Catalan-many tropical morphisms to trees; Part II: A space and a count

Abstract

In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus g and degree d = g/2 + 1, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number C_g/2 = 1/(g/2+1)binomgg/2. In this paper, we construct the tropical counterpart of this universal family: a polyhedral cone complex 𝒢_{g → 0, d}^trop parametrizing degree-d tropical morphisms from genus-g metric graphs to metric trees. For even g and d = g/2 + 1, we prove that the forgetful projection Π: 𝒢_{g → 0, d}^trop → ℳ_g^trop is a branched cover of degree C_g/2 equipped with natural determinantal multiplicities. We compute this degree by showing that on caterpillars of loops the morphisms are in bijection with ballot sequences, and we establish its global invariance across ℳ_g^trop via a tropical balancing condition across codimension-$1$ walls. Via deformation and path lifting, this yields an effective method to construct Catalan-many gonality-witnessing maps for any generic metric graph, establishing that the tree gonality of any genus-g metric graph is at most ⌈ g/2 ⌉ + 1.

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BibTeX RIS

BibTeX

@misc{vargas2026catalanmany,
  title = {{Catalan-many tropical morphisms to trees; Part II: A space and a count}},
  author = {Alejandro Vargas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09109},
  eprint = {2609.09109},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09109},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Catalan-many tropical morphisms to trees; Part II: A space and a count
AU  - Alejandro Vargas
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09109
UR  - https://arxiv.org/abs/2609.09109
AB  - In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus g and degree d = g/2 + 1, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number C_g/2 = 1/(g/2+1)binomgg/2. In this paper, we construct the tropical counterpart of this universal family: a polyhedral cone complex 𝒢_g → 0, d^trop parametrizing degree-d tropical morphisms from genus-g metric graphs to metric trees. For even g and d = g/2 + 1, we prove that the forgetful projection Π: 𝒢_g → 0, d^trop → ℳ_g^trop is a branched cover of degree C_g/2 equipped with natural determinantal multiplicities. We compute this degree by showing that on caterpillars of loops the morphisms are in bijection with ballot sequences, and we establish its global invariance across ℳ_g^trop via a tropical balancing condition across codimension-$1$ walls. Via deformation and path lifting, this yields an effective method to construct Catalan-many gonality-witnessing maps for any generic metric graph, establishing that the tree gonality of any genus-g metric graph is at most ⌈ g/2 ⌉ + 1.
ER  -

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