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On the syzygies of quasi-complete intersection space curves

Author: Youngook Choi
Journal: Proc. Amer. Math. Soc. 137 (2009), 3999-4006
MSC (2000): Primary 14M07, 14N05, 14M06
Published electronically: July 24, 2009
MathSciNet review: 2538560
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Abstract: In this paper, we discuss minimal free resolutions of the homogeneous ideals of quasi-complete intersection space curves. We show that if $ X$ is a quasi-complete intersection curve in $ \mathbb{P}^3$, then $ I_X$ has a minimal free resolution

$\displaystyle 0\to \oplus_{i=1}^{\mu-3} S(d_{i+3}+c_1)\to \oplus_{i=1}^{2\mu-4}S(-e_i)\to \oplus_{i=1}^\mu S(-d_i)\to I_X\to 0, $

where $ d_i,e_i\in \mathbb{Z}$ and $ c_1=-d_1-d_2-d_3$. Therefore the ranks of the first and the second syzygy modules are determined by the number of elements in a minimal generating set of $ I_X$. Also we give a relation for the degrees of syzygy modules of $ I_X$. Using this theorem, one can construct a smooth quasi-complete intersection curve $ X$ such that the number of minimal generators of $ I_X$ is $ t$ for any given positive integer $ t\in\mathbb{Z}^+$.

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  • 1. V. Beorchia, Ph. Ellia, On the equations defining quasicomplete intersection space curves, Arch. Math. (Basel) 70 (1998) 244-249. MR 1604080 (98k:14042)
  • 2. H. Bresinsky, Minimal free resolutions of monomial curves in $ \mathbb{P}^3$, Linear Alg. Appl. 59 (1984) 121-129. MR 743050 (85d:14042)
  • 3. 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)
  • 4. H. Bresinsky, P. Schenzel, W. Vogel, On liaison, arithmetical Buchsbaum curves and monomial curves in $ \mathbb{P}^3$, J. Algebra 86 (1984) 283-301. MR 732252 (85c:14031)
  • 5. W. Decker, Monads and cohomology modules of rank 2 vector bundles, Compositio Math. 76 (1990) 7-17. MR 1078855 (91k:14030)
  • 6. D. Franco, S. Kleiman, A. Lascu, Gherardelli linkage and complete intersections, Michigan Math. J. 48 (2000) 271-279. MR 1786490 (2002a:14057)
  • 7. M. Green, R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988) 301-314. MR 959214 (90d:14034)
  • 8. R. Hartshorne, Stable vector bundles of rank $ 2$ on $ \mathbb{P}^3$, Math. Ann. 238 (1978) 229-280. MR 514430 (80c:14011)
  • 9. F. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. 14 (3) (1964) 689-713. MR 0169877 (30:120)
  • 10. M. Martin-Deschamps, D. Perrin, Sur la classification des courbes gauches, Astérisque 184-185 (1990). MR 1073438 (91h:14039)
  • 11. J. Migliore, Introduction to Liaison Theory and Deficiency Modules, Birkhäuser Boston, 1998. MR 1712469 (2000g:14058)
  • 12. P. Rao, A note on cohomology modules of rank two bundles, J. Algebra 86 (1984) 23-34. MR 727366 (85b:14022)
  • 13. P. Rao, Liaison among curves in $ \mathbb{P}^3$, Invent. Math. 50 (1978/79) 205-217. MR 520926 (80e:14023)

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Additional Information

Youngook Choi
Affiliation: Department of Mathematics Education, Yeungnam University, 214-1 Daedong Gyeongsan, 712-749, Gyeongsangbuk-do, Republic of Korea

Keywords: Quasi-complete intersection, monomial curve, minimal free resolution, rank $2$ vector bundle, cohomology module.
Received by editor(s): July 1, 2008
Received by editor(s) in revised form: April 5, 2009
Published electronically: July 24, 2009
Additional Notes: This work was supported by a Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund), KRF-2007-521-C00002.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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