Constructing stability conditions on nodal curves from subdivisions of Lawrence polytopes

Abstract

A stability condition on a nodal curve $X$ is the assignment of integers to the nontrivial biconnected subcurves of $X$ satisfying some desired properties; equivalently, it is an assignment of integers to the biconnected subsets of $V(G)$, where $G$ is the graph dual to $X$. In this thesis, we study the combinatorics of these stability conditions, and provide a construction of these stability conditions from subdivisions of the Lawrence polytope of the cographic matroid $\M^\ast(G)$ using the theory of single-element extensions of oriented matroids. We also discuss connections to chip-firing on graphs through the theory of generalized break divisors.