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DAGA Seminar: Galois coverings of moduli spaces of curves and loci of curves with symmetry

December 15, 2011 - Marco Boggi

ARGOMENTI: Seminari

Marco Boggi (Universidad de los Andes, Bogotà

Title: "Galois coverings of moduli spaces of curves and loci of curves with symmetry"

December 15, 2011,
h. 14:30, room 2AB45

-Abstract
Let M?g,[n], for 2g?2+n>0, be the stack of genus g, stable algebraic curves over C, 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 M?g. We will introduce the notion of Looijenga level structure which generalizes Looijenga construction and provides a tower of Galois coverings of M?g,[n] equivalent to the tower of all geometric level structures over M?g,[n]. Then, 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, we will show that Looijenga level structures are smooth under relatively mild hypotheses. Time permitting, we will give a description of the nerve of the D--M boundary of abelian level structures and show 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├╝ller modular group.

Rif. int. A. Bertapelle

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