Divisorial and geometric gonality of higher-rank tropical curves

We consider a notion of metric graphs where 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 metric graph as the minimal degree of a horizontally conformal, non-degenerate morphism onto a metric tree, and prove that geometric gonality is an upper bound for divisorial gonality in the metric case. We relate this to the minimal degree of a map between logarithmic curves.Comment: 11 pages; comments very welcom