Reconstructing orbit closures from their boundaries

TL;DR

This paper introduces diamonds as a new tool to analyze and classify GL(2,R)-invariant subvarieties of Abelian and quadratic differentials by examining degenerations, leading to new classification results.

Contribution

The paper develops the concept of diamonds for studying invariant subvarieties, enabling classification of complex degenerations and advancing understanding of their structure.

Findings

Classified a rich collection of diamonds with trivial degenerations

Applied diamond techniques to classify large invariant subvarieties

Proved new results on orbit closure structures

Abstract

We introduce and study diamonds of GL(2,R)-invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneously considering two degenerations, and which provide a new tool for the classification of invariant subvarieties. We classify a surprisingly rich collection of diamonds where the two degenerations are contained in trivial invariant subvarieties. Our main results have been applied to classify large collections of invariant subvarieties; the statement of those results do not involve diamonds, but their proofs rely on them.