Tropicalized Quartics and Canonical Embeddings for Tropical Curves of Genus 3

In [8], it was shown that not all abstract non-hyperelliptic tropical curves of genus 3 can be realized as a tropicalization of a quartic in R-2. In this paper, 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 all curves but the tropicalizations of realizably hyperelliptic algebraic curves. Our approach is constructive: for a curve that is not the tropicalization of a hyperelliptic algebraic curve, we explicitly construct a realizable model of the tropical plane in R-n and a faithfully tropicalized quartic in it. These constructions rely on modifications resp. tropical refinements. Conversely, we prove that the tropicalizations of hyperelliptic algebraic curves cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems [3, 21] and recent advances in the realizability of sections of the tropical canonical divisor [30].