Cyclic symmetry and the Hodge conjecture for the fourfolds F(x)+F(y)=0

Abstract

For a ternary form F of degree d with smooth zero locus C_F, we study the fourfold X_F given by F(x) + F(y) = 0 in ℙ^5 and the surface Y_F given by F(x) + u^d = 0, the cyclic cover of the plane branched along C_F. Here F is very general among the forms invariant under a cyclic symmetry x_2 ↦ εx_2 with ε^n = 1; such forms with smooth zero locus exist exactly when n divides d or d − 1. For d ≥ 6 and every such n ≥ 2 we prove the Hodge conjecture for X_F and for the self-product Y_F × Y_F, and we compute the dimensions of their spaces of Hodge classes. In most cases the Hodge classes of X_F are spanned by the square of the hyperplane class and nd planes. For n = d, Y_F × Y_F carries further Hodge classes, which come from coincidences of Hodge types on the Fermat curve of degree d and are algebraic through a map of Shioda–Katsura type. By a theorem of Koblitz and Rohrlich they all come from evident automorphisms exactly when d is prime to 6, and for d ≥ 7 their number is determined by the Picard number of the Fermat surface of degree d. The upper bounds come from an infinitesimal computation at one special member, where the Jacobian ring has a monomial basis, and from the local monodromy of a nodal degeneration. A computer-assisted survey proves the Hodge conjecture for very general X_F in all 86 families of plane curves of degree 6 to 10 with a diagonal cyclic symmetry of order at most 2d that have moduli and smooth members; 59 of them are not covered by the theorems.