The Hodge conjecture for very general sums of binary quintics and sextics

Abstract

We prove the Hodge conjecture for the hypersurfaces F₀(u₀,v₀) + ⋯ + Fₘ(uₘ,vₘ) = 0 in ℙ²ᵐ⁺¹, where F₀, …, Fₘ are very general binary quintics, or very general binary sextics, in disjoint pairs of variables, and we compute the dimension of their space of Hodge classes. For quintics the Hodge classes are spanned by the 5ᵐ⁺¹ linear subspaces spanned by one zero of each form. For sextics these do not suffice, and the remaining classes are iterated joins of curves on the surfaces Fₛ + Fₜ = 0 with zeros of the other forms. The new cases are quintics of dimension at least 6 and sextics of dimension at least 4. The proof bounds the Hodge classes by the monodromy of the curves yᵈ + F(u,v) = 0, using the inductive structure of Shioda and Katsura, the algebraicity of Hodge loci and McMullen’s description of the braid group action. We also prove the Hodge conjecture for some special members and for products of independent very general members of degree at least 5, and the integral Hodge conjecture for very general members of every degree d ≥ 5 except 6.