Tropical trigonal curves

TL;DR

This paper establishes a correspondence between certain divisors and harmonic morphisms on tropical curves, defines moduli spaces of tropical trigonal curves, and compares their dimensions to algebraic counterparts.

Contribution

It introduces a new equivalence between divisors and harmonic morphisms for tropical trigonal curves and constructs their moduli spaces, linking tropical and algebraic geometry.

Findings

Existence of a divisor of degree 3 and rank ≥ 1 is equivalent to a harmonic morphism of degree 3.

Defined moduli spaces of tropical trigonal covers and curves.

Proved the moduli space dimension matches that of algebraic trigonal curves.

Abstract

We prove that the existence of a divisor of degree $3$ and Baker-Norine rank at least $1$ on a $3$-edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree $3$ from a tropical modification of it to a tropical rational curve. Using the second description, we define the moduli spaces of $3$-edge connected tropical trigonal covers and of $3$-edge connected tropical trigonal curves, the latter as a locus in the moduli space of tropical curves. Finally, we prove that the moduli space of $3$-edge connected genus $g$ tropical trigonal curves has the same dimension as the moduli space of genus $g$ algebraic trigonal curves.