On the irreducibility of the Hilbert scheme of space curves

Abstract:

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