## Algebra & Number Theory

### Lifting harmonic morphisms II: Tropical curves and metrized complexes

#### Abstract

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 nonvanishing 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 (nonaugmented) 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.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 2 (2015), 267-315.

Dates
Revised: 9 July 2014
Accepted: 6 December 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842281

Digital Object Identifier
doi:10.2140/ant.2015.9.267

Mathematical Reviews number (MathSciNet)
MR3320845

Zentralblatt MATH identifier
1312.14138

#### Citation

Amini, Omid; Baker, Matthew; Brugallé, Erwan; Rabinoff, Joseph. Lifting harmonic morphisms II: Tropical curves and metrized complexes. Algebra Number Theory 9 (2015), no. 2, 267--315. doi:10.2140/ant.2015.9.267. https://projecteuclid.org/euclid.ant/1510842281

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