Iso Edge Domains

TL;DR

This paper revisits iso-edge domains, providing new theoretical insights, including a mass-formula, projectivity of a related moduli space, proof of the Conway--Sloane conjecture in dimension 5, and characterization of certain quadratic forms.

Contribution

It introduces new results on iso-edge domains, including a mass-formula, projectivity proof, and solutions to longstanding conjectures in dimension 5.

Findings

Established a general mass-formula for iso-edge domains.

Proved the associated toroidal compactification is projective.

Confirmed the Conway--Sloane conjecture in dimension 5.

Abstract

Iso-edge domains are a variant of the iso-Delaunay decomposition introduced by Voronoi. They were introduced by Baranovskii & Ryshkov in order to solve the covering problem in dimension $5$. In this work we revisit this decomposition and prove the following new results: $\bullet$ We review the existing theory and give a general mass-formula for the iso-edge domains. $\bullet$ We prove that the associated toroidal compactification of the moduli space of principally polarized abelian varieties is projective. $\bullet$ We prove the Conway--Sloane conjecture in dimension $5$. $\bullet$ We prove that the quadratic forms for which the conorms are non-negative are exactly the matroidal ones in dimension $5$.