An inflectionary tangent to the Kummer variety and the Jacobian condition

Marini, Giambattista

doi:10.1007/s002080050123

Cited by 5 publications

(7 citation statements)

References 8 publications

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“…The strategy consists in showing that the presence of an inflectionary tangent line to the Kummer variety is equivalent to the fact that θ satisfies the KP equation, and then conclude using Theorem 1. First of all, in both [24], and [6], it is shown that the image φ(b) ∈ Y of a point b ∈ X, with a := 2b = 0 is a flex, if and only if there exist constant vector fields D 1 = 0, D 2 on X, and a constant d ∈ C such that the following bilinear differential equation ( 13)…”

Section: Kummer Varieties With One Flex Come From Jacobiansmentioning

confidence: 99%

“…Suppose that (13) holds but that (11) does not. We may then use Marini's Theorem 2 in [24], or else Dichotomy 1 in [6], to conclude that there exists an irreducible component W of D 1 Θ such that W red is D 1 -invariant and such that (13) holds on W , but…”

Section: Kummer Varieties With One Flex Come From Jacobiansmentioning

confidence: 99%

“…Our arguments build on the methods introduced by De Concini and the first author (see [1], [2], [3], [4], [5]), and on the subsquent developments mostly by Debarre and Marini (see [13], [24], [14], [25], [6]), which all provided algebro-geometric proofs of the above theorems, but only under certain additional assumptions.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we only deal with the first two cases. We treat the case of the flex in Subsection 5, where we use some of the ideas contained in Marini's paper [24], and in [6]. Finally, we treat the case and of a degenerate trisecant in Subsection 6, where we base our arguments on the work done by Debarre in [13], and [14].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach

Arbarello¹,

Codogni²,

Pareschi³

2020

Preprint

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We give completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota's characterization is given in terms of the KP equation. Krichever's characterization is given in terms of trisecant lines to the Kummer variety. Here we treat the case of flexes and degenerate trisecants. The basic tool we use is a theorem we prove asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result allows us to remove all the extra assumptions that were introduced in the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above.

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Section: Kummer Varieties With One Flex Come From Jacobiansmentioning

confidence: 99%

Section: Kummer Varieties With One Flex Come From Jacobiansmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach

Arbarello¹,

Codogni²,

Pareschi³

2020

Preprint

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show abstract

“…In this section we will explain a dichotomy that was first proved in [Mar97]. The dichotomy is the following.…”

Section: -Invariant Flowsmentioning

confidence: 99%

Characterizing Jacobians via flexes of the Kummer Variety

Arbarello

Krichever

Marini

2006

Mathematical Research Letters

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Abstract. Given an abelian variety X and a point a ∈ X we denote by < a > the closure of the subgroup of X generated by a. Let N = 2 g − 1. We denote by κ : X → κ(X) ⊂ P N the map from X to its Kummer variety. We prove that an indecomposable abelian variety X is the Jacobian of a curve if and only if there exists a point a = 2b ∈ X \ {0} such that < a > is irreducible and κ(b) is a flex of κ(X).

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Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach

Arbarello

Codogni

Pareschi

2021

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

We give completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota’s characterization is given in terms of the KP equation. Krichever’s characterization is given in terms of trisecant lines to the Kummer variety. Here we treat the case of flexes and degenerate trisecants. The basic tool we use is a theorem we prove asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result allows us to remove all the extra assumptions that were introduced in the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above.

show abstract

An inflectionary tangent to the Kummer variety and the Jacobian condition

Cited by 5 publications

References 8 publications

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach

Characterizing Jacobians via flexes of the Kummer Variety

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach

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