The Ceresa class and tropical curves of hyperelliptic type

TL;DR

This paper introduces the Ceresa-Zharkov class, an algebraic invariant of graphs and tropical curves, which characterizes hyperelliptic type graphs through minor exclusion, linking graph theory with algebraic geometry.

Contribution

It defines a new invariant for graphs and tropical curves that detects hyperelliptic type via minor exclusion, connecting combinatorics and algebraic geometry.

Findings

Ceresa-Zharkov class is trivial iff the graph is hyperelliptic.

Class specializes to an invariant of tropical curves related to the Ceresa cycle.

Characterizes hyperelliptic graphs through minor exclusion.

Abstract

We define a new algebraic invariant of a graph $G$ called the Ceresa-Zharkov class and show that it is trivial if and only if $G$ is of hyperelliptic type, equivalently, $G$ does not have as a minor the complete graph on 4 vertices or the loop of 3 loops. After choosing edge-lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph $G$ that is closely related to the Ceresa cycle for an algebraic curve defined over $\mathbb{C}(\!(t)\!)$.