On the complexity group of stable curves

TL;DR

This paper investigates the combinatorial properties of stable curves by analyzing the complexity group associated with their dual graphs, providing bounds and characterizations related to the maximal complexity of such curves.

Contribution

It introduces a partial characterization of stable curves with maximal complexity and establishes an asymptotically sharp upper bound depending on genus.

Findings

Provided an upper bound on the maximal complexity based on genus

Characterized stable curves with maximal complexity partially

Proposed conjectures and partial results on complexity behavior

Abstract

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.