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Let , for , be the stack of genus , stable algebraic curves over , endowed with  unordered marked points. In Looijenga (J Algebraic Geom 3:283–293, ), Looijenga introduced the notion of Prym level structures in order to construct smooth projective Galois coverings of the stack . In Sect.  of this paper, we introduce the notion of  which generalizes Looijenga construction and provides a tower of Galois coverings of  equivalent to the tower of all geometric level structures over . In Sect. , 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 Sect. , deals with the problem of describing the D–M boundary of Looijenga level structures. In Sect. , a description is given of the nerve of the D–M boundary of abelian level structures. In Sect. , 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 Teichmüller modular group along the lines of Boggi (Math Nach 279(9–10):953–987, ).

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