Linear series on metrized complexes of algebraic curves

Amini, Omid; Baker, Matthew

doi:10.1007/s00208-014-1093-8

Cited by 59 publications

(145 citation statements)

References 36 publications

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“…We refer to [BN07,MZ08,AC13,AB12] for the basic definitions concerning ranks of divisors on metric graphs, augmented metric graphs, and metrized complexes of curves.…”

Section: Gonality and Rankmentioning

confidence: 99%

Lifting harmonic morphisms II: Tropical curves and metrized complexes

Amini

Baker

Brugallé

et al. 2015

Algebra Number Theory

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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.

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“…We refer to [BN07,MZ08,AC13,AB12] for the basic definitions concerning ranks of divisors on metric graphs, augmented metric graphs, and metrized complexes of curves.…”

Section: Gonality and Rankmentioning

confidence: 99%

Lifting harmonic morphisms II: Tropical curves and metrized complexes

Amini

Baker

Brugallé

et al. 2015

Algebra Number Theory

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“…The curve X of that solution cannot be hyperelliptic since G n is not hyperelliptic. This follows from the specialisation Theorem from [2] or [3] already mentioned in the introduction. From [11] one obtains the following classification in case char.k/ ¤ 2 of non-hyperelliptic curves X of genus at least 6 satisfying dim.W , X is a smooth plane curve of degree 5 (hence has genus 6 and has a g 2 5 ) or X is bi-elliptic (there exists a double covering W X !…”

Section: The Lifting Problemmentioning

confidence: 77%

“…[2], since we restrict to the case of zero augmentation map this is in principle considered in [3]) dim.jDj/ Ä rk. .D// .…”

Section: Introductionmentioning

confidence: 99%

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

Coppens

2016

Open Mathematics

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For all integers g 6 we prove the existence of a metric graph G with w 1 4 D 1 such that G has Clifford index 2 and there is no tropical modification G 0 of G such that there exists a finite harmonic morphism of degree 2 from G 0 to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

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“…4 We recall from [1] that a lift of Ŵ M to a metrized complex means associating a P 1 k for each vertex v of the graph, which we denote C v , and a point on C v for each edge incident to v. A lift of the divisor D M is a choice of a point on C e for each element e of M.…”

Section: Proposition 22 the Divisor D M Has Rankmentioning

confidence: 99%

“…This bound can be sharpened by incorporating additional information about the components of the special fiber, giving augmented graphs [2] or metrized complexes [1]. All of these inequalities can be strict because there may be many algebraic curves and divisors with the same specialization.…”

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confidence: 99%

Lifting matroid divisors on tropical curves

Cartwright

2015

Mathematical Sciences

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The specialization inequality in tropical geometry gives an upper bound for the rank of a divisor on a curve in terms of a combinatorial quantity known as the rank of the specialization of the divisor on the dual graph of the special fiber of a degeneration [6]. This bound can be sharpened by incorporating additional information about the components of the special fiber, giving augmented graphs [2] or metrized complexes [1]. All of these inequalities can be strict because there may be many algebraic curves and divisors with the same specialization. Thus, the natural question is whether, for a given graph and divisor on that graph, the inequality is sharp for some algebraic curve and divisor. If R is the discrete valuation ring over which the degeneration of the curve is defined, we will refer to such a curve and divisor as a lifting of the graph with its divisor over R. In this paper, we show that the existence of a lifting can depend strongly on the characteristic of the field: lift over k[[t]] if and only if the characteristic of k is in P, and Ŵ ′ and D ′ lift over k[[t]] if and only if the characteristic of k is not in P.We also show that the existence of a lift depends on the field even beyond characteristic:Abstract Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain characteristics or over certain fields of definition. Our examples also apply to the theory of metrized complexes and weighted graphs. These examples arise by relating the lifting problem to matroid realizability. We also give a proof of Mnëv universality with explicit bounds on the size of the matroid, which may be of independent interest.

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Linear series on metrized complexes of algebraic curves

Cited by 59 publications

References 36 publications

Lifting harmonic morphisms II: Tropical curves and metrized complexes

Lifting harmonic morphisms II: Tropical curves and metrized complexes

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

Lifting matroid divisors on tropical curves

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