Cubic fourfolds admitting a cyclic group of symplectic automorphisms of order not a power of 2 are rational and lie in the Hassett divisors C_14 or C_42.
Planes in cubic fourfolds
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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.
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Rational cubic fourfolds with a symplectic group of automorphisms
Cubic fourfolds admitting a cyclic group of symplectic automorphisms of order not a power of 2 are rational and lie in the Hassett divisors C_14 or C_42.