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Description: |
In 1969 Deligne and Mumford constructed a remarkable modular compactification of the moduli space of smooth curves of given genus, whose points parametrize stable curves. The geometry of this moduli space is very rich and its study has motivated the development of several important branches of Algebraic Geometry. Stable curves have also a natural combinatoric nature: every stable curve has an associated weighted dual graph which governs the stratification of the moduli space of stable curves by topological type. To any graph, it is naturally associate a group, which we call the complexity group of the graph. So, we can define the complexity group of a stable curve as the complexity group of its dual graph. This group has different and surprising appearances in Geometry, for instance it arises naturally in the study of compactified Jacobians of families of curves. The problem of finding relations between the structure of the complexity group and the geometry of the curve seems to be as natural as difficult and intricate. In this talk I will report on joint work with Simone Busonero and Lidia Stoppino addressing some questions related to this problem. In particular, we give bounds for the order of the complexity group of stable curves of given genus, which, as we will see, has a geometric counterpart too.
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