Prym-Tjurin Constructions on Cubic Hypersurfaces

In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups of a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi mapping induces an isomorphism between the primitive cohomology of the cubic hypersurface and the Prym-Tjurin part of the above action. This also holds for Chow groups with rational coefficients. All the constructions are based on a natural relation among topological (resp. algebraic) cycles on $X$ modulo homological (resp. rational) equivalence.