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\documentclass[12pt]{minimal}
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\begin{document}$$\ne 2$$\end{document}, 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 CH2(X)\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}_2(X)$$\end{document} is generated, via the action of the universal P1\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {P}}^1$$\end{document}-bundle over F(X), by CH1(F(X))\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}_1(F(X))$$\end{document}. When the characteristic of the base field is 0, we use that result to prove that if dim(X)≥7\documentclass[12pt]{minimal}
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\begin{document}$$dim(X)\ge 7$$\end{document}, then CH2(X)\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}_2(X)$$\end{document} is generated by classes of planes contained in X and if dim(X)≥9\documentclass[12pt]{minimal}
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\begin{document}$$dim(X)\ge 9$$\end{document}, then CH2(X)≃Z\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}_2(X)\simeq {\mathbb {Z}}$$\end{document}. 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 CH2(X)\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}_2(X)$$\end{document} to torsion cycles in CH1(F(X))\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}_1(F(X))$$\end{document}. For complex cubic 5-folds, it allows to prove that the birational invariant provided by the group CH3(X)tors,AJ\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}^3(X)_{tors,AJ}$$\end{document} of homologically trivial, torsion codimension 3 cycles annihilated by the Abel–Jacobi morphism is controlled by the group CH1(F(X))tors,AJ\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {CH}}_1(F(X))_{tors,AJ}$$\end{document} which is a birational invariant of F(X), possibly always trivial for Fano varieties.
