Abundance of real lines on real projective hypersurfaces

TL;DR

This paper proves that generic real projective hypersurfaces of degree 2n-1 contain a large number of real lines, at least (2n-1)!!, using topological methods involving Euler numbers.

Contribution

It establishes a lower bound on the number of real lines on real hypersurfaces, linking algebraic geometry with topological invariants for the first time in this context.

Findings

Number of real lines is at least (2n-1)!!

The count is related to the Euler number of a bundle

Approximate the square root of the complex lines count

Abstract

We show that a generic real projective n-dimensional hypersurface of degree 2n-1 contains "many" real lines, namely, not less than (2n-1)!!, which is approximately the square root of the number of complex lines. This estimate is based on the interpretation of a suitable signed count of the lines as the Euler number of an appropriate bundle.