Generalized break divisors and triangulations of Lawrence polytopes

TL;DR

This paper introduces a new method for selecting representatives of divisor classes on graphs using Lawrence polytope triangulations, linking combinatorial graph theory with algebraic geometry and stability conditions.

Contribution

It constructs a novel connection between Lawrence polytope triangulations and Picard group representatives, extending classical stability concepts to graph divisors.

Findings

Sets of representatives correspond to stability conditions on dual nodal curves.

Regular triangulations of Lawrence polytope induce classical stability conditions.

The method bridges combinatorics, geometry, and algebraic geometry in graph theory.

Abstract

Let $G$ be a connected graph of genus $g$. The Picard group of degree $g$, $\text{Pic}^g(G)$, is the set of equivalence classes of divisors on $G$ of degree $g$, where two divisors are equivalent if one can be reached from the other through a sequence of chip-firing moves. We construct sets of representatives of the equivalence classes in $\text{Pic}^g(G)$ by defining a function $I_G$ on the spanning trees of $G$ from a triangulation of the Lawrence polytope of the cographic matroid $\mathcal{M}^\ast(G)$. Additionally, such sets of representatives correspond to stability conditions on the nodal curve dual to the graph $G$. We show that $I_G$ that are constructed from regular triangulations of Lawrence polytope correspond to classical stability conditions, which are induced by generic real-valued divisors on $G$.