$C$ is not equivalent to $C^-$ in its Jacobian: a tropical point of view

TL;DR

This paper demonstrates that the Abel-Jacobi image of a tropical curve in its Jacobian is not algebraically equivalent to its reflection, using tropical homology calculations to reveal a fundamental difference.

Contribution

It introduces a tropical homology-based method to distinguish the Abel-Jacobi image from its reflection in tropical Jacobians, highlighting a key difference in tropical geometry.

Findings

The Abel-Jacobi image of a tropical curve is not algebraically equivalent to its reflection.

Tropical homology provides a simple calculation to distinguish these images.

This result reveals a fundamental difference in the structure of tropical Jacobians.

Abstract

We show that the Abel-Jacobi image of a tropical curve $C$ in its Jacobian $J(C)$ is not algebraically equivalent to its reflection by using a simple calculation in tropical homology.