## On the irreducibility of the Hilbert scheme of space curves

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- by Hristo Iliev PDF
- Proc. Amer. Math. Soc.
**134**(2006), 2823-2832 Request permission

## 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).## References

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## Additional Information

**Hristo Iliev**- Affiliation: Department of Mathematics, Seoul National University, Seoul 151-747, Korea
- Email: itso@math.snu.ac.kr
- 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
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 2823-2832 - MSC (2000): Primary 14H10; Secondary 14C05
- DOI: https://doi.org/10.1090/S0002-9939-06-08516-9
- MathSciNet review: 2231604