Combinatorics of the tropical Torelli map

By Melody Chan. 2010

72 citations recorded by the source corpus.

This paper is a combinatorial and computational study of the moduli space of tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were introduced recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the category of stacky fans, of which the aforementioned moduli spaces are objects and the Torelli map is a morphism. We compute the poset of cells of tropical M_g and of the tropical Schottky locus for genus at most 5. We show that tropical A_g is Hausdorff, and we also construct a finite-index cover for A_3 which satisfies a tropical-type balancing condition. Many different combinatorial objects, including regular matroids, positive semidefinite forms, and metric graphs, play a role.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/63fcb3d46f6c205141300f6016328a6358cc6018

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Citations in the corpus

Moduli of tropical plane curves (2014, methodology): “The stacky fan M3 of all metric graphs has five maximal cones, as shown in [11], Figure four.”
A MODULI STACK OF TROPICAL CURVES (2017, unknown): “etry (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 Torelli map. A ne”