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Tautological cycles on Jacobian varieties. (English) Zbl 1148.14006
Let $$C$$ be a nonsingular projective curve of genus $$g$$ and let $$J(C)$$ denote its Jacobian variety. Let $$A_\bullet(J(C))_\mathbb{Q}$$ denote the group of rational algebraic cycles modulo algebraic equivalence, graded by dimension. The so-called tautological ring $${\mathcal R}(C)$$ is defined to be the subgroup of $$A_\bullet(J(C))_\mathbb{Q}$$ containing the class of $$C$$ and stable under the Fourier transform, the intersection product, the Pontryagin product, pullbacks and pushforwards through multiplication maps by integers.
In this paper the author constructs a set of generators of $${\mathcal R}(C)$$, which enables us to control all the possible structures of the tautological ring. Let $$[A_d(J(C))_{\mathbb{Q}}]_x= \{W\in A_d(J(C))_{\mathbb{Q}}; m^* W= m^{2g-2d-s} W$$ for any $$m\in\mathbb{Z}\}$$. He takes the set of $$s$$-components $$C_{(s)}\in[A_1(J(C))_{\mathbb{Q}}]_s$$ of the curve $$C$$ as a starting point, and constructs inductively a certain cycle $$\lambda_{\{i_1,\dots, i_d\}}\in [A_d(J(C))_{\mathbb{Q}}]_{i_1+\cdots+ i_d}$$ which is annihilated by the principal polarization divisor and is congruent to $$C_{(i_1)}*\cdots* C_{(i_d)}$$ modulo the subspace spanned by elements of lower degree. As an application he gives an exhaustive description of $${\mathcal R}(C)$$ for all the possibilities that may occur in the cases when $$g\leq 9$$.

##### MSC:
 14C25 Algebraic cycles 14C15 (Equivariant) Chow groups and rings; motives 14H40 Jacobians, Prym varieties 14K12 Subvarieties of abelian varieties
##### Keywords:
algebraic cycles; Chow group; Jacobian
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##### References:
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