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arXiv · 2112.04205

Divisorial and geometric gonality of higher-rank tropical curves

Josse van Dobben de Bruyn·David Holmes·David van der Vorm
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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.

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Josse van Dobben de Bruyn, David Holmes, David van der Vorm. 2021-12-08. Divisorial and geometric gonality of higher-rank tropical curves. https://arxiv.org/abs/2112.04205

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