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The high-dimensional cohomology of the moduli space of curves with level structures. (English) Zbl 07198108
Summary: We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least \(g - 2\). Well known conjectures of Looijenga would imply that this is sharp.

14D22 Fine and coarse moduli spaces
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
Full Text: DOI
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