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

Abstract

Fix a subscheme Z ⊂ ℙ^r, 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, d_i > 0, 0 ≤ g_i ≤ g, 1 ≤ i ≤ c, with ∑_i d_i ≥ Δ; if g_i > 0, then assume, say, d_i ≥ r + rg_i ∕ 2. Let X ⊂ ℙ^r be a general union of c disjoint curves X₁, ... , X_c with deg(X_i) = d_i and p_a(X_i) = g_i. Then Z ∪ X has maximal rank, i.e. its Hilbert function is the expected one.