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Transactions of the American Mathematical Society

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Sharp bounds on
Castelnuovo-Mumford regularity


Author: Chikashi Miyazaki
Journal: Trans. Amer. Math. Soc. 352 (2000), 1675-1686
MSC (1991): Primary 14B15; Secondary 13D45, 13H10, 14M05
DOI: https://doi.org/10.1090/S0002-9947-99-02380-6
Published electronically: October 21, 1999
MathSciNet review: 1621769
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Abstract: The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic measures such as dimension, codimension and degree. In this paper we consider an upper bound on the regularity $\operatorname{reg}(X)$ of a nondegenerate projective variety $X$, $\operatorname{reg}(X)\le\lceil(\deg (X) - 1)/\operatorname{codim}(X)\rceil +k \cdot \dim (X)$, provided $X$ is $k$-Buchsbaum for $k \ge 1$, and investigate the projective variety with its Castelnuovo-Mumford regularity having such an upper bound.


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  • 1. E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of algebraic curves I, Grundlehren der math. Wissenschaften 167, Springer, 1985. MR 86h:14019
  • 2. E. Ballico, On singular curves in positive characteristic, Math. Nachr. 141 (1989), 267 - 273. MR 90h:14042
  • 3. E. Ballico and K. Yanagawa, On the h-vector of a Cohen-Macaulay domain in positive characteristic, Comm. Algebra 26 (1998), 1745-1756. MR 99d:13018
  • 4. D. Bayer and D. Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (ed. D. Eisenbud and L. Robbiano), pp. 1 - 48, Cambridge University Press 1993. MR 95d:13032
  • 5. D. Eisenbud, Commutative algebra with a view toward algebraic geometry, GTM 150, Springer, 1995.
  • 6. D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89 - 133. MR 85f:13023
  • 7. T. Fujita, Classification theories of polarized varieties, London Math. Soc. Lecture Note Series 155, Cambridge University Press, 1990. MR 93e:14009
  • 8. A. Geramita and J. Migliore, A generalized liaison addition, J. Algebra 163 (1994), 139 - 164. MR 94m:14066
  • 9. S. Giuffrida and R. Maggioni, On the Rao-module of a curve lying on a smooth cubic surface in $\mathbb{P}^3$, II, Comm. Algebra, 20 (1992), 329 - 347. MR 93c:14030
  • 10. J. Harris (with D. Eisenbud), Curves in projective space, Les Presses de l'Université de Montréal, 1982. MR 84g:14024
  • 11. R. Hartshorne, Algebraic geometry, GTM 52, Springer, 1977. MR 57:3116
  • 12. L. T. Hoa, Personal communications.
  • 13. L. T. Hoa, R. Miró Roig and W. Vogel, On numerical invariants of locally Cohen-Macaulay schemes in $\mathbb{P}^n$, Hiroshima Math. J., 24 (1994), 299 - 316. MR 95m:14031
  • 14. L. T. Hoa and C. Miyazaki, Bounds on Castelnuovo-Mumford regularity for generalized Cohen-Macaulay graded rings, Math. Ann. 301 (1995), 587 - 598. MR 96e:13019
  • 15. L. T. Hoa and W. Vogel, Castelnuovo-Mumford regularity and hyperplane sections, J. Algebra 163 (1994), 348 - 365. MR 94m:13016
  • 16. P. Maroscia, Some problems and results on finite sets of points in $\mathbb{P}^n$, Lecture Notes in Math. 997, Algebraic Geometry - Open Problems, pp. 290 - 314, Springer, 1983. MR 85d:14013
  • 17. J. Migliore, An introduction to deficiency modules and liaison theory for subschemes of projective space, Lecture Note Series 24, Seoul National University, 1994. MR 95k:14070
  • 18. J. Migliore and R. Miró Roig, On $k$-Buchsbaum curves in $\mathbb{P}^3$, Comm. Algebra. 18 (1990), 2403 - 2422. MR 91i:14025
  • 19. C. Miyazaki and W. Vogel, Bounds on cohomology and Castelnuovo-Mumford regularity, J. Algebra, 185 (1996), 626 - 642. MR 98g:13016
  • 20. U. Nagel, On the defining equations and syzygies of arithmetically Cohen-Macaulay varieties in arbitrary characteristic, J. Algebra 175 (1995), 359 - 372. MR 96f:13023
  • 21. U. Nagel, Degree bounds for the defining equations and syzygies of arithmetically Buchsbaum varieties, Abstract of short communications at Summer school on commutative algebra, Centre de Recerca matemàtica, Bellaterra (Spain), 1996.
  • 22. U. Nagel and P. Schenzel, Cohomological annihilators and Castelnuovo-Mumford regularity, Comtemp. Math. 159 (1994), 307 - 328. MR 95c:14003
  • 23. U. Nagel and P. Schenzel, Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity, Nagoya Math. J. 152 (1998), 153-174. CMP 99:07
  • 24. P. Schenzel, Notes on liaison and duality, J. Math. Kyoto Univ. 22 (1982), 485 - 498. MR 84a:13012
  • 25. J. Stückrad and W. Vogel, Buchsbaum rings and applications. An interaction between algebra, geometry and topology, Springer, 1986. MR 88h:13011b
  • 26. N. V. Trung and G. Valla, Degree bounds for the defining equations of arithmetically Cohen-Macaulay varieties, Math. Ann. 281 (1988), 209 - 218. MR 89k:14083
  • 27. K. Yanagawa, Castelnuovo's Lemma and h-vectors of Cohen-Macaulay homogeneous domains, J. Pure Appl. Algebra 105 (1995), 107 - 116. MR 97a:13024
  • 28. K. Yanagawa, On the regularities of arithmetically Buchsbaum curves, Math. Z. 226 (1997), 155 - 163. MR 99b:14028

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Additional Information

Chikashi Miyazaki
Affiliation: Department of Mathematical Sciences, University of the Ryukyus, Nishihara-cho, Okinawa 903-0213, Japan
Email: miyazaki@math.u-ryukyu.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-99-02380-6
Received by editor(s): July 15, 1997
Received by editor(s) in revised form: February 28, 1998
Published electronically: October 21, 1999
Additional Notes: Partially supported by Grant-in-Aid for Scientific Research (no. 09740042), Ministry of Education, Science, Sports and Culture, Japan
Article copyright: © Copyright 2000 American Mathematical Society

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