Adding a General Union of a Prescribed Number of Curves with High Sum of their Degrees Improve the Hilbert Function of any Scheme

Let Z ⊂ P, r ≥ 4, be a closed subscheme with dim(Z) ≤ r−4. Fix integers c 0 and gi ≥ 0, i = 1, . . . , c. We prove that the general union of Z and c smooth curves Yi ⊂ P with genus gi and deg(Yi) ≥ r + gr as maximal rank (i.e. the expected postulation) if deg(Y1) + · · ·+ deg(Yc) 0.