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Periodic occurrence of complete intersection monomial curves


Authors: A. V. Jayanthan and Hema Srinivasan
Journal: Proc. Amer. Math. Soc. 141 (2013), 4199-4208
MSC (2010): Primary 13C40, 14H50
DOI: https://doi.org/10.1090/S0002-9939-2013-11991-X
Published electronically: August 23, 2013
MathSciNet review: 3105863
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Abstract: We study the complete intersection property of monomial curves in the family $ \Gamma _{\underline {a} + \underline {j}} = \{(t^{a_0 + j}, t^{a_1+j}, \ldots , t^{a_n + j}) ~ \vert ~ j \geq 0, ~ a_0 < a_1 < \cdots < a_n \}$. We prove that if $ \Gamma _{\underline {a}+\underline {j}}$ is a complete intersection for $ j \gg 0$, then $ \Gamma _{\underline {a}+\underline {j}+{\underline {a}_n}}$ is a complete intersection for $ j \gg 0$. This proves a conjecture of Herzog and Srinivasan on eventual periodicity of Betti numbers of semigroup rings under translations for complete intersections. We also show that if $ \Gamma _{\underline {a}+\underline {j}}$ is a complete intersection for $ j \gg 0$, then $ \Gamma _{\underline {a}}$ is a complete intersection. We also characterize the complete intersection property of this family when $ n = 3$.


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Additional Information

A. V. Jayanthan
Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai, India 600036
Email: jayanav@iitm.ac.in

Hema Srinivasan
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: srinivasanh@missouri.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11991-X
Received by editor(s): February 15, 2012
Published electronically: August 23, 2013
Additional Notes: The work was done during the first author’s visit to the University of Missouri-Columbia. He was funded by the Department of Science and Technology, Government of India. He sincerely thanks the funding agency and also the Department of Mathematics at the University of Missouri-Columbia for the great hospitality provided to him.
Communicated by: Irena Peeva
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.