Monodromy of cyclic covers and the Hodge conjecture for the very general hypersurfaces F(x^1)+...+F(x^r)=0 and the powers of F(x)+F(y)=0

Let F be a very general ternary form of degree d, C_F the plane curve F = 0 and Y_F the surface u^d + F(x) = 0, the cyclic cover of the plane branched along C_F. We study the hypersurfaces F(x^1) + ... + F(x^r) = 0 in r blocks of three variables, among them the fourfold X_F given by F(x) + F(y) = 0. Extending the inductive structure of Shioda and Katsura to families and to any number of blocks, we express their primitive middle cohomology, through algebraic correspondences and up to constant pieces of rank one, in terms of the eigenspaces of u ↦ ζu on Y_F and of H^1(C_F). For d ≥ 5 we determine the connected monodromy group of this structure (over the complex numbers a product of special linear groups, for even d an orthogonal group, and a symplectic group) and its Hodge group, which adds a central torus of dimension φ(d)/2. We deduce the Hodge conjecture for all products of the fourfolds X_F and the curve C_F (every d ≥ 3), for all products also involving Y_F when d is prime or d = 4, and for F(x^1) + ... + F(x^r) = 0 for every r when d is prime or d = 4 and below an explicit threshold r_0(d) otherwise. In this range, for d ≥ 5 and even r, the middle Hodge classes are spanned by a power of the hyperplane class and the classes of explicit linear subspaces. For composite d other than 4 some Hodge classes remain whose algebraicity is open. Analogous results hold for forms in any odd number of variables and for several independent forms. Code and data are at github.com/sallerk/HC-ts-powers.