(Q\) is not connected, and any configuration that appears in a hyperelliptic connected component appears also in the other connected components of \(Q\). Together with the result on configurations by \textit{H. Masur} and \textit{A. Zorich} [Geom. Funct. Anal. 18, No. 3, 919--987 (2008; Zbl 1169.30017)], the configurations are completely determined if \(g\geq 5\). The author also gives relations between the genus of a surface and the genera of the connected components of the decomposition of the surface by homologous saddle connections in terms of the pure ribbon graph.