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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
20/2018

Tropicalized quartics and canonical embeddings for tropical curves of genus 3

Marvin Anas Hahn, Hannah Markwig, Yue Ren and Ilya Tyomkin

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.

Received:
13.02.18
Published:
15.02.18
MSC Codes:
14T05, 14G22, 14C20
Keywords:
tropical geometry, moduli spaces of plane curves, tropical divisors and linear systems

Related publications

inJournal
2021 Repository Open Access
Marvin Anas Hahn, Hannah Markwig, Yue Ren and Ilya Tyomkin

Tropicalized quartics and canonical embeddings for tropical curves of genus 3

In: International mathematics research notices, 2021 (2021) 12, pp. 8946-8976