Remarks on the $\mathrm{CH}_2$ of cubic hypersurfaces

Abstract

This paper presents two approaches to reducing problems on $2$-cycles on a smooth cubic hypersurface $X$ over an algebraically closed field of characteristic $\neq 2$, to problems on $1$-cycles on its variety of lines $F(X)$. The first one relies on bitangent lines of $X$ and Tsen-Lang theorem. It allows to prove that $\mathrm{CH}_2(X)$ is generated, via the action of the universal $\mathbb P^1$-bundle over $F(X)$, by $\mathrm{CH}_1(F(X))$. When the characteristic of the base field is $0$, we use that result to prove that if $dim(X)\geq 7$, then $\mathrm{CH}_2(X)$ is generated by classes of planes contained in $X$ and if $dim(X)\geq 9$, then $\mathrm{CH}_2(X)\simeq \mathbb Z$. Similar results, with slightly weaker bounds, had already been obtained by Pan. The second approach consists of an extension to subvarieties of $X$ of higher dimension of an inversion formula developped by Shen in the case of curves of $X$. This inversion formula allows to lift torsion cycles in $\mathrm{CH}_2(X)$ to torsion cycles in $\mathrm{CH}_1(F(X))$. For complex cubic $5$-folds, it allows to prove that the birational invariant provided by the group $\mathrm{CH}^3(X)_{tors,AJ}$ of homologically trivial, torsion codimension $3$ cycles annihilated by the Abel-Jacobi morphism is controlled by the group $\mathrm{CH}_1(F(X))_{tors,AJ}$ which is a birational invariant of $F(X)$, possibly always trivial for Fano varieties.