Combinatorics of the tropical Torelli map
Melody Chan · Algebra & Number Theory · 2012
This paper is a combinatorial and computational study of the moduli space M tr g of tropical curves of genus g, the moduli space A tr g 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 tr g and A tr g are objects and the Torelli map is a morphism.We compute the poset of cells of M tr g and of the tropical Schottky locus for genus at most 5.We show that A tr g is Hausdorff, and we also construct a finite-index cover for the space A tr 3 which satisfies a tropical-type balancing condition.Many different combinatorial objects, including regular matroids, positive-semidefinite forms, and metric graphs, play a role.