Smoothness and singularities of the perfect form and the second Voronoi compactification of ${\mathcal A}_g$
Smoothness and singularities of the perfect form and the second Voronoi compactification of ${\mathcal A}_g$
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完美形式的平滑性和奇异性以及 ${mathcal A}_g$ 的第二次 Voronoi 紧致化
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发表时间：
2013
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通讯作者：
Achill Schurmann
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作者：
Mathieu Dutour Sikiri'c;K. Hulek;Achill Schurmann
我们研究的锥在第一Voronoi或完美的锥分解的二次形式的问题，这些锥是基本或单纯。作为结果，我们推导出奇异轨迹的模堆叠${\mathcal A_g^{\mathop{Perf}}}$，环面紧化的模空间的主要极化阿贝尔品种的维$g$由这个分解，有余维$10$如果$g \geq 4$。此外，我们还描述了余维$10$中的非单纯轨迹。我们还证明了第二个Voronoi紧化${\mathcal A_{g}^{\mathop{Vor}$对于$g\geq 5$具有余维$3$的奇异性。
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 $g\geq 5$.