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Description: |
The classical Torelli map is the modular map from the moduli space of curves of genus g to the moduli space of principally polarized abelian varieties of dimension g, sending a curve into its Jacobian. The Torelli map is known to be injective (Torelli theorem) and the image has been described in many different ways (the so called Schottky problem). We propose analogues of these classical theorems in tropical geometry. We construct moduli spaces of tropical curves and tropical principally polarized abelian varieties, and we define a tropical Torelli map. We study the fibers (tropical Torelli) and the image (tropical Schottky) of this map. Time permitting, we will see some applications to the compactification of the classical Torelli map. This is based on joint works with L. Caporaso and with S. Brannetti and M. Melo.
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