On metric graphs with prescribed gonality
Combinatorics
2017-10-10 v2 Algebraic Geometry
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.
Cite
@article{arxiv.1602.05542,
title = {On metric graphs with prescribed gonality},
author = {Filip Cools and Jan Draisma},
journal= {arXiv preprint arXiv:1602.05542},
year = {2017}
}
Comments
18 pages, 10 figures, corrected an erroneous lemma in previous version, many further improvements suggested by referees