Original version
Algebraic Geometry. 2023, 10 (2), 228-258, DOI: https://doi.org/10.14231/AG-2023-007
Abstract
We show that the maximal number of planes in a complex smooth cubic fourfold in P5 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.