Interpolation and Parallel Lines

Here we consider 3 interpolation problems for homogeneous polynomials in n n + 1 variables (i.e. for zero-dimensional subschemes Z of Pⁿ) in which the scheme Z is contained in a “ small number ” of “ parallel lines ”; here a finite union D₁ ∪ … ∪ D _x ? Pⁿ of lines is called a set of parallel lines if there is P ∈ Pⁿ such that P ∈ D _i for all i.