2009/05/11 by Matthew Baker, Baker, Matthew, Xander Faber +1 · 2 citations
Mathematics · #05C38 (secondary) #05C50 #14H40 (Primary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Geometry and complex manifolds #math.AG #math.CO #msc:05C38 #msc:05C50 #msc:14H40
paper · pdf · doi:10.48550/arxiv.0905.1679
29 pages, 4 figures; up to differences in formatting, this is the final version that will appear in the Journal of Algebraic Combinatorics
openalex publication_date 2009/05/11 · arxiv created 2010/10/10 · arxiv updated 2010/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a tropical curve (or metric graph), and fix a base point p on X. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(X) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for X. This result is useful for reducing certain questions about the Abel-Jacobi map Phip : X -> J(X), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over the rationals. As an application of our direct limit theorem, we derive some local comparison formulas between g and its pullback Phip^*(g) for three different natural "metrics" g on J(X). One of these formulas implies that Phip is a tropical isometry when X is 2-edge-connected. Another shows that the canonical measure on a metric graph X, defined by S. Zhang, measures lengths on the image Phip(X) with respect to the "sup-norm" on J(X).