#### Vol. 6, No. 6, 2012

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Combinatorics of the tropical Torelli map

### Melody Chan

Vol. 6 (2012), No. 6, 1133–1169
##### Abstract

This paper is a combinatorial and computational study of the moduli space ${M}_{g}^{tr}$ of tropical curves of genus $g$, the moduli space ${A}_{g}^{tr}$ of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were studied recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the category of stacky fans, of which ${M}_{g}^{tr}$ and ${A}_{g}^{tr}$ are objects and the Torelli map is a morphism. We compute the poset of cells of ${M}_{g}^{tr}$ and of the tropical Schottky locus for genus at most 5. We show that ${A}_{g}^{tr}$ is Hausdorff, and we also construct a finite-index cover for the space ${A}_{3}^{tr}$ which satisfies a tropical-type balancing condition. Many different combinatorial objects, including regular matroids, positive-semidefinite forms, and metric graphs, play a role.

##### Keywords
tropical geometry, tropical curves, metric graphs, Torelli map, moduli of curves, abelian varieties
##### Mathematical Subject Classification 2010
Primary: 14T05
Secondary: 14H10, 05C30