Configurations of saddle connections of quadratic differentials on CP1 and on hyperelliptic Riemann surfaces

TL;DR

This paper classifies saddle connection configurations in quadratic differentials on the Riemann sphere and hyperelliptic surfaces, revealing their distribution across different moduli space components and genera.

Contribution

It provides a classification of saddle connection configurations for quadratic differentials on CP1 and hyperelliptic surfaces, extending previous work and analyzing their occurrence across components.

Findings

Configurations in hyperelliptic components also appear in non-hyperelliptic ones for genera > 5.

Classification extends known results from Abelian to quadratic differentials.

Connected components distinguished by configurations are characterized for specific strata.

Abstract

Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and Zorich, although in that case the connected components were not distinguished. We classify the configurations for quadratic differentials on the Riemann sphere and on hyperelliptic connected components of the moduli space of quadratic differentials. We show that, in genera greater than five, any configuration that appears in the hyperelliptic connected component of a stratum also appears in the non-hyperelliptic one.