Postulation of General Unions of Lines and Multiplicity Two Points in P

Fix P ∈ P. The 2-point 2P of P is the closed subscheme of P with (I P ) 2 as its ideal sheaf. The scheme 2P is a zerodimensional scheme with deg(2P) = r + 1 and (2P)red = {P}. Now assume that r ≥ 3. For all (t, a) ∈ N, let Z(r, t, a) be the set of all disjoint unions X ⊂ P of t lines and a 2-points. Assume that (t, a) ̸ = (0, 0); that is, assume that Z(r, t, a) ̸ = 0. For each X ∈ Z(r, t, a) and each integer x ≥ 0, we have

Our interest in this topic (after, of course, [1-3, 8, 11]) was reborn by Carlini et al. who started a long project about the Hilbert functions of multiple structures on unions of linear subspaces of P  [9,12,13].

Preliminary Lemmas
Let Q ⊂ P 3 be a smooth quadric surface.For each finite set  ⊂ P 3 , set 2 := ∪ ∈ 2.For each closed subscheme  ⊂ P 3 , 2 ISRN Geometry the residual scheme Res  () of  with respect to  is the closed subscheme of  with I  : I  as its ideal sheaf.For each integer , we have the following exact sequence (often called Castelnuovo's sequence): From ( 1) we get ℎ 0 (I  ()) ≤ ℎ 0 (I Res  () ( − 2)) + ℎ 0 (, I ∩ ()) and ℎ 1 (I  ()) ≤ ℎ 1 (I Res  () ( − 2)) + ℎ 1 (, I ∩ ()).Let  ⊂  be a zero-dimensional subscheme, and let  ⊂  be a union of  distinct lines of type (0, 1).The residual scheme Res  () of  is the closed subscheme of  with I  : I  as its ideal sheaf.For each  ∈ Z, we have Castelnuovo's exact sequence of coherent sheaves on : Hence ℎ  (, Let (3, , )  denote the closure of (3, , ) in the Hilbert scheme of P 3 .Fix a line  ⊂ P 3 with  ∈  and a line  ⊂ P 3 with  ∈  and  ̸ = .Set V := 2 ∩ .Set  :=  ∪ V (it is the intersection of  ∪  with the scheme  ∪ 2).As in [10], we call  a +line with  as its support and  as the support of its nilradical.
For all (, ) ∈ N 2 \ {(0, 0)}, let (3, , ) be the set of all disjoint unions  ⊂ P 3 of  lines and  +lines.The algebraic set (3, , ) is irreducible and its general element has maximal rank [10,Theorem 1].For all integers (, ) ∈ N 2 , let (3, , )  denote the set of all  ∈ (3, , ) such that dim( ∩ ) = 0 and the support of the nilradical of O  is contained in .The algebraic set (3, , )  is irreducible.For a general  ∈ (3, , ), we have Res  () = ; that is, for each point  in the support of the nilradical of O  , the tangent vector V representing the nilradical of O  is not tangent to .
Proof.By Remark 5 to prove   , we may assume that  ≥ 9 and only check the case " =  + 1" of   .
(d) From now on we assume that  ≤   − 1.
Write  =   ⊔   with   the union of the lines of  containing a point of .Let  ⊂ P 3 be a disjoint union of  + 1 +lines with   =  red , and let  be the support of the nilradical of  and with  containing each tangent vector of a +lines of ; that is, we assume that deg( ∩ ) = 3( + 1) and that Res  () =   .Set  :=   ∪  ∪ .Deforming  to a general union of  3+2 lines of P 3 , we get that  is a flat limit of a family of elements of L(3, a 3x+2 − x − 1, x + 1) Q .