Degenerations of quadratic differentials on $\mathbb{CP}^1$

Abstract

We describe the connected components of the complement of a natural “diagonal” of real codimension $1$ in a stratum of quadratic differentials on ${ℂ\mathbb{P}}^1$. 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.