Generalized break divisors and triangulations of Lawrence polytopes

2025/05/14 by Natasha Crepeau, Crepeau, Natasha
Mathematics · #05E14 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2505.09719

openalex publication_date 2025/05/14 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

Let G be a connected graph of genus g. The Picard group of degree g, 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 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 M^∗(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.

Citations

• A complete theory of smoothable compactified Jacobians of nodal curves
• A framework unifying some bijections for graphs and its connection to Lawrence polytopes
• Geometry of genus one fine compactified universal Jacobians
• Riemann-Roch Theory for Graph Orientations
• Compactifying the relative Jacobian over families of reduced curves
• Moduli of representations of the fundamental group of a smooth projective variety I

Related

• A framework unifying some bijections for graphs and its connection to Lawrence polytopes
• Effective divisor classes on metric graphs
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