First homology of a real cubic is generated by lines

TL;DR

This paper provides a concise proof that the real lines on a real non-singular cubic hypersurface generate its first homology group with Z/2 coefficients, confirming a theorem by Benoist and Wittenberg.

Contribution

It offers a short, simplified proof of a known theorem regarding the generation of homology by lines on real cubic hypersurfaces.

Findings

Real lines generate H_1(X(R); Z/2) for non-singular cubic hypersurfaces.

The proof confirms the theorem by Benoist and Wittenberg.

The approach simplifies understanding of the homological properties of real cubic hypersurfaces.

Abstract

We suggest a short proof of O.Benoist and O.Wittenberg theorem (arXiv:1907.10859) which states that for each real non-singular cubic hypersurface $X$ of dimension $\ge 2$ the real lines on $X$ generate the whole group $H_1(X(\Bbb R);\Bbb Z/2)$.