Rationally Inequivalent Points on Generic Hypersurfaces

For a smooth surface of degree at least 4 in ${\mathbb P}^{^3}$, Mumford proved that its Chow group of 0-cycles are nontrivial. As the degree increases, we are expecting the group to become larger in some sense. Voisin proved that no two distinct points are rationally equivalent on a very general surface of degree at least 7.
We improved her result to degree 6, which is optimal. This is a joint work with James D. Lewis and Mao Sheng.