Combinatorics of the tropical Torelli map

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.