On the universal $$\mathrm {CH}_0$$ CH 0 group of cubic threefolds in positive characteristic

Abstract

We adapt for algebraically closed fields k of characteristic >2 two results of Voisin (On the universal $$\text {CH} _0$$ group of cubic hypersurfaces, arXiv:1407.7261 ), on the decomposition of the diagonal of a smooth cubic hypersurface X of dimension 3 over $${\mathbb {C}}$$ , namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the equivalence between the algebraicity (with $$\mathbb Z_2$$ -coefficients) of the minimal class $$\theta ^4/4!$$ of the intermediate Jacobian J(X) of X and the cohomological (hence Chow-theoretic) decomposition of the diagonal of X. Using the second result, the Tate conjecture for divisors on surfaces defined over finite fields predicts, via a theorem of Schoen (Math Ann 311(3), 493–500, 1998), that every smooth cubic hypersurface of dimension 3 over the algebraic closure of a finite field of characteristic >2 admits a Chow-theoretic decomposition of the diagonal.