Moduli of Tropical Plane Curves

TL;DR

This paper investigates the structure of the moduli space of tropical plane curves, revealing fewer such graphs than classical counterparts and providing explicit computations for low genus cases.

Contribution

It characterizes the moduli space as a stacky fan with cones indexed by unimodular triangulations, and explicitly computes these spaces for genus up to 5.

Findings

Fewer tropical plane curve graphs than classical tropicalizations.

Moduli space is a stacky fan with dimension 2g+1 (except for specific g).

Explicit computations of moduli spaces for g ≤ 5.

Abstract

We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with $g$ interior lattice points. It has dimension $2g+1$ unless $g \leq 3$ or $g = 7$. We compute these spaces explicitly for $g \leq 5$.