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

TL;DR

This paper investigates the relationship between gonality of tropical plane curves and properties of their Newton polygons, proposing a conjecture and providing dimension comparisons that suggest gonality is determined by the polygon's expected gonality in high genus.

Contribution

The paper introduces the locus of tropical plane curves with fixed gonality, conjectures its equality with a locus defined by Newton polygon parameters, and proves dimension equality for large genus.

Findings

Dimension of gonality locus matches that of expected gonality locus for large genus.

Gonality of tropical curves is conjecturally determined by Newton polygon properties.

Results support the conjecture relating gonality to lattice width of Newton polygons.

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.