  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.
