On the irreducibility of the Hilbert scheme of space curves

Abstract

Abstract. Denote by H d,g,r the Hilbert scheme parametrizing smooth irreducible complex curves of degree d and genus g embedded in P r . In 1921 SeveriAs it has turned out in recent years, the conjecture is true for r = 3 and 4, while for r ≥ 6 it is incorrect. We prove that H g,g,3 , H g+3,g,4 and H g+2,g,4 are irreducible, provided that g ≥ 13, g ≥ 5 and g ≥ 11, correspondingly. This augments the results obtained previously by Ein (1986Ein ( ), (1987 and by Keem and Kim (1992).

The generality of a section of a curve

“…In particular, the above descriptions show f (C) ∩ Q is not a general collection of 2d points on Q. And conversely, for (d, g) = (6,4), the intersection f (C) ∩ H is a general collection of six points on a conic in H P 2 ; in particular, it is not a general collection of d = 6 points. Theorem 1.6.…”

“…Let f : C → P 3 be a general nondegenerate map of degree d from a general curve of genus g, and Q be a general quadric. Then f (C) ∩ Q is a general collection of 2d points on Q unless (d, g) ∈ {(4, 1), (5,2), (6,2), (6,4), (7,5), (8,6)}.…”

“…Both the results and the techniques developed here play a critical role in the author's proof of the Maximal Rank Conjecture [10], as explained in [11]. (5,2), (6,2), (6,4), (7,5), (8,6)}.…”

“…• If (d, g) = (8,6), then f (C) ∩ Q is a general collection of sixteen points on a curve of bidegree (3,3).…”

“…Theorem 1.6. Let f : C → P 4 be a general BN-curve of degree d and genus g. Then the intersection f (C) ∩ H, of C with a general hyperplane H, consists of a general collection of d points on H, unless (d, g) ∈ { (8,5), (9,6), (10, 7)}.…”