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

Baker, Matthew; Jensen, David

doi:10.1007/978-3-319-30945-3_11

Cited by 39 publications

(40 citation statements)

References 101 publications

(110 reference statements)

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“…The local lifting multiplicity for the cases (1), (2) and (4) can be found in Table 1. The lifting multiplicity for the case (3) can be found in Proposition 3.7 (see also Lemma 3.9 and Table 1), and for case (5) in Proposition 3.11. Theorem 3.1.…”

Section: Lifting Bitangent Linesmentioning

confidence: 97%

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Lifting tropical bitangents

Len

Markwig

2020

Journal of Symbolic Computation

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We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents; however the conditions to determine whether a tropical bitangent has real lifts are not purely combinatorial.2010 Mathematics Subject Classification. 14T05.

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Section: Lifting Bitangent Linesmentioning

confidence: 97%

“…In order to state the genericity condition 3.3, we first need to define an invariant of tangency points which we call the initial lift. Propositions 3.6 and 3.7 motivate our choice of terminology: the initial lift equals the residue m of the constant coefficient of the line equation (5) in both cases (2) and (3).…”

Section: Lifting Bitangent Linesmentioning

confidence: 99%

Lifting tropical bitangents

Len

Markwig

2020

Journal of Symbolic Computation

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“…In the following two sections, we will use the theory of linear systems of divisors on metric graphs/tropical curves. For the definitions, we refer to [1,2,10,16]. We will often use the terminology of chip firing (see e.g.…”

Section: Characterization Of Reduced Divisors On Metric Graphsmentioning

confidence: 99%

Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches

Cools¹,

D'Adderio²,

Jensen³

et al. 2019

Algebraic Combinatorics

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The gonality sequence (dr) r≥1 of a smooth algebraic curve comprises the minimal degrees dr of linear systems of rank r. We explain two approaches to compute the gonality sequence of smooth curves in P 1 × P 1 : a tropical and a classical approach. The tropical approach uses the recently developed Brill-Noether theory on tropical curves and Baker's specialization of linear systems from curves to metric graphs [1]. The classical one extends the work [11] of Hartshorne on plane curves to curves on P 1 × P 1 .

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“…The question of whether a divisor D ∈ Div Q (Γ ′ ) of rank r(D) lifts to a divisor in X K of the same rank is not completely solved. A survey for partial results can be find in [BJ16]. The case when Γ ′ is a chain of loops with generic edge length is proved liftable in [CJP15] by a thoroughly examination of divisors on Γ ′ .…”

Section: Lifting Divisors On Metric Graphsmentioning

confidence: 99%

Smoothing of Limit Linear Series on Curves and Metrized Complexes of Pseudocompact Type

He¹

2018

Can. J. Math.-J. Can. Math.

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We investigate the connection between Osserman limit series [Oss] (on curves of pseudocompact type) and Amini-Baker limit linear series [AB15] (on metrized complexes with corresponding underlying curve) via a notion of pre-limit linear series on curves of the same type. Then, applying the smoothing theorems of Osserman limit linear series, we deduce that, fixing certain metrized complexes, or for certain types of Amini-Baker limit linear series, the smoothability is equivalent to a certain "weak glueing condition". Also for arbitrary metrized complexes of pseudocompact type the weak glueing condition (when it applies) is necessary for smoothability. As an application we confirm the lifting property of specific divisors on the metric graph associated to a certain regular smoothing family, and give a new proof of [CJP15, Theorem 1.1] for vertex avoiding divisors, and generalize loc.cit. for divisors of rank one in the sense that, for the metric graph, there could be at most three edges (instead of two) between any pair of adjacent vertices.

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

Cited by 39 publications

References 101 publications

Lifting tropical bitangents

Lifting tropical bitangents

Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches

Smoothing of Limit Linear Series on Curves and Metrized Complexes of Pseudocompact Type

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