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 Mgtr of tropical curves of genus g, the moduli space Agtr 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 Mgtr and Agtr are objects and the Torelli map is a morphism. We compute the poset of cells of Mgtr and of the tropical Schottky locus for genus at most 5. We show that Agtr is Hausdorff, and we also construct a finite-index cover for the space A3tr 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
Milestones
Received: 23 February 2011
Revised: 11 July 2011
Accepted: 13 August 2011
Published: 12 August 2012
Authors
Melody Chan
Department of Mathematics
University of California Berkeley
970 Evans Hall #3840
Berkeley, CA 94720-3840
United States
http://math.berkeley.edu/~mtchan/