COMBINATORIAL GEOMETRY OF GENERIC DEGENERATIONS OF QUADRATIC DIFFERENTIALS HOWARD MASUR AND ANTON ZORICH Abstract. We describe typical degenerations of quadratic differentials thus describing “generic cusps” of the moduli space of meromorphic quadratic dif- ferentials with at most simple poles. The part of the boundary of the moduli space which does not arise from “generic” degenerations is often negligible in problems involving information on compactification of the moduli space. However, even for a typical degeneration one may have several short loops on the Riemann surface which shrink simultaneously. We explain this phe- nomenon, describe all rigid configurations of short loops, present a detailed description of analogs of desingularized stable curves arising here, and show how one can reconstruct a Riemann surface endowed with a quadratic differ- ential which is close to a “cusp” by the corresponding point at the principal boundary. Contents Introduction 1 1. ˆ Homologous saddle connections 3 2. Graph of connected components 5 3. Boundary singularities 8 4. Configurations of ˆ homologous saddle connections 13 5. Principal boundary 16 6. Main Theorems 17 7. Final comments, open problems, applications 18 Appendix A. Long saddle connections 19 Appendix B. List of configurations in genus 2 21 References 23 Introduction 0.1. Saddle connections on flat surfaces. We study flat metrics on a closed orientable surface of genus g, which have isolated conical singularities and linear holonomy restricted to {Id, −Id}. If the linear holonomy group is trivial, then the surface is referred to as a translation surface, such a flat surface corresponds to an Abelian differential ω on a Riemann surface. If the holonomy group is nontrivial, then such a flat surface arises from a meromorphic quadratic differential q with at most simple poles on a Riemann surface. In this paper, unless otherwise stated, a quadratic differential is not the square of an Abelian differential and a flat surface Research of the first author is partially supported by NSF grant 0244472. 1