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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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  • [AC81] E. Arbarello, M.Cornalba, Su una congetura di Petri, Comment. Math. Helv., 56 (1981), 1-38. MR 82k:14029
  • [AC83] E. Arbarello, M. Cornalba, A few remarks about the variety of irreducible plane curves of given degree and genus, Ann. Scient. Ec. Norm. Sup. (4), 16 (1983), no. 3, 467-488. MR 86a:14020
  • [ACGH] E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of algebraic curves, Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, 1985. MR 86h:14019
  • [Ein86] L. Ein. Hilbert scheme of smooth space curves, Ann. Scient. Ec. Norm. Sup. (4), 19 (1986), no. 4, 469-478. MR 88c:14009
  • [Ein87] L. Ein. The irreducibility of the Hilbert scheme of complex space curves, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 83-87, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. MR 89f:14006
  • [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [Kee94] C. Keem. Reducible Hilbert scheme of smooth curves with positive Brill-Noether number, Proc. Amer. Math. Soc., 122 (1994), no. 2, 349-354. MR 95a:14026
  • [KK92] C. Keem S. Kim, Irreducibility of a subscheme of the Hilbert scheme of complex space curves, J. Algebra, 145 (1992), no. 1, 240-248. MR 93a:14004
  • [Ser84] E. Sernesi, On the existence of certain families of curves, Invent. Math., 75 (1984), no. 1, 25-57. MR 85e:14035
  • [Sev21] F. Severi. Vorlesungen uber algebraische Geometrie, Teubner, Leipzig, 1921.

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Additional Information

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.

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