On the universal CH$_ 0$ group of cubic hypersurfaces

Source abstract

We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its CH _0 group. We prove that for odd-dimensional cubic hypersurfaces or for cubic fourfolds, this is equivalent to the existence of a cohomological decomposition of the diagonal, and we translate geometrically this last condition. For cubic threefolds X , this turns out to be equivalent to the algebraicity of the minimal class \theta^4/4! of the intermediate Jacobian J(X) . In dimension 4, we show that a special cubic fourfold with discriminant not divisible by 4 has universally trivial CH _0 group.