Trigonal and embedded tropical curves of low genus

TL;DR

The article develops a tropical analogue of trigonal curves via embeddings into Hirzebruch surfaces, focusing on genus $3$ and $4$ with maximal combinatorial type. It links smooth planar embeddings to realizable degree $3$ tropical covers and Maroni-type invariants, showing obstructions that prevent smooth embeddings and detailing how tropical modifications can unfold contracted features to reflect trigonal morphisms. The work combines combinatorial, geometric, and tropical techniques to characterize when tropical curves of low genus admit Hirzebruch dual embeddings, and illustrates unfolding as a mechanism to recover trigonal structures in non-smooth cases. Overall, it advances the correspondence between tropical gonality notions and Hirzebruch-surface embeddings, with concrete classifications and illustrative examples for low-genus cases.

Abstract

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.

Figures (44)

• Figure 1: The polygon of a curve of class $3E_n + \frac{g + 3n + 2}{2} F_n$ in $\mathbb{F}_n$.
• Figure 2: Tropicalization of a curve with a morphism to $\mathbb{P}^1$ which factors through the projection of a Hirzebruch surface to $\mathbb{P}^1$.
• Figure 3: A Newton subdivision and a dual tropical plane curve. On the right, an abstract tropical curve that can be embedded into the plane, as the skeleton of the plane tropical curve, as depicted in color on the second to right picture.
• Figure 4: On the left a well-contracted cover and in the middle the admissible cover from which we can recover it. On the right instead is depicted an admissible cover which does not define a well-contracted cover: for any edge in the target tree, the contraction of the edges in its preimage would decrease the genus of the graph.
• Figure 5: The polygons defining a genus $3$ curve embedded with degree $4$ in the plane (on the left) and in the first Hirzebruch surface (on the right).

Theorems & Definitions (36)

• Proposition 1.1: See \ref{['lm:realizability_cover']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9