On metric graphs with prescribed gonality
Filip Cools, Jan Draisma

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.
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