The $d$-gonal locus in the moduli space of tropical plane curves

TL;DR

The paper investigates two tropical moduli loci inside the genus g nondegenerate plane-curve space: the gonality-d constrained locus and the expected-gonality locus derived from Newton polygons. Central to the approach is translating geometric information about polygons (lattice width, interior genus, and regular unimodular triangulations) into combinatorial and metric-graph data (gonality, scramble number, and skeletons) via the duality between tropical curves and polygon subdivisions. The main result establishes that for fixed d \ge 3 and large genus g (specifically g \ge \max\{d^3,32\}), the dimensions of the gonality-based and the expected-gonality-based tropical loci coincide, and the paper provides matching upper and lower bounds using constructions with crystals and beehive triangulations. This yields evidence that, in high genus, tropical gonality is determined by the Newton polygon's expected gonality, bridging tropical and algebraic perspectives on gonality and expanding our understanding of moduli spaces in tropical geometry. The work also furnishes explicit dimension formulas for small genus and low expected gonality, and outlines future directions toward proving the conjectured equality for all genus and gonality pairs.

Abstract

We introduce and study the locus $\mathbb{M}_{g,d}^\textrm{nd}$ of genus $g$ tropical plane curves of gonality $d$ inside the moduli space $\mathbb{M}^{\textrm{nd}}_{g}$ of tropical plane curves of genus $g$. Each such tropical curve arises from a Newton polygon, and we conjecture that the gonality of the tropical curve is equal to an easily computed parameter of this polygon called the expected gonality, closely related to the lattice width of the polygon. Let $\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}$ denote the locus of tropical curves whose associated Newton polygon has expected gonality $d$. We prove that for fixed $d$ and sufficiently large genus $g$, the dimensions of these two loci agree: \[ \\dim\left(\mathbb{M}_{g,d}^\textrm{nd}\right) =\dim\left(\mathbb{M}_{g,{\underline{d}}}^\textrm{nd}\right). \] Our results provide evidence that, in sufficiently high genus compared to expected gonality, the gonality of a tropical curve is determined by the expected gonality of the Newton polygon from which it arises.

Figures (30)

• Figure 1: The triangles $4\Sigma$ and $2\Upsilon$.
• Figure 2: Polygons and their relaxations. Note that on the right, $P^{(-1)}$ is not a lattice polygon, meaning that $P$ is not the interior polygon of any lattice polygon
• Figure 3: Polygons and their column vectors.
• Figure 4: A triangulation of a Newton polygon, a tropical curve, and its skeleton.
• Figure 5: A polygon of genus $5$, a dual tropical curve, and the skeleton $\Gamma$

Theorems & Definitions (74)

• Conjecture 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Proposition 2.1: Koelman_Dissertation, § 2.2 and HS09, Lemmas 9 and 10
• Definition 2.2
• Proposition 2.3
• Proposition 2.4: echavarria2021scramble, Lemma 2.15