Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus

Details Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus

2011-06-16

arxiv.org/abs/1106.3291v2

Mathematics

Combinatorics

27 pages, 2 figures, final version, to appear in Mathematische Annalen

Scientific paper

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.

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