A Non-Hyperelliptic Curve with Torsion Ceresa Cycle Modulo Algebraic Equivalence

Beauville, Arnaud; Schoen, Chad

doi:10.1093/imrn/rnab344

Cited by 5 publications

(10 citation statements)

References 17 publications

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“…By this computation, formula (3.6) and the fact that (𝛿 𝐺 − 𝐼) 2 | 𝐹 2 (𝐿/𝐻 ) = 0, we deduce that (𝛿 𝐺 − 𝐼) 2 | 𝐹 1 (𝐿/𝐻 ) = 𝜓 𝐺 | 𝐹 1 (𝐿/𝐻 ) . Therefore,…”

Section: Proposition 41mentioning

confidence: 68%

“…Nevertheless, it is always trivial for hyperelliptic curves, and it has long been conjectured that these are the only curves with trivial Ceresa cycle [13,Question 8.5], [4,Remark 1.2]. Although several recent results [1,2,4,14] present nonhyperelliptic curves whose Ceresa cycles give rise to torsion classes in the Griffiths group, this problem remains open. The goal of this paper is to study hyperellipticity of tropical curves and graphs using cohomological invariants arising from the study of Ceresa triviality of algebraic curves defined over C((𝑡)).…”

Section: Introductionmentioning

confidence: 99%

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The Ceresa class and tropical curves of hyperelliptic type

Corey,

2024

Forum of Mathematics, Sigma

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We define a new algebraic invariant of a graph G called the Ceresa–Zharkov class and show that it is trivial if and only if G is of hyperelliptic type, equivalently, G does not have as a minor the complete graph on four vertices or the loop of three loops. After choosing edge lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph G that is closely related to the Ceresa cycle for an algebraic curve defined over $\mathbb {C}(\!(t)\!)$ .

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Section: Proposition 41mentioning

confidence: 68%

Section: Introductionmentioning

confidence: 99%

The Ceresa class and tropical curves of hyperelliptic type

Corey,

2024

Forum of Mathematics, Sigma

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“…Suppose G 1 and G 2 are Ceresa-Zharkov trivial. By Proposition 4.2, there are elements 2 ; this follows from Formula (3.5) and the fact that…”

Section: Proposition 55 a Connected Graph G Is Ceresa-zharkov Trivial...mentioning

confidence: 86%

“…Nevertheless, it is always trivial for hyperelliptic curves, and it has long been conjectured that these are the only curves with trivial Ceresa cycle [10,Question 8.5], [3,Remark 1.2]. Although several recent results [1,2,3,11] present nonhyperelliptic curves whose Ceresa cycles give rise to torsion classes in the Griffiths group, this problem remains open. The goal of this paper is to probe the relationship between Ceresa triviality and hyperellipticity from the tropical viewpoint.…”

Section: Introductionmentioning

confidence: 99%

“…2 (F 1 (L/H)) is spanned by2β 1 ∧ β 2 ∧ β 3 + 2β 1 ∧ β 2 ∧ β 4 + 2β 2 ∧ β 3 ∧ β 4 , 4β 1 ∧ β 2 ∧ β 4 , 2β 1 ∧ β 3 ∧ β 4 + 2β 2 ∧ β 3 ∧ β 4 , 4β 2 ∧ β 3 ∧ β 4 ,andw τ (Γ) = −2β 1 ∧ β 2 ∧ β 3 − 2β 1 ∧ β 2 ∧ β 4 . One can readily check that −2β 1 ∧ β 2 ∧ β 3 − 2β 1 ∧ β 2 ∧ β 4 isnot in the Z-span of the vectors listed above, and hence L 3 is not Ceresa-Zharkov trivial.…”

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confidence: 99%

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The Ceresa class and tropical curves of hyperelliptic type

Corey¹,

Li²

2022

Preprint

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We define a new algebraic invariant of a graph G called the Ceresa-Zharkov class and show that it is trivial if and only if G is of hyperelliptic type, equivalently, G does not have as a minor the complete graph on 4 vertices or the loop of 3 loops. After choosing edge-lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph G that is closely related to the Ceresa cycle for an algebraic curve defined over C((t)).

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The Ceresa Class: Tropical, Topological and Algebraic

Corey,

Ellenberg,

2023

J. Inst. Math. Jussieu

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The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for nonhyperelliptic curves is in general not easy. We present a ‘combinatorialization’ of this problem, explaining how to define a Ceresa class for a tropical algebraic curve and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over $\mathbb {C}(\!(t)\!)$ and show that the Ceresa class in each of these settings is torsion.

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A Non-Hyperelliptic Curve with Torsion Ceresa Cycle Modulo Algebraic Equivalence

Abstract: We exhibit a non-hyperelliptic curve $C$ of genus $3$ such that the class of the Ceresa cycle $[C]-[C^-]$ in $JC$ modulo algebraic equivalence is torsion.

Cited by 5 publications

References 17 publications

The Ceresa class and tropical curves of hyperelliptic type

The Ceresa class and tropical curves of hyperelliptic type

The Ceresa class and tropical curves of hyperelliptic type

The Ceresa Class: Tropical, Topological and Algebraic

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