Degtyarev, AlexItenberg, I.Ottem, J. C.2024-03-152024-03-15202323131691https://hdl.handle.net/11693/114785We 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 © 2023,Algebraic Geometry. All Rights Reserved.en2-planesCubic fourfoldDiscriminant formIntegral latticeNiemeier latticeArticle10.14231/AG-2023-007