Cubic surfaces, Eckardt points and the Hodge conjecture for sums of cubic forms

When the equation of a smooth cubic hypersurface is a sum of cubic forms in disjoint sets of variables, its Hodge classes are built from those of the blocks. This note studies the Hodge conjecture for such sums when every block has at most five variables. For ternary blocks, which define plane cubic curves, the inductive structure of Shioda and Katsura reduces everything to products of elliptic curves. This gives the Hodge conjecture for every such hypersurface and for products of them, a case stated earlier in an unrefereed paper on the rStation forum, as well as a formula for the number of Hodge classes; for the cubic tenfolds F1+F2+F3+F4=0 with the smallest number, 487, these classes are spanned by joins of planes and a power of the hyperplane class. The main new cases have blocks in four variables, which define smooth cubic surfaces. Their Hodge classes are governed by the intermediate Jacobian of the cubic threefold G+y^3=0, an abelian fivefold with multiplication by Q(√−3) of signature (4,1), whose Hodge group for a very general block was determined by Achter. For very general blocks the Hodge conjecture follows for all such hypersurfaces and their products in which each block occurs at most four times. Five copies of a block first meet Hodge classes of Weil type, in dimension 22, and for any multiplicity the question reduces to the algebraicity of the Weil classes on the abelian eightfolds J_G × E_ω^3. When the cubic surface has an Eckardt point, the corresponding classes are Weil classes on abelian sixfolds with a split polarization, which are algebraic by work of Schoen and a recent theorem of Markman (a preprint), so very general Eckardt blocks may occur any number of times. Blocks in five variables are treated with the Hodge groups of cyclic cubic fourfolds from an earlier note, with at most ten copies of each block. Code and data are at github.com/sallerk/HC-ternary-cubics.