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A Torelli theorem for graphs via quasistable divisors

Alex Abreu, Marco Pacini

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.

A Torelli theorem for graphs via quasistable divisors

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.
Paper Structure (9 sections, 23 theorems, 64 equations, 4 figures)

This paper contains 9 sections, 23 theorems, 64 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Posets
  4. Graphs
  5. Divisors on graphs
  6. The poset of quasistable divisors
  7. Special posets
  8. Torelli theorem for graphs
  9. Torelli Theorem for tropical curves

Key Result

Theorem 1

Let $\Gamma$ and $\Gamma'$ be graphs with set of bridges $\mathop{\mathrm{Br}}\nolimits(\Gamma)$ and $\mathop{\mathrm{Br}}\nolimits(\Gamma')$. The posets $\mathbf{QD}(\Gamma)$ and $\mathbf{QD}(\Gamma')$ are isomorphic if and only if there is a bijection between the biconnected components of $\Gamma/

Figures (4)

  • Figure 1: The Hasse diagram of $\mathbf P$.
  • Figure 2: The Hasse diagram of $\mathbf R$.
  • Figure 3: The two possibility for the Hasse diagram of the poset $g(\mathbf P)$.
  • Figure 4: The Hasse diagram of the poset $\mathbf{R}_{e_1,e_2}(D)$

Theorems & Definitions (62)

  • Theorem : Theorem \ref{['thm:main1']}
  • Theorem : Theorem \ref{['thm:main2']}
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 3.1
  • ...and 52 more