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

Moonen, B.J.; Polishchuk, Alexander

doi:10.1017/s147474801000006x

Cited by 6 publications

(6 citation statements)

References 19 publications

(86 reference statements)

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“…Examples of monoids to which our result applies are J (ungraded case) and C [•] := n 0 C [n] with C [n] the nth symmetric power of C (graded case). In [14] we have obtained a natural isomorphism β : CH * (C [•] ) ∼ − → CH * (J) t u , where CH * (C [•] ) and CH * (J) are both equipped with the Pontryagin product, and where R u denotes the PD-polynomial algebra in the variable u over a commutative ring R. Also we prove that the β is a PD-isomorphism with regard to the PD-structures on source and target as defined in the present paper.…”

Section: Introductionmentioning

confidence: 99%

Divided powers in Chow rings and integral Fourier transforms

Moonen

Polishchuk

2010

Advances in Mathematics

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We prove that for any monoid scheme M over a field with proper multiplication maps M × M → M , we have a natural PD-structure on the ideal CH>0(M ) ⊂ CH * (M ) with regard to the Pontryagin ring structure. Further we investigate to what extent it is possible to define a Fourier transform on the motive with integral coefficients of the Jacobian of a curve. For a hyperelliptic curve of genus g with sufficiently many k-rational Weierstrass points, we construct such an integral Fourier transform with all the usual properties up to 2 N -torsion, where N = 1 + ⌊log 2 (3g)⌋. As a consequence we obtain, over k =k, a PD-structure (for the intersection product) on 2 N · a, where a ⊂ CH(J) is the augmentation ideal. We show that a factor 2 in the properties of an integral Fourier transform cannot be eliminated even for elliptic curves over an algebraically closed field.the Pontryagin product and the usual product on CH(X) Q are interchanged by the Fourier transform CH(X) Q → CH(X t ) Q , we are led to ask whether one can define an integral version of the Fourier transform. An integral Fourier transform having all the usual properties would allow us to transport the PD-structure on CH * (X) for the Pontryagin product to a PDstructure on CH * (X t ) for the intersection product. Our second main result is that when X is the Jacobian J = J(C) of a hyperelliptic curve C of genus g, this plan works up to 2 N -torsion, with N = 1 + ⌊log 2 (3g)⌋. This relies on two elementary results in intersection theory of J and of powers of C that we establish in section 2. Namely, complementing the results of Collino in [4] we prove that up to 2-torsion the intersections of classes of the Brill-Noether loci W i in J are given by the same formula as in cohomology. Also, we check that a modified diagonal class on C × C × C introduced by Gross and Schoen in [11] is 2-torsion.The idea for the construction of an integral Fourier transform F for hyperelliptic Jacobians is as follows. Working with Q-coefficients one can express F entirely in terms of *products. To be precise, one has F(a) = (−1) g ·j * 2 E(τ ) * j 1, * (a) , where j 1 , j 2 : J → J 2 are the inclusions of the coordinate axes, τ ∈ CH(J 2 ) Q is the class (−1) g · ∆ * F(θ)−j 1, * F(θ)−j 2, * (θ) , and where E(τ ) is the * -exponential of τ . (See (3.7.3) in the text.) Our theorem on divided powers for the Pontryagin product allows us to define the * -exponential integrally. The next ingredient we need is the class F(θ), which, in general, we do not know how to define integrally. However, if C is hyperelliptic and ι : C → J is the embedding associated to a Weierstrass point p 0 ∈ C then we have F(θ) = (−1) g+1 · ι(C) . This puts us in a position to define, for hyperelliptic C, an integral endomorphism F : h(J) → h(J) of the ungraded motive h(J). We prove in Theorem 3.9 that it has all the expected properties up to 2 N -torsion. As a corollary we obtain a natural PD-structure on a large ideal in CH(J) for the intersection product; see Cor. 3.10.Our final result is Theorem 3.11, in which ...

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Section: Introductionmentioning

confidence: 99%

Divided powers in Chow rings and integral Fourier transforms

Moonen

Polishchuk

2010

Advances in Mathematics

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“…Part (ii) is due to Polishchuk; see [11], Corollary 4.4(iv). Part (iii) is essentially due to Polishchuk and the first author in [9] (see especially the proof of loc. cit.…”

Section: 2mentioning

confidence: 98%

“…The maps u d : C [d] → J give us a morphism u: C [•] → J, which induces a homomorphism u * : CH C [•] → CH(J). By [9], Theorem 3.4, there is a Q-subalgebra K ⊂ CH C [•] such that the restriction of u * to K gives an isomorphism K ∼ −→ CH(J). Further, by ibid., Lemma 8.4 and the proof of Theorem 8.5, all classes Γ n (C, a) lie in this subalgebra K and we have, for n 2,…”

Section: 3mentioning

confidence: 99%

Some remarks on modified diagonals

Moonen

Yin

2016

Commun. Contemp. Math.

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“…For a curve C which is smooth over a quasi-projective smooth base variety S, Moonen and Polischuk [MP10] have computed ⊕ n≥0 A * (C [n] ) in terms of A * (J), using a similar strategy to that of this paper. Their computation holds in Chow groups with integral coefficients.…”

Section: Relation To Existing Workmentioning

confidence: 99%

Homology of Hilbert schemes of points on a locally planar curve

Rennemo¹

2018

J. Eur. Math. Soc.

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Let C be a proper, integral, locally planar curve, and consider its Hilbert schemes of points C [n] . We define 4 creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface.As a corollary, we compute the cohomology of C [n] in terms of the cohomology of the compactified Jacobian of C together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently obtained in a different way by Maulik and Yun and independently Migliorini and Shende.Here Q[2] denotes the space Q with homological degree 2. A very similar statement was recently shown by Maulik and Yun [MY14] and Migliorini and Shende [MS13]. See Section 1.5 for a discussion of how these papers relate to this one.

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

Cited by 6 publications

References 19 publications

Divided powers in Chow rings and integral Fourier transforms

Divided powers in Chow rings and integral Fourier transforms

Some remarks on modified diagonals

Homology of Hilbert schemes of points on a locally planar curve

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