[Paper Review] Divisorial condition for the stable gonality of tropical curves

Yuki Kageyama|arXiv (Cornell University)|Jan 23, 2018

Polynomial and algebraic computation3 citations

TL;DR

This paper establishes a divisorial condition for stable $d$-gonality in tropical curves by proving that a tropical curve is stably $d$-gonal if and only if it admits a finitely generated linear system of degree $d$, rank one, and geometric dimension one. The key contribution is a characterization of stable gonality via divisor theory on tropical curves, using harmonic morphisms from modified graphs to trees.

ABSTRACT

Let d be a positive integer. There are several versions of d-gonality for tropical curves, stable d-gonality and divisorial d-gonality, which are both inspired by d-gonality for compact Riemann surfaces. However, that conditions are not equivalent. We have a condition of divisors equivalent to stable d-gonality for tropical curves.

Motivation & Objective

• To clarify the distinction between stable $d$-gonality and divisorial $d$-gonality in tropical curves.
• To identify a precise divisor-theoretic condition equivalent to stable $d$-gonality.
• To establish a bridge between tropical geometry and algebraic geometry via linear systems and harmonic morphisms.
• To provide a constructive characterization of stable $d$-gonality using modifications and tropical morphisms.

Proposed method

• Define the geometric dimension of a finitely generated linear system on a tropical curve.
• Construct a tropical modification $\widetilde{\varGamma}$ of the original curve $\varGamma$ by attaching subtrees $T_{p,n}$ at indeterminacy points $p$ of a linear system.
• Define a finite harmonic morphism $\widetilde{\varphi}: \widetilde{\varGamma} \to \varPhi_\Lambda(\varGamma)$ to a tree, using the image of the linear system under a tropical map.
• Prove that the morphism $\widetilde{\varphi}$ is harmonic by verifying degree conditions at all vertices, including indeterminacy points.
• Show that the degree of $\widetilde{\varphi}$ equals $d$, the degree of the divisor, using summation over preimages with multiplicity.
• Use the harmonic morphism to conclude that $\varGamma$ is stably $d$-gonal if and only if such a linear system exists.

Experimental results

Research questions

• RQ1 What divisorial condition characterizes stable $d$-gonality in tropical curves?
• RQ2 How does stable $d$-gonality relate to the existence of linear systems of degree $d$, rank one, and geometric dimension one?
• RQ3 Can stable $d$-gonality be characterized purely through divisor theory on tropical curves?
• RQ4 Is there a constructive way to realize stable $d$-gonality via modifications and harmonic morphisms?

Key findings

• A tropical curve $\varGamma$ is stably $d$-gonal if and only if it admits a finitely generated linear system of degree $d$, rank one, and geometric dimension one.
• The construction of a harmonic morphism $\widetilde{\varphi}: \widetilde{\varGamma} \to \varPhi_\Lambda(\varGamma)$ from a modified graph $\widetilde{\varGamma}$ ensures that the degree of the morphism is exactly $d$, matching the divisor degree.
• The morphism $\widetilde{\varphi}$ is harmonic at all points, including indeterminacy points, due to balanced degree conditions derived from divisor coefficients.
• The image $\varPhi_\Lambda(\varGamma)$ is a tree, which confirms that the morphism realizes stable $d$-gonality.
• The equivalence between stable $d$-gonality and the existence of such a linear system provides a divisorial characterization of stable gonality.
• The proof relies on a careful analysis of slopes and degrees in tropical linear systems, particularly at points where the map is not well-defined.