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 $g\gg 0$. Eventually, we state some conjectures on the behavior of stable curves with maximal complexity, and prove partial results in this direction.