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This paper presents two approaches to reducing problems on 2-cycles on a smooth cubic hypersurface X over an algebraically closed field of characteristic $$\ne 2$$ ≠ 2 , to problems on 1-cycles on its variety of lines F(X). The first one relies on osculating lines of X and Tsen-Lang theorem. It allows to prove that $${\mathrm {CH}}_2(X)$$ CH 2 ( X ) is generated, via the action...
We adapt for algebraically closed fields k of characteristic >2 two results of Voisin (On the universal $$\text {CH} _0$$ CH 0 group of cubic hypersurfaces, arXiv:1407.7261 ), on the decomposition of the diagonal of a smooth cubic hypersurface X of dimension 3 over $${\mathbb {C}}$$ C , namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal...
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