Catalan-many tropical morphisms to trees; Part I: Constructions. (May 2021)

• Record Type: Journal Article
• Authors: Draisma, Jan; Vargas, Alejandro
• Is Part Of: Journal of symbolic computation. Volume 104(2021)
• Page Start: 580
• Page End: 629
• Publication Date: 2021-05
• Subjects: Tropical geometry -- Gonality -- Metric graph -- Tropical morphism -- Balancing condition; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.09.005
• Languages: English
• ISSNs: 0747-7171

Abstract

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.