Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus

TL;DR

This paper compares two rational polyhedral decompositions of the cone of positive definite quadratic forms, analyzing their intersection and implications for the associated toroidal compactifications of moduli spaces.

Contribution

It identifies which cones are common to both decompositions, confirming a conjecture, and compares the resulting compactifications of moduli spaces.

Findings

Identified cones belonging to both decompositions

Confirmed a conjecture of Alexeev and Brunyate

Compared the perfect cone and 2nd Voronoi compactifications

Abstract

We compare two rational polyhedral admissible decompositions of the cone of positive definite quadratic forms: the perfect cone decomposition and the 2nd Voronoi decomposition. We determine which cones belong to both the decompositions, thus providing a positive answer to a conjecture of V. Alexeev and A. Brunyate. As an application, we compare the two associated toroidal compactifications of the moduli space of principal polarized abelian varieties: the perfect cone compactification and the 2nd Voronoi compactification.