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 $ \mathrm {CH}_1(X)$ is always generated by lines and that it is isomorphic to $ \mathbb{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. - See more at: http://www.ams.org/journals/jag/2014-23-03/S1056-3911-2014-00631-7/home.html#sthash.T8vmMhvP.dpuf