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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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).
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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