Degenerations of quadratic differentials on ℂℙ¹

Abstract

We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP 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.

Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials

“…This construction was generalized by the author to polygonal curves in [Boi08], section 3. Such curve must still be transverse to the vertical foliation in a neighborhood of the singularity P and must have nontrivial linear holonomy (if k is odd).…”

“…We obtain a flat surface, which is a translation surface if and only if all the identifications are done by translation. One can show that any flat surface can be represented by such a polygon (see [Boi08], Section 2).…”

Labeled Rauzy classes and framed translation surfaces

“…In particular, it means that up to a small deformation of the surface in the ambient stratum, there is no other saddle connection in the surface parallel to γ. Then, deforming suitably the surface with the Teichmüller geodesic flow (see [Boi08,Boi09] for instance), one gets a surface for which the saddle connection corresponding to γ is very short compared to the other ones. Then, one can show that such surface is obtained by the breaking up a zero surgery (see [EMZ03]).…”

Labeled Rauzy classes and framed translation surfaces

“…In particular, it means that up to a small deformation of the surface in the ambient stratum, there is no other saddle connection in the surface parallel to γ. Then, deforming suitably the surface with the Teichmüller geodesic flow (see [3,4] for instance), one gets a surface for which the saddle connection corresponding to γ is very short compared to the other ones. Then, one can show that such surface is obtained by the breaking up a zero surgery (see [10]).…”