An asymptotic vanishing theorem for generic unions of multiple points

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

“…Take k = 2, a > 0 and d > 0. It is sufficient to do the following cases (d, a): (1, 1), (2, 1), (1,2). The first one is not defective, because h 0 (I Y (1,1) (2)) = 3 and hence h 1 (I Y (1,1) (2)) = 0.…”

“…Since a general rational curve of degree d has maximal rank, it is sufficient to check the following pairs (d, a): (1, 4), (2,3), (2,4), (3,2), (3,3), (4, 1), (4, 2), (5, 1), (6,1). For the case (d, a) = (1, 4), see [4,Corollary 2].…”

“…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). …”

“…Hence h i (Q, I (U ∩Q)∪S ′ (k)) = 0, i = 0, 1. By the differential Horace lemma for double points ( [1], [11,Lemma 5], [2] in characteristic = 2) to prove that h 1 (I W (k)) = 0 it is sufficient to prove …”

“…Hence to prove Theorem 1 for the triple (d, a, k) it is sufficient to prove that h 1 (I W (k)) = 0, where W is the union of U and f general 2-points. We have [2] in characteristic = 2) to prove that h 1 (I W (k)) = 0 it is sufficient to prove that…”