Galois coverings of moduli spaces of curves and loci of curves with symmetry

TL;DR

This paper introduces a generalized notion of Looijenga level structures for moduli spaces of curves, providing a tower of Galois coverings, and explores their geometric properties and boundary structures to advance understanding of the congruence subgroup problem.

Contribution

The paper generalizes Looijenga level structures, offers a geometric interpretation involving symmetry, and analyzes their boundary structures to facilitate progress on the congruence subgroup problem.

Findings

Looijenga level structures are generalized via Looijenga level structures.

A criterion for the smoothness of these structures is established.

The boundary of level structures is described and used to approach the congruence subgroup problem.

Abstract

Let $\ccM_{g,[n]}$, for $2g-2+n>0$, be the stack of genus $g$, stable algebraic curves, endowed with $n$ unordered marked points. Looijenga introduced the notion of Prym level structures in order to construct smooth projective Galois coverings of the stack $\ccM_{g}$. In \S 2 of this paper, we introduce the notion of Looijenga level structure which generalizes Looijenga construction and provides a tower of Galois coverings of $\ccM_{g,[n]}$ equivalent to the tower of all geometric level structures over $\ccM_{g,[n]}$. In \S 3, Looijenga level structures are interpreted geometrically in terms of moduli of curves with symmetry. A byproduct of this characterization is a simple criterion for their smoothness. As a consequence of this criterion, it is shown that Looijenga level structures are smooth under mild hypotheses. The second part of the paper, from \S 4, deals with the problem of describing the D-M boundary of level structures. In \S 6, a description is given of the nerve of the D-M boundary of abelian level structures. In \S 7, it is shown how this construction can be used to "approximate" the nerve of Looijenga level structures. These results are then applied to elaborate a new approach to the congruence subgroup problem for the Teichm\"uller modular group.