On the universal CH${}_0$ group of cubic hypersurfaces

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.