Stability conditions and subdivisions of Lawrence polytopes

@misc{crepeau2026stability,
  title = {{Stability conditions and subdivisions of Lawrence polytopes}},
  author = {Natasha Crepeau},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00591},
  eprint = {2610.00591},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00591},
}

TY  - UNPB
TI  - Stability conditions and subdivisions of Lawrence polytopes
AU  - Natasha Crepeau
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00591
UR  - https://arxiv.org/abs/2610.00591
AB  - Let X be a nodal curve, and G be the graph dual to X. A stability condition on a graph G is an assignment of integers to the biconnected subsets of vertices of G satisfying some desired properties. Stability conditions yield degeneracy sets, which are certain collections of biconnected subsets of G, but degeneracy sets can also be described without a reference stability condition. We show that the degeneracy sets of a graph G correspond to the single-element extensions of the graphic matroid M(G). Some single-element extensions of the graphic matroid M(G) can be oriented to be single-element extensions of the oriented graphic matroid ℳ(G). Single-element extensions of oriented matroids are in bijection with subdivisions of the Lawrence polytope of the dual oriented matroid. We construct stability conditions on G from subdivisions of the Lawrence polytope of the cographic matroid ℳ^∗(G). Finally, we show that any degeneracy set 𝒟 of a graph G that has the set of all biconnected subsets of G as a lower bound in the poset of degeneracy sets corresponds to an orientable extension of M(G).
ER  -