
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 . In this work we revisit this decomposition and prove the following new results: We review the existing theory and give a general mass-formula for the iso-edge domains. We prove that the associated toroidal compactification of the moduli space of principally polarized abelian varieties is projective. We prove the Conway--Sloane conjecture in dimension . We prove that the quadratic forms for which the conorms are non-negative are exactly the matroidal ones in dimension .
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