On the irreducibility of the Hilbert scheme of space curves

Denote by H-d,H-g,H-r the Hilbert scheme parametrizing smooth irreducible complex curves of degree d and genus g embedded in P-r. In 1921 Severi claimed that Hd, g, r is irreducible if d >= g + r. As 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,H-g,H-3, H-g+3,H-g,H-4 and H-g+2,H-g,H-4 are irreducible, provided that g >= 13, g >= 5 and g >= 11, correspondingly. This augments the results obtained previously by Ein (1986), (1987) and by Keem and Kim (1992).