Trigonal and embedded tropical curves of low genus

In algebraic geometry, trigonal curves can always be embedded into Hirzebruch surfaces. In tropical geometry, the notion of trigonality does not have a unique translation. We focus on the characterization in terms of the existence of a degree 3 morphism to a line, and discuss relations to possible embeddings into $\mathbb R^2$ reflecting an embedding into a Hirzebruch surface. Our results can be divided into three parts: for tropical curves of low genus 3 and 4, we discuss the relation between a trigonal morphism and an embedding dual to the polygon of a Hirzebruch surface, building on works on embeddings of hyperelliptic tropical curves and curves of low genus. We compare obstructions for embeddings with obstructions for the existence of a degree 3 morphism to a line. Finally, we showcase examples where a non-smooth embedding can be unfolded to reflect certain features of a degree 3 morphism to a line.

💡 Research Summary

The paper investigates how the classical notion of a trigonal curve— a non‑hyperelliptic algebraic curve admitting a degree‑3 map to ℙ¹— translates into tropical geometry. In the algebraic setting such curves always embed into a Hirzebruch surface Fₙ, where the Maroni invariant n satisfies 0 ≤ n ≤ ⌊(g+2)/3⌋ and g ≡ n (mod 2). For genus 3 the invariant is forced to be 1, while for genus 4 it can be 0 or 2. The embedding class is