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

TL;DR

This paper extends Voisin's results on the decomposition of the diagonal for cubic threefolds from complex numbers to algebraically closed fields of characteristic greater than 2, linking algebraic cycles, intermediate Jacobians, and the Tate conjecture.

Contribution

It adapts key results about the decomposition of the diagonal for cubic threefolds to positive characteristic fields, connecting algebraic and cohomological properties.

Findings

Equivalence between Chow-theoretic and cohomological decompositions in positive characteristic.

Algebraicity of the minimal class of the intermediate Jacobian implies Chow-theoretic decomposition.

Chow-theoretic decomposition holds for cubic threefolds over algebraic closures of finite fields of characteristic > 2.

Abstract

We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, 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 fact that the algebraicity (with $\mathbb Z_2$-coefficients) of the minimal class $\theta^4/4!$ of the intermediate jacobian $J(X)$ of $X$ implies the 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, 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.