The high-dimensional cohomology of the moduli space of curves with level   structures
  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. 
Neil Fullarton, Andrew Putman
math.GT math.AG math.AT