The Tropical Moduli Space of Degree-3 Rational Maps

TL;DR

This paper constructs and classifies the tropical moduli space of degree-3 rational maps on the tropical projective line, providing a combinatorial and polyhedral model with explicit stratification and connections to tropical Hurwitz theory.

Contribution

It introduces a combinatorial classification and polyhedral model of the tropical moduli space  of degree-3 rational maps, including automorphisms and degenerations.

Findings

Exactly ten combinatorial types of degree-3 tropical rational maps.

A polyhedral parametrization of the moduli space based on gap lengths.

Explicit stratification and relation to tropical Hurwitz theory.

Abstract

We construct and study the tropical moduli space \(\mathcal{M}_3^{\mathrm{trop}}\) of degree-$3$ tropical rational maps \(\mathbb{T}\PP^1 \to \mathbb{T}\PP^1\) up to post-composition. Using a combinatorial description in terms of slope sequences, we classify all such maps and show that there are exactly ten combinatorial types. This yields a polyhedral model of \(\mathcal{M}_3^{\mathrm{trop}}\) parametrized by gap lengths between break points. We determine the automorphism groups and obtain a stratification by explicit linear conditions. We also relate the construction to tropical Hurwitz theory and describe a natural compactification via degenerations of the parameters.