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

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-[center dot]/S) := circle plus(n) CH*(C-[n]/S) as a ring using the Pontryagin product. We prove that CH*(C-[center dot]/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-[center dot]/S) and we give an explicit geometric description of how the operators partial derivative([m])(t) and partial derivative(u) act. This builds upon the study of certain geometrically defined operators P-i,P-j(a) that was undertaken by one of us.
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(Fil)(m) as multiplication by N-m. Hence, after - circle times 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.