Trigonal and embedded tropical curves of low genus

TL;DR

This paper explores the concept of trigonal tropical curves of low genus, examining their embeddings and morphisms, and compares geometric obstructions to understand their structure and relationships to algebraic counterparts.

Contribution

It provides a detailed characterization of trigonal tropical curves of genus 3 and 4, linking morphisms to lines with embeddings into tropical surfaces, and offers new insights into their geometric obstructions.

Findings

Relation between trigonal morphisms and embeddings into tropical surfaces.

Obstructions for embeddings compared to obstructions for degree 3 morphisms.

Examples of non-smooth embeddings reflecting degree 3 morphism features.

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.