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 ${ \mathcal A_g^{ \mathop{ Perf}}}$, the toroidal compactification of the moduli space of principally polarized abelian varieties of dimension $g$ given by this decomposition, has codimension $10$ if $g \geq 4$. Moreover we describe the non-simplicial locus in codimension $10$. We also show that the second Voronoi compactification ${ \mathcal A_{ g}^{ \mathop{ Vor}}}$ has singularities in codimension $3$ for $g\geq 5$.
Categories: math.AG math.CO math.MG