Tropicalized quartics and canonical embeddings for tropical curves of genus 3
Marvin Anas Hahn, Hannah Markwig, Yue Ren, and Ilya Tyomkin
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Submission date: 13. Feb. 2018 (revised version: May 2019)
MSC-Numbers: 14T05, 14G22, 14C20
Keywords and phrases: tropical geometry, moduli spaces of plane curves, tropical divisors and linear systems
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Link to arXiv:See the arXiv entry of this preprint.
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.