Algebra & Number Theory

Lifting harmonic morphisms II: Tropical curves and metrized complexes

Omid Amini, Matthew Baker, Erwan Brugallé, and Joseph Rabinoff

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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.

This article is the second in a series of two.

Article information

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

Received: 25 June 2013
Revised: 9 July 2014
Accepted: 6 December 2014
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G22: Rigid analytic geometry
Secondary: 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20] 11G20: Curves over finite and local fields [See also 14H25]

tropical lifting skeleton Berkovich space analytic curve harmonic morphism Hurwitz number metrized complex


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.

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