On the irreducibility of the Hilbert scheme of space curves

Abstract

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