Principal boundary of moduli spaces of abelian and quadratic differentials

Abstract

The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted differentials over Deligne-Mumford pointed stable curves. We also describe similarly the principal boundary of moduli spaces of quadratic differentials originally studied by Masur-Zorich. Our main technique is the flat geometric degeneration and smoothing developed by Bainbridge-Chen-Gendron-Grushevsky-M\"{o}ller.