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On metric graphs with prescribed gonality. (English) Zbl 1381.05012
Summary: 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 $\min \{3 g - 3, 2 g + 2 d - 5 \} .$ 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.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C12 Distance in graphs 05C05 Trees 14T05 Tropical geometry (MSC2010)
##### Keywords:
metric graphs; tropical geometry; Brill-Noether theory; gonality
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##### References:
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