On the complexity group of stable curves

In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the N ́eron model of the generalized Jacobian of the curve. We study the order of this group, called the complexity. In particular, we provide a partial characterization of the stable curves having maximal complexity, and we provide an upper bound, depending only on the genus g of the curve, on the maximal complexity of stable curves; this bound is asymptotically sharp for g ≫ 0. Eventually, we state some conjectures on the behavior of stable curves with maximal complexity, and prove partial results in this direction.