Journal of the Institute of Mathematics of Jussieu

Table of Contents - 2010 - Volume 9, Issue 04  

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Journal of the Institute of Mathematics of Jussieu (2010), 9:799-846 Cambridge University Press
Copyright © Cambridge University Press 2010
doi:10.1017/S147474801000006X

Research Article

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

Ben Moonena1 and Alexander Polishchuka2

a1 Department of Mathematics, University of Amsterdam, PO Box 94248, 1090 GE Amsterdam, The Netherlands, (b.j.j.moonen@uva.nl)
a2 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA, (apolish@uoregon.edu)
Article author query
moonen b [Google Scholar]
polishchuk a [Google Scholar]

Abstract

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[∙]/S) := ⊕n CH*(C[n]/S) as a ring using the Pontryagin product. We prove that CH*(C[∙]/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[∙]/S) and we give an explicit geometric description of how the operators $\smash{\partial_t^{[m]}}\$ and ∂u act. This builds upon the study of certain geometrically defined operators Pi,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 $\operatorname{gr}^m_{\mathrm{Fil}}$ as multiplication by Nm. 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.

(Received April 25 2009)

(Accepted August 30 2009)

Key Wordssymmetric products of curves; Jacobians; Chow rings; algebraic cycles; motives

AMS 2010 Mathematics subject classificationPrimary 14C15; 14H40; Secondary 14C25