Cycle relations on Jacobian varieties

Geer, Gerard van der; Kouvidakis, Alexis

doi:10.1112/s0010437x07002849

Cited by 7 publications

(18 citation statements)

References 3 publications

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“…Herbaut [10] proved that in this case we get new relations between the classes p m modulo algebraic equivalence. This result was reproved and simplified by van der Geer and Kouvidakis in [9]. In itself, it is not very hard to lift the result of Herbaut and van der Geer-Kouvidakis to the Chow level.…”

Section: Introductionmentioning

confidence: 99%

“…P be the two projections. As in van der GeerKouvidakis [9] we want to apply Grothendieck-Riemann-Roch to the line bundle M WD .q 1 id/ L on Y J and the morphism .q 2 id/ W Y J ! P J .…”

Section: Cycle Relations In the Chow Ringmentioning

confidence: 99%

“…In itself, it is not very hard to lift the result of Herbaut and van der Geer-Kouvidakis to the Chow level. We do this in Section 4, following the method of [9] which is based on a Grothendieck-RiemannRoch calculation. The relation we obtain (see Theorem 4.6) is the following.…”

Section: Introductionmentioning

confidence: 99%

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Relations between tautological cycles on Jacobians

Moonen¹

2009

Comment. Math. Helv.

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Abstract. We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Further we lift a result of Herbaut and van der Geer-Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuk's operator D lifts to the tautological subalgebra of CH.J /.

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Section: Introductionmentioning

confidence: 99%

“…P be the two projections. As in van der GeerKouvidakis [9] we want to apply Grothendieck-Riemann-Roch to the line bundle M WD .q 1 id/ L on Y J and the morphism .q 2 id/ W Y J ! P J .…”

Section: Cycle Relations In the Chow Ringmentioning

confidence: 99%

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Relations between tautological cycles on Jacobians

Moonen¹

2009

Comment. Math. Helv.

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“…They proved that if C admits a g 1 d , then p i is algebraically equivalent to zero if i ≥ d. Recently the second named author computed some relations between the cycles p i in T / alg for a curve admitting a base point free g r d . In [6], van der Geer and Kouvidakis gave another proof of the main result in [7], by using the Grothendieck-Riemann-Roch theorem. They gave simpler relations; however, as Zagier showed the set of relations is equivalent to those of Herbaut.…”

Section: 4mentioning

confidence: 97%

“…We present two proofs: the first follows the method in [6] and uses an argument in [7]. The second one follows the methods of [7].…”

Section: 5mentioning

confidence: 98%

On the tautological ring of a Jacobian modulo rational equivalence

Herbaut

2007

Geom Dedicata

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Abstract. We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g r d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P 1 whose ramification points are all of order d, and then for hyperelliptic curves.

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Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

Moonen

Polishchuk

2010

J. Inst. Math. Jussieu

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Abstract. Let C be a family of curves over a non-singular variety S. We study algebraic cycles on the relative symmetric powers C [n] and on the relative Jacobian J. We consider the Chow homology CH * (C [•] /S) := ⊕n CH * (C [n] /S) as a ring using the Pontryagin product. We prove that CH * (C [•] /S) is isomorphic to CH * (J/S)[t] u , the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over CH * (J/S). We give two such isomorphisms that over a general base are different. Further we give precise results on how CH * (J/S) sits embedded in CH * (C [•] /S) and we give an explicit geometric description of how the operators ∂ [m] t and ∂u act. This builds upon the study of certain geometrically defined operators Pi,j (a) that was undertaken by one of us in Part 1 of this work, [18].Our results give rise to a new grading on CH * (J/S). The associated descending filtration is stable under all operators [N ] * and [N ] * acts on gr m Fil as multiplication by N m . Hence, after − ⊗ Q this filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's.Finally we give a version of our main result for tautological classes, and we show how our methods give a simple geometric proof of some relations obtained by Herbaut and van der Geer-Kouvidakis, as later refined by one of us in [14].

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Cycle relations on Jacobian varieties

Cited by 7 publications

References 3 publications

Relations between tautological cycles on Jacobians

Relations between tautological cycles on Jacobians

On the tautological ring of a Jacobian modulo rational equivalence

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

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