On metric graphs with prescribed gonality

Filip Cools, Jan Draisma

Research output: Contribution to journalArticleAcademicpeer-review

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.

Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalJournal of Combinatorial Theory, Series A
Volume156
DOIs
Publication statusPublished - 1 May 2018

Fingerprint

Gonality
Metric Graphs
Harmonic Maps
Graph in graph theory
Moduli Space
Locus
Genus
Count
Lower bound
Upper bound

Keywords

  • Brill–Noether theory
  • Gonality
  • Metric graphs
  • Tropical geometry

Cite this

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On metric graphs with prescribed gonality. / Cools, Filip; Draisma, Jan.

In: Journal of Combinatorial Theory, Series A, Vol. 156, 01.05.2018, p. 1-21.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Draisma, Jan

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N2 - 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.

AB - 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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KW - Tropical geometry

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