Postulation of General Unions of Decorated Lines

Abstract

For all integers a ≥ 0 and r ≥ 3 an a-decorated line D ⊂ P r is a scheme union of a line D red and a tangent vectors of P r at points of D red . Here we study the postulation of general disjoint unions of a-decorated lines.

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

“…We use certain nilpotent structures on reducible conics (called sundials in [9]) and on lines (called +lines in [10]; see Section 2 for them).…”

“…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.…”

“…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 .…”

“…Take a general ⊂ P 3 with ♯( ) ≤ 6. is a contained in the union of 3 disjoint lines and is contained in a smooth quadric with, say, of type (3, 0). Since a general element of (3, − 1 − , y) has maximal rank [10], to prove , it is sufficient to check the existence of for all > 7. In particular is true if ≤ 6.…”

Postulation of Unions of Tangent Vectors

“…In particular it is true in many interesting cases. We collect in this note all the cases we will need in [1].…”