Combinatorics of the tropical Torelli map

Chan, Melody

doi:10.2140/ant.2012.6.1133

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Combinatorics of the tropical Torelli map

Melody Chan

Vol. 6 (2012), No. 6, 1133–1169

DOI: 10.2140/ant.2012.6.1133

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.

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/

Vol. 6, No. 6, 2012