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Comparing perfect and 2nd Voronoi decompositions: the matroidal locus

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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 Alexeev and Brunyate (Invent. Math. doi:10.1007/s00222-011-0347-2, 2011). 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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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal

    Margarida Melo

  2. Department of Mathematics, University of Roma Tre, Rome, Italy

    Filippo Viviani

  3. CMUC, University of Coimbra, Coimbra, Portugal

    Filippo Viviani

Authors
  1. Margarida Melo
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  2. Filippo Viviani
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Correspondence to Filippo Viviani.

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Melo, M., Viviani, F. Comparing perfect and 2nd Voronoi decompositions: the matroidal locus. Math. Ann. 354, 1521–1554 (2012). https://doi.org/10.1007/s00208-011-0774-9

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  • DOI: https://doi.org/10.1007/s00208-011-0774-9

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