Postulations in projective spaces of unions of a fixed schemes and several high degree general curves

Abstract

Fix a subscheme Z ⊂ ℙr, r ≥ 5, with dim(Z) ≤ r - 5 and an integer g ≥ 0. We prove the existence of an integer Δ (depending only on r, g and the Hilbert polynomial of Z) with the following property. Fix positive integers c > 0, di > 0, 0 ≤ gi ≤ g, 1 ≤ i ≤ c, with ∑i di ≥ Δ; if gi > 0, then assume, say, di ≥ r + rgi ∕ 2. Let X ⊂ ℙr be a general union of c disjoint curves X1, ... , Xc with deg(Xi) = di and pa(Xi) = gi. Then Z ∪ X has maximal rank, i.e. its Hilbert function is the expected one.