On metric graphs with prescribed gonality

TL;DR

This paper characterizes the dimension of the space of genus-g metric graphs with gonality at most d, providing bounds and constructions based on harmonic maps and combinatorial data.

Contribution

It establishes the precise dimension of the gonality-limited locus in the moduli space of metric graphs, combining parameter counts and explicit constructions.

Findings

Dimension of gonality ≤ d locus is min{3g-3, 2g+2d-5}

Constructs graphs with prescribed gonality using harmonic maps

Introduces combinatorial data for harmonic maps to trees

Abstract

We prove that in the moduli space of genus-g metric graphs the locus of graphs with gonality at most d has the classical dimension min{3g-3,2g+2d-5}. This follows from a careful parameter count to establish the upper bound and a construction of sufficiently many graphs with gonality at most d to establish the lower bound. Here, gonality is the minimal degree of a non-degenerate harmonic map to a tree that satisfies the Riemann-Hurwitz condition everywhere. Along the way, we establish a convenient combinatorial datum capturing such harmonic maps to trees.