Degenerations of quadratic differentials on CP1

TL;DR

This paper characterizes the topology of quadratic differential strata on the Riemann sphere, identifying connected components, their configurations, and properties like having a single topological end, with implications for Siegel-Veech constants.

Contribution

It provides a detailed description of the connected components of quadratic differential strata on CP1 and introduces tools for calculating Siegel-Veech constants.

Findings

Connected components correspond to generic configurations on flat spheres

The stratum has only one topological end

Developed toolkit for Siegel-Veech constant evaluation

Abstract

We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel-Veech constants.