A Torelli theorem for graphs via quasistable divisors

TL;DR

This paper presents a combinatorial Torelli theorem for graphs, showing that the poset of quasistable divisors uniquely determines the graph's structure and extends to tropical curves, linking graph invariants to algebraic geometry.

Contribution

It introduces a combinatorial Torelli theorem based on quasistable divisors, connecting graph posets to the structure of algebraic curves and tropical geometry.

Findings

Poset of quasistable divisors determines biconnected components of a graph.

The theorem extends to tropical curves.

Graph invariants encode curve and tropical curve structures.

Abstract

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.