On Relations among 1-cycles on Cubic Hypersurfaces

In this paper we give two explicit relations among 1-cycles modulo rational equivalence on a smooth cubic hypersurface X. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjape’s theorem that CH1(X) is always generated by lines and that it is isomorphic to Z if the dimension of X is at least 5. Another application is to the intermediate jacobian of a cubic threefold X. To be more precise, we show that the intermediate jacobian of X is naturally isomorphic to the Prym–Tjurin variety constructed from the curve parameterizing all lines meeting a given rational curve on X. The incidence correspondences play an important role in this study. We also give a description of the Abel–Jacobi map for 1-cycles in this setting.