Generalized break divisors and triangulations of Lawrence polytopes

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 $G$.