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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2505.09719 |
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| _version_ | 1866908364221448192 |
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| author | Crepeau, Natasha |
| author_facet | Crepeau, Natasha |
| contents | 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 $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized break divisors and triangulations of Lawrence polytopes Crepeau, Natasha Combinatorics 05E14 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 $G$. |
| title | Generalized break divisors and triangulations of Lawrence polytopes |
| topic | Combinatorics 05E14 |
| url | https://arxiv.org/abs/2505.09719 |