Abstract.
We describe typical degenerations of quadratic differentials thus describing “generic cusps” of the moduli space of meromorphic quadratic differentials 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 phenomenon, 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 differential which is close to a “cusp” from the corresponding point at the principal boundary.
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Received: July 2006, Accepted: February 2007
Research of the first author is partially supported by NSF grant 0244472. Research of the second author is partially supported by ANR grant BLAN06-3 138280 and by PICS grant 4170.
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Masur, H., Zorich, A. Multiple Saddle Connections on Flat Surfaces and the Principal Boundry of the Moduli Spaces of Quadratic Differentials. GAFA Geom. funct. anal. 18, 919–987 (2008). https://doi.org/10.1007/s00039-008-0678-3
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DOI: https://doi.org/10.1007/s00039-008-0678-3
Keywords and phrases:
- Teichmüller geodesic flow
- moduli space of quadratic differentials
- flat structure
- saddle connection
- thick part