{
  "id": "2505.09719",
  "version": "v1",
  "published": "2025-05-14T18:24:32.000Z",
  "updated": "2025-05-14T18:24:32.000Z",
  "title": "Generalized break divisors and triangulations of Lawrence polytopes",
  "authors": [
    "Natasha Crepeau"
  ],
  "categories": [
    "math.CO"
  ],
  "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 $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-14T18:24:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E14"
    ],
    "keywords": [
      "generalized break divisors",
      "triangulation",
      "equivalence classes",
      "stability conditions",
      "nodal curve dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}