Postulation of General Unions of Lines and Multiplicity Two Points in

We prove that a general union of prescribed numbers of lines and double points has maximal rank, except a few well-known exceptional cases. 1. Introduction Fix . The 2-point of is the closed subscheme of with as its ideal sheaf. The scheme is a zero-dimensional scheme with and . Now assume that . For all , let be the set of all disjoint unions of lines and 2-points. Assume that ; that is, assume that . For each and each integer , we have . The algebraic set is an irreducible subset of the Hilbert scheme. We recall that a closed subscheme is said to have maximal rank if for each integer , the restriction map is a linear map with maximal rank; that is, it is either injective or surjective. has maximal rank if and only if for each integer either or . In this paper we prove the following result. Theorem 1. Fix and an integer . Let be a general element of . Then either or , unless either or . The case is one of the exceptional cases in the famous Alexander-Hirschowitz theorem [1–5]. See [6, Example 1] for the case . Theorem 1 is obviously false for , but it is false in a controlled way [6, Lemma 1], because a disjoint union of 3 lines of is contained in a unique quadric surface and for each , the linear system is the set of all quadric cones with vertex containing . Therefore as an immediate corollary of Theorem 1, we get the following result. Corollary 2. Fix such that , , and . Let be a general element of . Then has maximal rank. The case of Theorem 1 was proved in [6, Propositions ] and a weaker form of Theorem 1 was proved in [6, Proposition 5] (we required that ). This weaker version was enough to prove the statement corresponding to Theorem 1 first in and then in . Then we proved by induction on the corresponding statement in , , but only if [7]. For the proof, we use the case proved by Hartshorne and Hirschowitz [8] and the case , that is, the Alexander-Hirschowitz theorem [1, 2, 4, 5]. We use certain nilpotent structures on reducible conics (called sundials in [9]) and on lines (called +lines in [10]; see Section 2 for them). 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 [9, 12, 13]. 2. Preliminary Lemmas Let be a smooth quadric surface. For each finite set , set . For each closed subscheme , the residual scheme of with respect to is the closed subscheme of with as its ideal sheaf. For each integer , we have the following exact sequence (often called Castelnuovo's sequence): From (1) we get and .