Combinatorics of the tropical Torelli map
Abstract
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 Mg and of the tropical Schottky locus for genus at most 5. We show that tropical Ag is Hausdorff, and we also construct a finite-index cover for A3 which satisfies a tropical-type balancing condition. Many different combinatorial objects, including regular matroids, positive semidefinite forms, and metric graphs, play a role.