Main Article Content
Let Y ⊂ ℙr, r ≥ 3, be a scheme with dim(Y) ≤ r - 2. We prove the existence of an integer d0 such that for all d ≥ d0 a general union X of Y and either a general degree d rational curve or (if r ≠ 3) d lines has maximal rank, i.e. for each t ∈ ℕ X gives the expected number of conditions to the set of all degree t hypersurfaces of ℙr. We also consider unions of Y and a general curve with a prescribed genus and high degree.