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

TL;DR

This paper investigates the rich rational cohomology of the moduli space of curves with level structures, establishing a lower bound on its coherent cohomological dimension and discussing implications of existing conjectures.

Contribution

It provides new results on the cohomological properties of moduli spaces of curves with level structures, including a lower bound on their coherent cohomological dimension.

Findings

The moduli space has extensive rational cohomology in its top degree.

The coherent cohomological dimension is at least g-2.

Conjectures suggest this bound may be sharp.

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.