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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
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