Rank of Divisors on Hyperelliptic Curves and Graphs Under Specialization

Kawaguchi, Shu; Yamaki, Kazuhiko

doi:10.1093/imrn/rnu059

Cited by 12 publications

(27 citation statements)

References 29 publications

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“…We will address the problem whether the divisors constructed in some of the combinatorial results of Section are liftable. Other lifting problems addressed in the literature question themselves if given a metric graph there exists a curve such that any divisor on the graph is liftable, see for example . Question Let Γ be a metric graph.…”

Section: Specialization Map Triangulated Punctured Curves and Skeletamentioning

confidence: 99%

Gonality of complete graphs with a small number of omitted edges

Panizzut

2016

Mathematische Nachrichten

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Let K d be the complete metric graph on d vertices. We compute the gonality of graphs obtained from K d by omitting edges forming a K h , or general configurations of at most d − 2 edges. We also investigate if these graphs can be lifted to curves with the same gonality. We lift the former graphs and the ones obtained by removing up to d − 2 edges not forming a K 3 using models of plane curves with certain singularities. We also study the gonality when removing d − 1 edges not forming a K 3 . We use harmonic morphism to lift these graphs to curves with the same gonality because in this case plane singular models can no longer be used due to a result of Coppens and Kato.

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Section: Specialization Map Triangulated Punctured Curves and Skeletamentioning

confidence: 99%

Gonality of complete graphs with a small number of omitted edges

Panizzut

2016

Mathematische Nachrichten

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“…In [KY1] and [KY2, Thm 1.1 and Thm 1.2] the authors prove the inequality r alg (G, δ) ≥ r G (δ) in certain cases; combining with Theorem 4.2, we have the following partial answer to Problem 1.…”

Section: Now We Definementioning

confidence: 84%

Algebraic and combinatorial rank of divisors on finite graphs

Caporaso

Len

Melo

2015

Journal de Mathématiques Pures et Appliquées

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We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove that it is at most equal to the (usual) combinatorial rank, and that equality holds in many cases, though not in general.2000 Mathematics Subject Classification. Primary 14Hxx, 05Cxx .

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“…Our basic references are Amini, Baker, Brugallé, and Rabinoff [2], Baker and Norine [6], Amini and Caporaso [3], and Baker and Rabinoff [9]. See also [35,36].…”

Section: 2mentioning

confidence: 99%

“…For example, Cools, Draisma, Payne and Robeva [19] gave a tropical proof of the Brill-Noether theorem. See also [17,18,35,36], for example. Remark 1.9.…”

Section: Introductionmentioning

confidence: 99%

Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

Kawaguchi

Yamaki

2018

International Mathematics Research Notices

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Let R be a complete discrete valuation ring of equi-characteristic zero with fractional field K. Let X be a connected, smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model X of X over R and a relatively ample line bundle L over X with L | X ∼ = L. Let S(X ) be the skeleton associated to X in the Berkovich analytification X an of X. In this article, we study when S(X ) is faithfully tropicalized into tropical projective space by the adjoint linear system |L ⊗m ⊗ ω X |. Roughly speaking, our results show that, if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of S(X ).

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Rank of Divisors on Hyperelliptic Curves and Graphs Under Specialization

Cited by 12 publications

References 29 publications

Gonality of complete graphs with a small number of omitted edges

Gonality of complete graphs with a small number of omitted edges

Algebraic and combinatorial rank of divisors on finite graphs

Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

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