On metric graphs with prescribed gonality

Filip Cools; Jan Draisma

doi:10.1016/j.jcta.2017.11.017

On metric graphs with prescribed gonality

Filip Cools, Jan Draisma

Discrete Mathematics

Research output: Contribution to journal › Article › Academic › peer-review

6 Citations (Scopus)

70 Downloads (Pure)

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 language	English
Pages (from-to)	1-21
Number of pages	21
Journal	Journal of Combinatorial Theory, Series A
Volume	156
DOIs	https://doi.org/10.1016/j.jcta.2017.11.017
Publication status	Published - 1 May 2018

Funding

We thank Aart Blokhuis, who provided the proof of Lemma 9 reproduced in this paper. JD was partially supported by Vidi 639.032.019 and Vici 639.033.514 grants from the Netherlands Organisation for Scientific Research (NWO).

Keywords

Brill–Noether theory
Gonality
Metric graphs
Tropical geometry

Access to Document

10.1016/j.jcta.2017.11.017

author versionFinal author version , 445 KB

Cite this

@article{28b599196ef7476785ded7f99b42d166,

title = "On metric graphs with prescribed gonality",

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.",

keywords = "Brill–Noether theory, Gonality, Metric graphs, Tropical geometry",

author = "Filip Cools and Jan Draisma",

year = "2018",

month = may,

day = "1",

doi = "10.1016/j.jcta.2017.11.017",

language = "English",

volume = "156",

pages = "1--21",

journal = "Journal of Combinatorial Theory, Series A",

issn = "0097-3165",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - On metric graphs with prescribed gonality

AU - Cools, Filip

AU - Draisma, Jan

PY - 2018/5/1

Y1 - 2018/5/1

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.

KW - Brill–Noether theory

KW - Gonality

KW - Metric graphs

KW - Tropical geometry

UR - https://www.scopus.com/pages/publications/85038222901

U2 - 10.1016/j.jcta.2017.11.017

DO - 10.1016/j.jcta.2017.11.017

M3 - Article

AN - SCOPUS:85038222901

SN - 0097-3165

VL - 156

SP - 1

EP - 21

JO - Journal of Combinatorial Theory, Series A

JF - Journal of Combinatorial Theory, Series A

ER -

On metric graphs with prescribed gonality

Abstract

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this