Linear subspaces and the Hodge classes of very general hypersurfaces F(x)+F(y)=0

In its rational and integral forms, we prove the Hodge conjecture for the hypersurfaces F(x) + F(y) = 0 in ℙ²ᵛ⁻¹, where F is a very general form of degree d in v variables, v ≥ 3, d ≥ 3 and (v,d) ≠ (3,3), (3,4), (4,3). These hypersurfaces contain the d linear subspaces y = ηx with η^d = −1, and we show that their rational Hodge classes in the middle degree form a space of dimension d + 1, spanned by these subspaces and the power of the hyperplane class. The main step is an exact computation, at every smooth F, of the first-order (infinitesimal) condition for Hodge classes: the Thom–Sebastiani decomposition of the Jacobian ring turns it into commutation relations between endomorphisms of the graded pieces of the Jacobian ring of F, which a symmetrizer lemma of Donagi–Green type solves. Moonen’s form of the infinitesimal method of Carlson and Griffiths then gives the very general member, and a theorem of Degtyarev and Shimada at the Fermat point gives the integral statement. The same method shows that very general even-dimensional hypersurfaces F(x) + G(y) = 0 have no Hodge classes beyond the hyperplane class when (v + w)(d − 2) ≥ 4d − 2. It also treats, in a range of v and d, families in which {F = 0} contains a fixed linear subspace of half its dimension, or a fixed smooth subvariety of that dimension and d is large: there the joins of its components give further algebraic classes, and we obtain the Hodge conjecture together with the exact dimension of the space of Hodge classes. For (v,d) = (3,6) the dimension statement was obtained by computer in our note on the sextic fourfolds of a family of K3 surfaces with real multiplication; the proof here replaces those calculations. The proofs use no computer calculations; code checking the formulas in many cases is at github.com/sallerk/HC-ts-diagonal.