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Tropical trigonal curves

Margarida Melo, Angelina Zheng

TL;DR

The paper proves a tropical analogue of the trigonal description for genus $g$ curves by showing that, on a $3$-edge-connected tropical curve, the existence of a divisor of degree $3$ with Baker-Norine rank at least $1$ is equivalent to the existence of a non-degenerate harmonic morphism of degree $3$ from a tropical modification to a tropical tree. It develops moduli spaces for 3-edge-connected tropical trigonal covers and curves, proving they have the same dimension as the algebraic trigonal locus and identifying maximal cells via 3-ladders. The approach blends divisor theory on metric graphs with harmonic morphisms, using tropical modifications to mirror admissible covers in the algebraic setting. The results pave the way for a tropical boundary analysis analogous to the boundary of $ar M_g$ and suggest further links to cohomology through tropical moduli spaces and tropical admissible covers.

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.

Tropical trigonal curves

TL;DR

The paper proves a tropical analogue of the trigonal description for genus curves by showing that, on a -edge-connected tropical curve, the existence of a divisor of degree with Baker-Norine rank at least is equivalent to the existence of a non-degenerate harmonic morphism of degree from a tropical modification to a tropical tree. It develops moduli spaces for 3-edge-connected tropical trigonal covers and curves, proving they have the same dimension as the algebraic trigonal locus and identifying maximal cells via 3-ladders. The approach blends divisor theory on metric graphs with harmonic morphisms, using tropical modifications to mirror admissible covers in the algebraic setting. The results pave the way for a tropical boundary analysis analogous to the boundary of and suggest further links to cohomology through tropical moduli spaces and tropical admissible covers.

Abstract

We prove that the existence of a divisor of degree and Baker-Norine rank at least on a -edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree from a tropical modification of it to a tropical rational curve. Using the second description, we define the moduli spaces of -edge connected tropical trigonal covers and of -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 -edge connected genus tropical trigonal curves has the same dimension as the moduli space of genus algebraic trigonal curves.
Paper Structure (12 sections, 20 theorems, 29 equations, 29 figures)

This paper contains 12 sections, 20 theorems, 29 equations, 29 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Metric graphs
  5. Harmonic morphisms of graphs and metric graphs
  6. Divisor theory on metric graphs
  7. Gonality of graphs and metric graphs
  8. Trigonal and divisorially trigonal metric graphs: the $3$-edge connected case
  9. The loopless case
  10. The case with loops
  11. Moduli of tropical trigonal curves and tropical trigonal covers
  12. Maximal cells for the moduli space of $3$-edge connected trigonal curves

Key Result

Theorem 1.1

Let $\Gamma$ be a $3$-edge connected metric graph with canonical loopless model $(G_{-},l_{-}).$ The following are equivalent.

Figures (29)

  • Figure 2.1: A non-degenerate non-harmonic morphism.
  • Figure 2.2: A degenerate morphism.
  • Figure 2.3: A non-degenerate harmonic morphism of degree $2.$
  • Figure 2.4: A non-degenerate harmonic morphism of degree $3$ and a tropical modification without edge contractions .
  • Figure 2.5: A divisorially trigonal metric graph of genus $5$ with two non linearly equivalent divisors of degree $3$ and rank $1$.
  • ...and 24 more figures

Theorems & Definitions (84)

  • Theorem 1.1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Definition 6
  • Definition 7
  • Remark 2
  • ...and 74 more