The $L^\infty$-isodelaunay decomposition of strata of abelian differentials

TL;DR

This paper introduces an $L^$-isodelaunay decomposition of strata of abelian differentials, classifies adjacency relations between regions, and constructs a finite simplicial complex homotopy equivalent to the stratum.

Contribution

It provides a novel classification of isodelaunay regions and a method to compute a finite simplicial complex with the same homotopy type as the stratum.

Findings

Classified all adjacencies between isodelaunay regions.

Constructed a finite simplicial complex homotopy equivalent to the stratum.

Proved a stronger version of the Nerve Lemma for equivariant cases.

Abstract

We study the decomposition of a stratum $\mathcal H(\kappa)$ of abelian differentials into regions of differentials that share a common $L^\infty$-Delaunay triangulation. In particular, we classify the infinitely many adjacencies between these isodelaunay regions, a phenomenon whose observation is attributed to Filip in work of Frankel. This classification allows us to construct a finite simplicial complex with the same homotopy type as $\mathcal H(\kappa)$, and we outline a method for its computation. We also require a stronger equivariant version of the traditional Nerve Lemma than currently exists in the literature, which we prove.