Lifting harmonic morphisms II: Tropical curves and metrized complexes

Abstract

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 (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.Throughout this paper, unless explicitly stated otherwise, K denotes a complete algebraically closed nonArchimedean field with nontrivial valuation val : K → R ∪ {∞}. Its valuation ring is denoted R, its maximal ideal is mR, and the residue field is k = R/mR. We denote the value group of K by Λ = val(K × ) ⊂ R.

Degeneration of Linear Series from the Tropical Point of View and Applications

“…We call such a pair minimal if there is no vertex v with val(v) = 1 and ω(v) = 0. Theorem 10.3 ( [Cap12,ABBR14b]). Let (Γ, ω) be a minimal vertex-weighted metric graph.…”

Lifting matroid divisors on tropical curves

“…5]. While the existence of a tame harmonic morphism depends on the characteristic, the dependence is only when the characteristic is at most the degree of the divisor [3,Rmk. 3.9].…”

“…In the case of rank 1 divisors, lifts can be constructed using the theory of harmonic maps of metrized complexes, which gives a complete theory for divisors defining tamely ramified maps to P 1 [3]. A sufficient condition for lifting a rank 1 divisor is for it to be the underlying graph of a metrized complex which has a tame harmonic morphism to a tree (see [4,Sec.…”

“…Now we consider the case that e contained in more than two flats, in which case M\e satisfies ρ > 0. By induction on the number of elements, we can assume that M\e is on our list, in which case the possibilities with ρ > 0 are (1) and case (3). For the former matroid, if e is contained in a flat of M\e, then M is a matroid of the type from the previous paragraph, with a equal to 1 or 2.…”

A metric graph satisfying w41=1$w_4^1 = 1$ that cannot be lifted to a curve satisfying dim⁡ (W41)=1$\dim \;(W_4^1 ) = 1$

“…We say that a linear system g As an example, from [13,Theorem 4.8] it is known that the linear system g 1 2 on a hyperelliptic graph having a vertex v adjacent to at least 3 different bridges (a bridge of a graph G is an edge e such that G n e is disconnected) is not liftable to a hyperelliptic curve because of the violation of some Hurwitz condition. In [14,Example 5.13] one finds an example due to Luo of a graph with a g 1 3 that cannot be lifted to a curve. Also in [7] one finds lots of types of linear systems g r d that cannot be lifted to curves, e.g.…”