https://arxiv.org/abs/2011.08807

Reconstructing orbit closures from their boundaries
Paul Apisa, Alex Wright

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 investigate the surprisingly rich range of situations where the two degenerations are contained in "trivial" invariant subvarieties. Our main results will be applied in forthcoming work to classify large collections of invariant subvarieties; the statement of those results will not involve diamonds, but their proofs will use them as a crucial tool.