Lifting harmonic morphisms II: tropical curves and metrized complexes
Abstract
In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.
Cite
@article{arxiv.1404.3390,
title = {Lifting harmonic morphisms II: tropical curves and metrized complexes},
author = {Omid Amini and Matthew Baker and Erwan Brugallé and Joseph Rabinoff},
journal= {arXiv preprint arXiv:1404.3390},
year = {2016}
}
Comments
35 pages, 18 figures. This article used to be the second half of arXiv:1303.4812, and is now its sequel