TORELLI THEOREM FOR GRAPHS AND TROPICAL CURVES
By Lucia Caporaso, Filippo Viviani. 2009
82 citations recorded by the source corpus.
Algebraic curves have a discrete analog 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. By contrast, we prove that, in a suitably defined sense, the tropical Torelli map has degree one. Finally, we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/876943cd97815b2d50d3205dd9f0c6fc08e5b7dd
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In Computational Theory and Mathematics
- David Jensen
- Melody Chan
- Martin Ulirsch
- Matthew Baker
- Renzo Cavalieri
- Jonathan Wise
- Dhruv Ranganathan
- Lucia Caporaso
In Geometry and Topology
- David Jensen
- Melody Chan
- Martin Ulirsch
- Matthew Baker
- Renzo Cavalieri
- Jonathan Wise
- Dhruv Ranganathan
- Lucia Caporaso
Citations in the corpus
- Combinatorics of the tropical Torelli map (2010, unknown): “Combinatorics of the tropical Torelli map 1135 The starting point of this paper is the article [BMV 2011], by Brannetti, Melo, and Viviani, where the authors rigorously define a plausible category for tropical moduli spaces called stacky fans.”
- A MODULI STACK OF TROPICAL CURVES (2017, background): “ in tropical geometry (see e.g. [Cap13, CMV13, Cap16, Cha16] for survey papers). In particular, we refer the reader to [Mik06], [Mik07], [GKM09], and [GM08] for the case of genus g = 0, as well as to [CV10], [BMV11], [Cha12], and [Viv13] for the higher genus case and the connections to the tropical ”
- Degeneration of Linear Series from the Tropical Point of View and Applications (2015, background): “In fact, there is an analogue of the Torelli theorem in this context saying that up to certain “Whitney flips”, the Jacobian as a principally polarized real torus determines the metric graph Γ; see [CV10].”