Combinatorics of the tropical Torelli map

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.