Comparing perfect and 2nd Voronoi decompositions: the matroidal locus

2011/06/30 by Margarida Melo, Filippo Viviani · 1 citation
Mathematics · #Abelian group #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Compactification (mathematics) #Cone (formal languages) #Conjecture #Homotopy and Cohomology in Algebraic Topology #Locus (genetics) #Voronoi diagram #math.AG #math.CO

paper · pdf · doi:10.1007/s00208-011-0774-9

published as Math. Ann. 354 (2012), no. 4, 1521--1554 · 27 pages, 2 figures, final version, to appear in Mathematische Annalen

arxiv created 2011/12/09 · openalex publication_date 2012/01/02 · arxiv updated 2012/11/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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.

Citations

• Toroidal Compactification of Siegel Spaces
• Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs.
• The irreducibility of the space of curves of given genus

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• Complete moduli of cubic threefolds and their intermediate Jacobians