The WYSIWYG compactification

TL;DR

This paper demonstrates that a certain partial compactification of Abelian differential strata is not algebraic, yet it provides an alternative proof for a key formula related to GL(2,R) orbit closures and explores boundary structures.

Contribution

It offers an unconditional proof of Mirzakhani and Wright's tangent space formula using mixed methods and presents new insights into boundary structures, despite the non-algebraic nature of the compactification.

Findings

Partial compactification is not an algebraic variety.

Unconditional proof of Mirzakhani and Wright's tangent space formula.

New results on boundary structure of orbit closures.

Abstract

We show that the partial compactification of a stratum of Abelian differentials previously considered by Mirzakhani and Wright is not an algebraic variety. Despite this, we use a combination of algebro-geometric and other methods to provide a short, unconditional proof of Mirzakhani and Wright's formula for the tangent space to the boundary of a GL(2,R) orbit closure, and give new results on the structure of the boundary.