Geodesic
On the universal CH$_ 0$ group of cubic hypersurfaces
Journal of the European Mathematical Society, Tome 19 (2017) no. 6, pp. 1619-1653 Cet article a éte moissonné depuis la source EMS Press

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We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its CH0​ 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 θ4/4! 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 CH0​ group.
DOI : 10.4171/jems/702
Classification : 14-XX
Keywords: Cubic hypersurfaces, decomposition of the diagonal, stable rationality 
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     author = {Claire Voisin},
     title = {On the universal {CH}$_ 0$  group of cubic hypersurfaces},
     journal = {Journal of the European Mathematical Society},
     pages = {1619--1653},
     year = {2017},
     volume = {19},
     number = {6},
     doi = {10.4171/jems/702},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/702/}
}
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Claire Voisin. On the universal CH$_ 0$  group of cubic hypersurfaces. Journal of the European Mathematical Society, Tome 19 (2017) no. 6, pp. 1619-1653. doi: 10.4171/jems/702

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