Postulation of General Unions of Lines and Double Points in a Higher Dimensional Projective Space

Abstract

We prove that a general union X ⊂ P r , r ≥ 6, of lines and double points has the expected postulation with respect to the hypersurfaces of degree ≥ 4.

On the Hilbert Function of General Unions of a Rational Curve and 2-Points

“…Since (1,8), (2,6), (2,7), (3,5), (3,6), (4,4), (4,5), (5,3), (5,4), (6,2), (6, 3), (7, 1), (7,2), (8,1). …”

“…Since 10 4 = 120, we need to test the following triples (d, a, φ) with 1 ≤ φ ≤ 2: (16, 2, 1), (15,4,2), (12, 9, 1), (11,11,2), (8,16,1), (7,18,2), (4,23,1), (3,25,2). (b3.3) Assume a = 9, 11.…”

“…Take the set-up of step (a1) with e := ⌊(64 − 2d)/3⌋ and f = 64 − 2d − 3e. We have a ≥ e + f (with (e, f ) = (16, 2) if (d, a) = (7,18) and (e, f ) = (16, 0) if (d, a) = (8, 16)). …”

“…Take the following pairs (d, a): (1,8), (2,8), (3,8), (4,8), (5,5), (6,5), (7,5), (8,5), (9, 2), (10, 2), (11,2), (12,2). (6,7), (7,6), (8,5), (9,5), (10,4), (11,3), (12,2).…”

“…Take the following pairs (d, a): (1,8), (2,8), (3,8), (4,8), (5,5), (6,5), (7,5), (8,5), (9, 2), (10, 2), (11,2), (12,2). (6,7), (7,6), (8,5), (9,5), (10,4), (11,3), (12,2). (16, 1, 1), (17, 2, 2), (18, 3, 3), (19,4,4), (20,4,1), (21,5,2), (22,6,3), (23,6,4), (24,…”