Geodesic
The Ceresa class and tropical curves of hyperelliptic type
Forum of Mathematics, Sigma, Tome 12 (2024)

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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 four vertices or the loop of three 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 $\mathbb {C}(\!(t)\!)$. 
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     author = {Daniel Corey and Wanlin Li},
     title = {The {Ceresa} class and tropical curves of hyperelliptic type},
     journal = {Forum of Mathematics, Sigma},
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Daniel Corey; Wanlin Li. The Ceresa class and tropical curves of hyperelliptic type. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.36

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