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Smoothness and singularities of the perfect form and the second Voronoi compactification of Ag

We prove that the singular locus of the moduli stack Perf given by the perfect cone or first Voronoi decomposition has codimension 10 if g>=4 and we describe the non-simplicial locus in codimension 10. We also show that the second Voronoi compactification Vor has singularities in codimension 3 for g>=5.

compactification ; perfect cone ; enumeration

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