Generalized break divisors and triangulations of Lawrence polytopes

TL;DR

The paper develops a framework to classify degree $g$ divisors on a connected graph $G$ via generalized break divisors $\mathcal{BD}_{I_G}(G)$, constructed from regular triangulations of the Lawrence polytope of the cographic matroid $\mathcal{M}^*(G)$. It defines $I_G$ from these triangulations to produce complete sets of chip-firing representatives, linking to stability conditions on the dual nodal curve and showing that regular-triangulation-derived $I_G$ recover classical stability coming from generic real divisors. It situates the construction within matroid-theoretic bijections between bases, orientations up to cycle/cocycle reversal, and Lawrence polytope triangulations, and proves that nondegenerate $V$-stability conditions correspond to BD sets with equal cardinalities to spanning trees. The work therefore unifies graph-theoretic divisor theory, polyhedral geometry, and stability notions through explicit combinatorial and geometric correspondences, enabling systematic construction of stability data from polytope triangulations.

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 $G$.

Figures (4)

• Figure 1: Break divisors of the graph $G$, with an example $\mathcal{O}_{\mathcal{E}_T}$ for each.
• Figure 2: Triangulating internal atlas of $\mathcal{M}(G)$
• Figure 3: Construction of $I_G^{\sigma^\ast}$ and the generalized break divisors in $\mathcal{BD}_{I_G^{\sigma^\ast}}$, with $q$ being the lower blue vertex.
• Figure 4: Two triangulating cocycle signatures that differ by a cocycle flip in the fourth cocycle.

Theorems & Definitions (28)

• Theorem A
• Theorem B
• Theorem 2.1: BACKMAN2017655
• Lemma 2.1: ding2023framework
• Definition 2.1: santos2006geometric
• Definition 2.2
• Theorem 2.2
• Definition 3.1: viviani2023new
• Theorem 3.1: viviani2023new
• Theorem 4.1: ding2023framework