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arXiv · 1802.02440

Tropicalized quartics and canonical embeddings for tropical curves of genus 3

Marvin Anas Hahn·Hannah Markwig·Yue Ren·Ilya Tyomkin
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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.

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Marvin Anas Hahn, Hannah Markwig, Yue Ren, Ilya Tyomkin. 2019-05-16. Tropicalized quartics and canonical embeddings for tropical curves of genus 3. https://doi.org/10.1093/imrn%2Frnz084

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