Torelli theorem for graphs and tropical curves

Details Torelli theorem for graphs and tropical curves Torelli theorem for graphs and tropical curves

2009-01-10

arxiv.org/abs/0901.1389v3

Duke Mathematical Journal Vol.153, no. 1 (2010), 129--171

Mathematics

Combinatorics

Final version incorporating the referee's suggestions. To appear in DMJ. 30 pages

Scientific paper

Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Finally we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.

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