Tropicalized quartics and canonical embeddings for tropical curves of genus 3

TL;DR

This paper classifies genus 3 tropical curves that can be faithfully embedded as quartics in tropical planes, showing that non-hyperelliptic curves are realizable while hyperelliptic ones are not, using explicit constructions and tropical divisor theory.

Contribution

It provides a complete classification of realizable genus 3 tropical curves in tropical planes, with explicit constructions for non-hyperelliptic cases and theoretical proof for hyperelliptic cases.

Findings

Non-hyperelliptic curves are realizable as faithful tropicalizations in tropical planes.

Hyperelliptic curves cannot be embedded faithfully in such tropical planes.

Explicit constructions rely on tropical modifications and refinements.

Abstract

Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus $3$ can be realized as a tropicalization of a quartic in the euclidean plane. In this article, we focus on the interior of the maximal cones in the moduli space and classify all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, we show that these are exactly those which are not realizably hyperelliptic. Our approach is constructive: For any not realizably hyperelliptic curve, we explicitly construct a realizable model of the tropical plane and a faithfully tropicalized quartic in it. These constructions rely on modifications resp. tropical refinements. Conversely, we prove that any realizably hyperelliptic curve cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems, and recent advances in the realizability of sections of the tropical canonical divisor.