[Paper Review] Moduli spaces of rational tropical curves

Grigory Mikhalkin|ArXiv.org|Apr 6, 2007

Polynomial and algebraic computation7 references85 citations

TL;DR

This paper provides a corrected and detailed construction of the moduli space of rational tropical curves with $n$ marked points, establishing it as a smooth tropical variety of dimension $n-3$. It introduces the Deligne-Mumford compactification and defines tropical $ψ$-class divisors, with explicit analysis of $\overline{\mathcal{M}}_{0,5}$ as a union of 15 quadrants forming a Petersen graph structure, confirming balancing conditions for divisors.

ABSTRACT

This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification of this space and tropical $ψ$-class divisors.

Motivation & Objective

• To correct a previously published oversimplification in the coordinate description of the moduli space of rational tropical curves.
• To define and construct the Deligne-Mumford compactification of the moduli space $\mathcal{M}_{0,n}$ in the tropical setting.
• To introduce and rigorously define tropical $\psi$-class divisors on the moduli space.
• To provide a detailed geometric and combinatorial description of $\overline{\mathcal{M}}_{0,5}$ as a key example.
• To verify the balancing condition for $\psi_k$-divisors using double ratio functions and symmetry.

Proposed method

• The moduli space $\mathcal{M}_{0,n}$ is constructed as a polyhedral complex of dimension $n-3$, parameterizing 3-valent trees with $n$ marked leaves.
• Tropical modification is used to define the structure of the space, incorporating weights on facets via regular functions and their zero loci.
• The Deligne-Mumford compactification $\overline{\mathcal{M}}_{0,n}$ is obtained by adding strata corresponding to curves with higher-valence vertices.
• Tropical $\psi$-class divisors are defined as the closures of loci where a marked point is adjacent to a 4-valent vertex.
• The balancing condition for divisors is verified using double ratio functions and their gradients on rays of the moduli space.
• The universal curve $\operatorname{ft}_5: \mathcal{M}_{0,5} \to \mathcal{M}_{0,4}$ is described via fiber and section structures in the link of the origin.

Experimental results

Research questions

• RQ1 How can the moduli space of rational tropical curves be correctly parameterized, correcting a prior oversimplification?
• RQ2 What is the geometric and combinatorial structure of $\overline{\mathcal{M}}_{0,5}$, and how does it relate to the Petersen graph?
• RQ3 How are tropical $\psi$-class divisors defined and what is their role in the moduli space?
• RQ4 What conditions ensure that $\psi_k$-divisors are balanced in the tropical sense?
• RQ5 How does the universal curve $\operatorname{ft}_5: \mathcal{M}_{0,5} \to \mathcal{M}_{0,4}$ decompose into fibers and sections?

Key findings

• The moduli space $\mathcal{M}_{0,n}$ is a smooth tropical variety of dimension $n-3$, constructed as a polyhedral complex of 3-valent trees with $n$ marked leaves.
• The space $\overline{\mathcal{M}}_{0,5}$ consists of 15 quadrants $\mathbb{R}_{\geq 0}^2$, glued along 10 rays corresponding to 4-valent vertices, forming a Petersen graph structure.
• The $\psi_k$-divisor is a union of 6 rays corresponding to curves where the $k$-th marked point is adjacent to a 4-valent vertex, and it satisfies the tropical balancing condition.
• The balancing condition for $\psi_1$ is verified by checking that the sum of gradients of double ratio functions over its 6 rays vanishes, using symmetry and explicit gradient computation.
• The universal curve $\operatorname{ft}_5: \mathcal{M}_{0,5} \to \mathcal{M}_{0,4}$ has three fibers and four sections, with the link of the origin in $\mathcal{M}_{0,5}$ decomposing into the fibers and sections of this map.
• The link of the origin in $\mathcal{M}_{0,5}$ is homeomorphic to the Petersen graph, with vertices corresponding to rays and edges to quadrants.