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

On the universal $$\mathrm {CH}_0$$ CH 0 group of cubic threefolds in positive... 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 of those hypersurfaces and the equivalence between the algebraicity (with $$\mathbb Z_2$$ Z 2 -coefficients) of the minimal class $$\theta ^4/4!$$ θ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

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

, Volume 154 (2) – Jan 31, 2017
22 pages

/lp/springer_journal/on-the-universal-mathrm-ch-0-ch-0-group-of-cubic-threefolds-in-qGaJjJ0qHs
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-016-0912-5
Publisher site
See Article on Publisher Site

### Abstract

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 of those hypersurfaces and the equivalence between the algebraicity (with $$\mathbb Z_2$$ Z 2 -coefficients) of the minimal class $$\theta ^4/4!$$ θ 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.

### Journal

Manuscripta MathematicaSpringer Journals

Published: Jan 31, 2017

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