The Hodge conjecture for sums of binary quartics

We observe that the Hodge conjecture holds for every smooth hypersurface F_0(u_0,v_0) + … + F_m(u_m,v_m) = 0 in P^(2m+1), where F_0, …, F_m are binary quartic forms in disjoint pairs of variables, and more generally for products of such hypersurfaces and curves whose Jacobians are isogenous to products of elliptic curves. The proof combines the inductive structure of Shioda and Katsura with the theorem of Tate, Murasaki and Imai on products of elliptic curves, and also gives a formula for dim Hdg^m(X). Not peer reviewed; written with the assistance of an AI system (see the note).