A Torelli theorem for graphs via quasistable divisors

Abreu, Alex ; Pacini, Marco

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e25

Résumé

The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a curve comes with a natural stratification that can be regarded as a poset. Furthermore, this poset is entirely determined by the dual graph of the curve and is referred to as the poset of quasistable divisors on the graph. We present a combinatorial version of the Torelli theorem, which demonstrates that the poset of quasistable divisors of a graph completely determines the biconnected components of the graph (up to contracting separating edges). Moreover, we achieve a natural extension of this theorem to tropical curves.

Détail

DOI : 10.1017/fms.2024.135

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