Abundance of 3-planes on real projective hypersurfaces

TL;DR

The paper demonstrates that generic real projective hypersurfaces of odd degree contain a large number of real 3-planes, with their count growing at a rate comparable to the complex case, using Euler number interpretations.

Contribution

It establishes a new asymptotic estimate for the number of real 3-planes on hypersurfaces, linking real and complex enumerations through topological methods.

Findings

Real hypersurfaces contain many real 3-planes.

Number of real 3-planes grows as d^3 log d.

Growth rate matches that of complex 3-planes.

Abstract

We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$, as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle.