Catalan-many tropical morphisms to trees; Part I: Constructions. (May 2021) - Draisma, Jan; Vargas, Alejandro
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 3 g − 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.