Smoothness and singularities of the perfect form and the second Voronoi compactification of ${\mathcal{A}}_g$

Abstract

We study the cones in the first Voronoi or perfect cone decomposition of quadratic forms with respect to the question which of these cones are basic or simplicial. As a consequence we deduce that the singular locus of the moduli stack ${\mathcal{A}}_g^{\mathrm{Perf}}$, the toroidal compactification of the moduli space of principally polarized abelian varieties of dimension $g$ given by this decomposition, has codimension $10$ if $g\ge 4$. Moreover we describe the non-simplicial locus in codimension $10$. We also show that the second Voronoi compactification ${\mathcal{A}}_g^{\mathrm{Vor}}$ has singularities in codimension $3$ for $g\ge 5$.

16 pages; to appear in Algebraic Geometry

References in corpus (2)

• Perfect forms and the moduli space of abelian varieties
• Stable cohomology of the perfect cone toroidal compactification of ${\mathcal{A}}_g$

Cited by in corpus (3)

• Computing symmetry groups of polyhedra
• Stable cohomology of the perfect cone toroidal compactification of ${\mathcal{A}}_g$
• Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties