REMARKS ON THE CH_2 OF CUBIC HYPERSURFACES

Rene Mboro (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

This paper presents two approaches to reducing problems on 2-cycles on a smooth cubic hypersurface X over an algebraically closed field of characteristic = 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 CH 2 (X) is generated, via the action of the universal P 1-bundle over F (X), by CH 1 (F (X)). When the characteristic of the base field is 0, we use that result to prove that if dim(X) ≥ 7, then CH 2 (X) is generated by classes of planes contained in X and if dim(X) ≥ 9, then CH 2 (X) Z. Similar results, with slightly weaker bounds, had already been obtained by Pan([27]). The second approach consists of an extension to subvarieties of X of higher dimension of an inversion formula developped by Shen ([30], [31]) in the case of curves of X. This inversion formula allows to lift torsion cycles in CH 2 (X) to torsion cycles in CH 1 (F (X)). For complex cubic 5-folds, it allows to prove that the birational invariant provided by the group CH 3 (X) tors,AJ of homologically trivial, torsion codimension 3 cycles annihilated by the Abel-Jacobi morphism is controlled by the group CH 1 (F (X)) tors,AJ which is a birational invariant of F (X), possibly always trivial for Fano varieties.

Mots clés: Algebraic Geometry; Rationality problems; Algebraic cycles; Hypersurface

Domaines: Géométrie algébrique [math.AG]

Rene Mboro. REMARKS ON THE CH_2 OF CUBIC HYPERSURFACES. 2017. ⟨hal-01478975⟩