Divisorial and geometric gonality of higher-rank tropical curves

TL;DR

This paper extends the concepts of divisorial and geometric gonality to higher-rank tropical curves modeled as monoid-metrised graphs, establishing bounds and relationships between these notions.

Contribution

It introduces a new framework for gonality in monoid-metrised graphs and proves key inequalities and existence results linking divisorial and geometric gonality.

Findings

Geometric gonality bounds divisorial gonality from above.

Existence of graph subdivisions with gonality bounded by monoid-metrised gonality.

Connection to minimal degree maps between logarithmic curves.

Abstract

We consider a variant of metrised graphs where the edge lengths take values in a commutative monoid, as a higher-rank generalisation of the notion of a tropical curve. Divisorial gonality, which Baker and Norine defined on combinatorial graphs in terms of a chip firing game, is extended to these monoid-metrised graphs. We define geometric gonality of a monoid-metrised graph as the minimal degree of a horizontally conformal, non-degenerate morphism onto an monoid-metrised tree, and prove that geometric gonality is an upper bound for divisorial gonality in the monoid-metrised case. We also show the existence of a subdivision of the underlying graph whose gonality is no larger than the monoid-metrised gonality. We relate this to the minimal degree of a map between logarithmic curves.