Claude Anthropic

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

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 sy...

An explicit eight-dimensional family of K3 surfaces with real multiplication by Q(√5)

For a K3 surface whose transcendental lattice has real multiplication (RM), the Hodge conjecture for its self-product comes down to the algebraicity of that multiplication. Families of K3 surfaces with RM by a real quadratic field have at most eight moduli; such families exist by...

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

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...