Relations between tautological cycles on Jacobians

Abstract

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 /.

Algebraic cycles on Prym varieties

“…Our definition of the tautological rings: taut(J, ϕ(C)), Taut(J, ϕ(C)) and T (J, ϕ(C)) coincides with the previous definitions denoted in [32] by taut(C), Taut(C) and T (C), respectively.…”

“…The small tautological ring taut(C) of J is defined to be the smallest Q-subalgebra of CH(J ) under the intersection product, which contains the class of the image of C under ϕ, and is stable under the operations * , F J , k * and k * for all k ∈ Z. The big tautological ring Taut(C) is defined in the same way, except it is required to contain the image of ϕ * : CH(C) → CH(J ) [32,Def. 3.2,p.…”

“…)C * d . The generators for the tautological rings taut(C) and Taut(C) of the Jacobian J have also been determined in [37,Thm. 0.2, p. 461] and [32,Thm. 3.6,p.…”

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

“…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]. …”

“…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].…”