Tropicalized Quartics and Canonical Embeddings for Tropical Curves of Genus 3

Abstract

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 which 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].

Tropically planar graphs

“…We would have also included the graphs arising from the hyperelliptic polygons, although it turns out Fig. 16 The maximal nonhyperelliptic polygons of genus 6 and genus 7 all such graphs also arose from nonhyperelliptic polygons for g = 6 and g = 7 . In the end we found that there are 152 troplanar graphs of genus 6, and 672 troplanar graphs of genus 7.…”

“…Instead of considering tropical curves that are a subset of ℝ 2 , one could more generally a n = A n + 2Dr n cos(n − ) ≥ A 2 n consider tropical curves that are subsets of 2-dimensional tropical linear spaces embedded in some ℝ n . The authors of [16] take this approach, and show that every metric graph of genus 3 (with a low-dimensional set of exceptions) arises in such a 2-dimensional tropical linear space in ℝ n for n ≤ 5 ; this includes metric graphs whose underlying graph is the non-troplanar graph of genus 3. A natural question to ask is which graphs of genus g appear in tropical curves embedded in such tropical linear spaces for g ≥ 4.…”

Smooth tropical complete intersection curves of genus $3$ in $\mathbb{R}^3$

“…On the other hand, the lollipop graph can be realized on a tropical plane in R 5 ( [2]). However, it is an open question if the lollipop graph can be realized on a tropical plane in R 3 or R 4 .…”

Forbidden patterns in tropical plane curves

“…It would be interesting to know how the tropically plane curves of a fixed genus fit into the moduli space of all tropical curves. For genus 3 this was recently answered in terms of modifications by Hahn et al (2019).…”