The high-dimensional cohomology of the moduli space of curves with level structures

Abstract

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.

Equivariant cohomology of moduli spaces of genus three curves with level two structure

“…For instance, taking k = 7 the above sequence gives that dim(H 7 (M 3 [2])) ≥ 7680. This bound is in fact far from optimal, as Fullarton and Putman [FP16] recently have shown that dim(H 7 (M 3 [2])) ≥ 11520 via completely different methods. In particular, we see that the cohomology of M 3 [2] is not the smallest possible fitting in a four term exact sequence of the above type.…”

“…Acknowledgements. The author would like to thank Carel Faber and Jonas Bergström for helpful discussions and comments and Orsola Tommasi for pointing out the papers [FP16] and [LM14]. Some of the contents in this note is part of the PhD thesis [Ber16a], written at Stockholms universitet, and parts of the research was carried out at Humboldt-Universität zu Berlin and made possible by the Einstein foundation.…”

Congruence subgroups of braid groups

“…For example, when m ≥ 3, the congruence subgroup Mod(S g )[m] is III-9 torsion-free and (by definition) finite-index, which enables us to come to grips with geometric group theory invariants such as virtual cohomological dimension and duality [24]. As another example, in the algebro-geometric setting, the regular cover of M(S g ) corresponding to Mod(S g )[m] is the moduli space of surfaces of genus g equipped with a full level m structure, that is, a basis for the m-torsion in their Jacobian; see Fullarton-Putman for an overview of this viewpoint [19]. See Putman's lecture notes [45] for further exposition of this topic.…”

The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary

“…However, unlike for PMod g its top rational cohomology group is not zero. In fact, a recent theorem of Fullarton-Putman [12] says that if p is a prime dividing , then the dimension of the rational cohomology of PMod g [ ] in its virtual cohomological dimension is at least…”

“…Applications to algebraic geometry. The moduli space M g,n is a quasi-projective complex variety of dimension 3g − 3 + n. In [12], Fullarton-Putman applied their theorem (1.2) to deduce an interesting result about the algebraic geometry of M g . As we describe now, our Theorem A allows a generalization of this to M g,n .…”