Planes on Fourfolds Defined by Sums of Binary Forms

We study planes on complex projective fourfolds defined by sums of three squarefree binary forms of the same degree d ≥ 3 in disjoint pairs of variables. We first establish a three-block rigidity theorem: if a plane has rank-two projection to each of the three binary blocks, then all three binary forms must be of Fermat type. This yields a complete classification of the planes on these fourfolds and shows that the Fano scheme of planes F2(X) is finite and reduced. Outside the Fermat case, every plane is either a root plane, determined by one root line from each binary block, or a graph plane, arising from a cancelling linear isomorphism between two blocks together with a root line in the remaining block. If νij denotes the number of projective equivalences between the corresponding root sets, then F2(X) = d3 + d2 (ν01 + ν02 + ν12). In the Fermat case, a uniform Hessian argument gives 15d3 planes. We determine the sharp maximum for every d ≥ 3. Up to projective equivalence, the unique extremizer is the Fermat fourfold except for d = 12, where the icosahedral model has 27,648 planes. We also describe the integral Chow subgroup generated by root planes and prove, by an intersection-theoretic argument, that every plane class spans a distinct extremal ray of the cone generated by plane classes.