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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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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Additional Information

Youngook Choi
Affiliation: Department of Mathematics Education, Yeungnam University, 214-1 Daedong Gyeongsan, 712-749, Gyeongsangbuk-do, Republic of Korea
Email: ychoi824@ynu.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09996-1
PII: S 0002-9939(09)09996-1
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.