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arXiv · 1407.7261

On the universal $CH_0$ group of cubic hypersurfaces

Claire Voisin
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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.

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Claire Voisin. 2014-07-27. On the universal $CH_0$ group of cubic hypersurfaces. https://arxiv.org/abs/1407.7261

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