Postulation of General Unions of Lines and Multiplicity Two Points in ℙ3

E Ballico

doi:10.1155/2014/120850

Abstract

We prove that a general union X⊂ℙ3 of prescribed numbers of lines and double points has maximal rank, except a few well-known exceptional cases.

Highlights

We recall that a closed subscheme W ⊂ Pr is said to have maximal rank if for each integer x ≥ 0, the restriction map H0(OPr (x)) → H0(OX(x)) is a linear map with maximal rank; that is, it is either injective or surjective
In this paper we prove the following result
Theorem 1 is obviously false for k = 2, but it is false in a controlled way [6, Lemma 1], because a disjoint union of 3 lines of P3 is contained in a unique quadric surface and for each P ∈ P3, the linear system |I2P(2)| is the set of all quadric cones with vertex containing P

Summary

Introduction

Theorem 1 is obviously false for k = 2, but it is false in a controlled way [6, Lemma 1], because a disjoint union of 3 lines of P3 is contained in a unique quadric surface and for each P ∈ P3, the linear system |I2P(2)| is the set of all quadric cones with vertex containing P. 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, [1,2,3, 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 Pr [9, 12, 13]

Preliminary Lemmas

Proof of Theorem 1

Conflict of Interests