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 $\ne 2$, to problems on 1-cycles on its variety of lines F(X). The first one relies on osculating 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 $dim(X)\ge 7$, then ${\mathrm {CH}}_2(X)$ is generated by classes of planes contained in X and if $dim(X)\ge 9$, then ${\mathrm {CH}}_2(X)\simeq {\mathbb {Z}}$. Similar results, with slightly weaker bounds, had already been obtained by Pan (Math Ann 1–28, 2016). The second approach consists of an extension to subvarieties of X of higher dimension of an inversion formula developped by Shen (J Algebraic Geom 23:539–569, 2014, Rationality, universal generation and the integral Hodge conjecture, arXiv:1602.07331) in the case of 1-cycles 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.