Planes in cubic fourfolds

TL;DR

This paper determines the maximum number of planes in complex and real smooth cubic fourfolds, identifying unique examples with the highest counts and classifying those with over 350 planes.

Contribution

It establishes the maximum number of planes in smooth cubic fourfolds over complex and real fields and classifies the unique cubics with over 350 planes.

Findings

Maximum of 405 planes in complex cubic fourfolds, achieved by Fermat cubic.

Maximum of 357 real planes in real cubic fourfolds, achieved by Clebsch--Segre cubic.

Only three cubics have more than 350 planes, up to projective equivalence.

Abstract

We show that the maximal number of planes in a complex smooth cubic fourfold in ${\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the so-called Clebsch--Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than $350$ planes.