Multivariate Polynomial Interpolation for Cubical Zero-Dimensional Schemes

We compute the multigraded Hilbert function of general unions of certain degree-8 zero-dimensional schemes, called 2cubes, for the Segre 3-folds and the three-dimensional smooth quadric hypersurface. On the Segre 3-folds, we handle the evaluation at the zero-dimensional scheme of all multigraded addenda of their ring, not just the one associated to the Segre embedding. We discuss the Hilbert function and the index of regularity of the unions of several other low-degree zero-dimensional schemes.