Principal boundary of moduli spaces of abelian and quadratic differentials

TL;DR

This paper characterizes the principal boundary of moduli spaces of abelian and quadratic differentials using twisted differentials and flat geometric degeneration techniques, extending previous descriptions to all configurations.

Contribution

It provides a comprehensive description of the principal boundary for all configurations of differentials via twisted differentials and flat geometric methods.

Findings

Describes principal boundary for each configuration of abelian differentials.

Extends boundary description to quadratic differentials.

Utilizes flat geometric degeneration and smoothing techniques.

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.