The moduli space of tropical curves with fixed Newton polygon

TL;DR

The paper studies the moduli space of tropical plane curves with a fixed Newton polygon $\Delta$, relating tropical and algebraic dimensions. It develops a combinatorial framework to compute the dimension of a tropical moduli piece $\mathbb{M}_\mathcal{T}$ via data from regular unimodular triangulations, yielding the formula $\dim(\mathbb{M}_\mathcal{T})=3g-3-2g^{(1)}-2b_1-b_2$ for nonhyperelliptic $\Delta$, with $g^{(1)}$ the interior polygon's lattice points and $b_1,b_2$ boundary-incident radial-edge counts. The authors construct optimal triangulations (beehive) to maximize the tropical dimension and prove that, for nonhyperelliptic $\Delta$ and for maximal hyperelliptic $\Delta$, the tropical dimension matches the algebraic dimension $\dim(\mathcal{M}_\Delta)$, thus establishing a broad tropical–algebraic correspondence. They also provide explicit dimension formulas, such as $\dim(\mathbb{M}_\Delta)=g-3-c(\Delta)+r$ in the maximal nonhyperelliptic case, and show that maximal hyperelliptic polygons attain $\dim(\mathbb{M}_\Delta)=2g-1$. These results yield both precise dimension counts and classification data for when tropical and algebraic moduli spaces align, offering a robust combinatorial toolkit for tropical geometry of plane curves.

Abstract

Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. We prove that if this polygon is nonhyperelliptic or maximal and hyperelliptic, then this formula matches the dimension of the moduli space of nondegenerate algebraic curves with that given Newton polygon.

Figures (16)

• Figure 1: A triangulated lattice polygon, a dual tropical curve, and its skeleton
• Figure 2: A lattice polygon $\Delta$ whose relaxed polygon is not a lattice polygon; and a lattice polygon with a face $\tau$ such that $\tau^{(-1)}$ is a vertex of $\Delta^{(-1)}$
• Figure 3: A nonmaximal polygon, its interior polygon, and the corresponding maximal polygon
• Figure 4: Column vectors associated to the faces of two polygons
• Figure 5: A lattice polygon with a height function and corresponding regular triangulation

Theorems & Definitions (50)

• Proposition 1.1: BJMS, Lemma 4.2
• Definition 1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: Koelman, § 2.2 and hs, Lemmas 9 and 10
• Lemma 2.2
• proof
• Definition 2
• Proposition 2.3