On the universal $CH_0$ group of cubic hypersurfaces

TL;DR

This paper investigates the conditions under which smooth cubic hypersurfaces have a universally trivial ${ m CH}_0$-group, linking geometric, cohomological, and algebraic properties across different dimensions.

Contribution

It establishes equivalences between Chow-theoretic decompositions, cohomological decompositions, and algebraicity conditions for cubic hypersurfaces, especially in dimensions three and four.

Findings

For odd-dimensional cubics and fourfolds, Chow and cohomological decompositions are equivalent.

The algebraicity of a specific class in the intermediate Jacobian characterizes cubic threefolds.

Certain special cubic fourfolds with specific discriminants have universally trivial ${ m CH}_0$ groups.

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 ${\rm CH}_0$-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 ${\rm CH}_0$ group.