## Abstract

We investigate the tree gonality of a genus-g metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this number is at most ⌈g/2⌉+1, a fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, the tropical morphism which realizes the bound belongs to a family of tropical morphisms that is pure of dimension 3g−3 and that has a generically finite-to-one map onto the moduli space of genus-g metric graphs. Our methods focus on the study of such families. This is part I in a series of two papers: in part I we fix the combinatorial type of the metric graph to show a bound on tree-gonality, while in part II we vary the combinatorial type and show that the number of tropical morphisms, counted with suitable multiplicities, is the same Catalan number that counts morphisms from a general genus-g curve to the projective line.

Original language | English |
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Pages (from-to) | 580-629 |

Number of pages | 50 |

Journal | Journal of Symbolic Computation |

Volume | 104 |

DOIs | |

Publication status | Published - 1 May 2021 |

## Keywords

- Balancing condition
- Gonality
- Metric graph
- Tropical geometry
- Tropical morphism