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On quasi-complete intersections of codimension $ 2$


Author: Youngook Choi
Journal: Proc. Amer. Math. Soc. 134 (2006), 1249-1256
MSC (2000): Primary 14M07, 14N05, 14M06
DOI: https://doi.org/10.1090/S0002-9939-05-08425-X
Published electronically: December 14, 2005
MathSciNet review: 2199166
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Abstract: In this paper, we prove that if $ X\subset\mathbb{P}^n$, $ n\ge 4$, is a locally complete intersection of pure codimension $ 2$ and defined scheme-theoretically by three hypersurfaces of degrees $ d_1\ge d_2\ge d_3$, then $ H^1(\mathbb{P}^n,\mathcal{I}_X(j))=0$ for $ j<d_3$ using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold $ X\subset\mathbb{P}^5$ is projectively normal if $ X$ is defined by three quintic hypersurfaces.


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  • 1. D. Arapura, D. Jaffe, On Kodaira vanishing for singular varieties, Proc. Amer. Math. Soc. 105 (1989) 911-916. MR 0952313 (89h:14013)
  • 2. A. Aure, W. Decker, K. Hulek, S. Popescu, K. Ranestad, Syzygies of abelian and bielliptic surfaces in $ \mathbb{P}^4$, Internat. J. Math. 7 (1997) 849-919. MR 1482969 (99a:14049)
  • 3. V. Beorchia, Ph. Ellia, On the equations defining quasi complete intersection space curves, Arch. Math. 70 (1998) 244-249. MR 1604080 (98k:14042)
  • 4. A. Bertram, L. Ein, R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. of Amer. Math. Soc. 4 (1991) 587-602. MR 1092845 (92g:14014)
  • 5. H. Bresinsky, P. Schenzel, J. Stückrad, Quasi-complete intersection ideals of height $ 2$, J. Pure Appl. Algebra 127 (1998) 137-145. MR 1620704 (99d:13014)
  • 6. Y. Choi, S. Kwak, Remarks on the defining equations of smooth threefolds in $ \mathbb{P}^5$, Geom. Dedicata 96 (2003) 151-159. MR 1956837 (2003k:14052)
  • 7. W. Decker, S. Popescu, On surfaces in $ \mathbb{P}^4$ and $ 3$-folds in $ \mathbb{P}^5$, Vector bundles in algebraic geometry (Durham, 1993), 69-100, London Math. Soc. Lecture Note Ser. 208 Cambridge Univ. Press, Cambridge, 1995. MR 1338413 (96d:14046)
  • 8. L. Ein, R. Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993) 51-67. MR 1193597 (93m:13006)
  • 9. G. Faltings, Ein Kriterium fur vollständig Durschnitte, Invent. Math. 62 (1981) 393-401. MR 0604835 (82f:14050)
  • 10. D. Franco, L. Kleiman, T. Lascu, Gherardelli Linkage and Complete Intersections, Michigan Math. J. 48 (2000) 271-279. MR 1786490 (2002a:14057)
  • 11. M. Fiorentini, A. Lascu, A criterion for quasi complete intersections and related embedding questions, Ann. Univ. Ferrara 28 (1982) 153-166. MR 0701894 (85b:14066)
  • 12. G. Horrocks, D. Mumford, Rank $ 2$ vector bundle on $ \mathbb{P}^4$ with 15,000 symmetries, Topology 12 (1973) 63-81. MR 0382279 (52:3164)
  • 13. S. Kwak, Castelnuovo-Mumford regularity for smooth threefolds in $ \mathbb{P}^5$ and extremal examples, J. Reine Angew. Math. 509 (1999) 21-34. MR 1679165 (2000e:14064)
  • 14. N. Kumar, Monads on projective spaces, Manuscripta Math. 112 (2003) 183-189. MR 2064915 (2005g:14083)
  • 15. J. Migliore, Introduction to liasion theory and deficiency modules, Birkhäuser, 1998. MR 1712469 (2000g:14058)
  • 16. D. Peskine, L. Szpiro, Liaison des variétés algébriques I, Invent. Math. 26 (1974) 271-302. MR 0364271 (51:526)

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Additional Information

Youngook Choi
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong Yusung-gu, Daejeon, Korea
Email: ychoi@math.kaist.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-05-08425-X
Keywords: Quasi-complete intersections, liaison, normality, defining equations
Received by editor(s): September 10, 2004
Published electronically: December 14, 2005
Additional Notes: The author was supported in part by KRF (grant No. KRF-2002-070-C00003)
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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