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On the irreducibility of the Hilbert scheme of space curves

Author: Hristo Iliev
Journal: Proc. Amer. Math. Soc. 134 (2006), 2823-2832
MSC (2000): Primary 14H10; Secondary 14C05
Published electronically: April 11, 2006
MathSciNet review: 2231604
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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).

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Hristo Iliev
Affiliation: Department of Mathematics, Seoul National University, Seoul 151-747, Korea

Received by editor(s): December 10, 2003
Received by editor(s) in revised form: April 22, 2005
Published electronically: April 11, 2006
Additional Notes: The author was supported in part by NIIED and KOSEF (R01-2002-000-00051-0).
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.