Mathematics – Algebraic Geometry
Scientific paper
2007-04-21
Mathematics
Algebraic Geometry
31 pages, uses Paul Taylor's diagrams.sty. Added in v2: construction of a new grading on the algebra of 0-cycles on the Jacobi
Scientific paper
In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers ${\cal C}^{[\bullet]}$ and the relative Jacobian ${\cal J}$ of a family of curves ${\cal C}/S$. As one of the applications, we show that in the case of a single curve $C$ this action induces a integral form of a Lefschetz $\operatorname{sl}_2$-action on the Chow groups of $C^{[N]}$. Another application gives a new grading on the ring of 0-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in $CH^*({\cal C}^{[\bullet]})$ and in $CH^*({\cal J})$ and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in math.AG/0204188. We also show that the our algebras of operators preserve the subrings of tautological cycles and act on them via some explicit differential operators.
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