Tropicalized quartics and canonical embeddings for tropical curves of genus 3

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.

24 pages, 18 figures

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• The tropicalization of the moduli space of curves
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• First Steps in Tropical Intersection Theory
• Brief introduction to tropical geometry
• Lifting harmonic morphisms II: tropical curves and metrized complexes
• Tropical curves with a singularity in a fixed point
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Cited by in corpus (3)

• Combinatorics and real lifts of bitangents to tropical quartic curves
• Forbidden Patterns in Tropical Plane Curves
• Tropical Graph Curves