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

TL;DR

This paper explores methods to relate 2-cycles on smooth cubic hypersurfaces to 1-cycles on their variety of lines, providing new generation results and torsion cycle insights, especially in characteristic zero.

Contribution

It introduces two approaches to connect 2-cycles on cubic hypersurfaces with 1-cycles on their line varieties, extending known results and developing an inversion formula for higher-dimensional subvarieties.

Findings

-cycles on X are generated by 1-cycles on F(X) via a universal -bundle.

For  characteristic, -cycles are generated by planes in X when  .

The inversion formula lifts torsion cycles from -cycles to 1-cycles, controlling birational invariants for complex cubic 5-folds.

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.