  In this paper, we give a Prym-Tjurin construction for the cohomology and the
Chow groups for a cubic hypersurface. On the space of lines meeting a given
rational curve, there is the incidence correspondence. This correspondence
induces an action on both of the primitive cohomology and the primitive Chow
groups. We first show that this action satisfies a quadratic equation. Then the
Abel-Jacobi homomorphism 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.
