Gonality of algebraic curves and graphs

TL;DR

This paper explores the relationship between algebraic curves and graphs through gonality, establishing equivalences and introducing harmonic indexed morphisms, with implications for tropical geometry and hyperelliptic graphs.

Contribution

It proves the duality between d-gonal graphs and d-gonal curves and introduces harmonic indexed morphisms to define d-gonality on graphs.

Findings

Every d-gonal weighted graph of Hurwitz type is the dual of a d-gonal curve.

The dual graph of a d-gonal curve is equivalent to a d-gonal graph.

Generalizations to higher dimensions and applications to tropical curves are provided.

Abstract

We study the interplay between the classical theory of linear series on curves, and the recent theory of linear series on graphs. We prove that every d-gonal (weighted) graph of Hurwitz type is the dual graph of a d-gonal curve. Conversely the dual graph of a d-gonal curve is equivalent to a d-gonal graph. We define d-gonal graphs by what we call harmonic indexed morphisms. Generalizations to higher dimensional linear series, and applications to tropical curves and hyperelliptic graphs are given.