# Droites sur les hypersurfaces cubiques

## TL;DR

This paper proves that on any complex cubic hypersurface of dimension at least 2, the Chow group of 1-cycles is generated by lines, extending known results to possibly singular cases using algebraic K-theory.

## Contribution

It extends the known generation of the Chow group of 1-cycles by lines to all complex cubic hypersurfaces of dimension at least 2, including singular ones, using algebraic K-theory techniques.

## Key findings

- Chow group of 1-cycles is generated by lines on complex cubic hypersurfaces.
- The result applies to hypersurfaces with singularities, not just smooth.
- Uses algebraic K-theory to establish the generation of the Chow group.

## Abstract

Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. -- On montre que sur toute hypersurface cubique complexe de dimension au moins 2, le groupe de Chow des cycles de dimension 1 est engendr\'e par les droites. Le cas lisse est un th\'eor\`eme connu. La d\'emonstration ici donn\'ee repose sur un r\'esultat sur les surfaces g\'eom\'etriquement rationnelles sur un corps quelconque (1983), obtenu via la K-th\'eorie alg\'ebrique.