Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing the tight closure in dimension two

Author: Holger Brenner
Journal: Math. Comp. 74 (2005), 1495-1518
MSC (2000): Primary 13A35; Secondary 14H60
Published electronically: January 27, 2005
MathSciNet review: 2137014
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of relations for generators of the ideal to compute its tight closure. In particular, our method gives an algorithm to compute the tight closure of three elements under the condition that we are able to compute the Harder-Narasimhan filtration. We apply this to the computation of $(x^{a},y^{a},z^{a})^*$ in $K[x,y,z]/(F)$, where $F$ is a homogeneous polynomial.

References [Enhancements On Off] (What's this?)

  • 1. H. Brenner, Tight closure and projective bundles, J. Algebra 265 (2003), 45-78. MR 1984899 (2004h:13008)
  • 2. H. Brenner, The slope of vector bundles and applications to tight closure problems, Trans. Amer. Math. Soc. 356 (1) (2004), 371-392. MR 2020037
  • 3. D. Gieseker, p-ample bundles and their Chern classes, Nagoya Math. J. 43 (1971), 91-116. MR 0296078 (45:5139)
  • 4. A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique II. Inst. Hautes Études Sci. Publ. Math. 8 (1961).MR 0217084 (36:177b)
  • 5. N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. of Math. 120, 5 (1998), 981-996.MR 1646049 (99h:13005)
  • 6. G. Harder, M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215-248.MR 0364254 (51:509)
  • 7. R. Hartshorne, Ample vector bundles, Publ. Math. I.H.E.S. 29 (1966), 63-94.MR 0193092 (33:1313)
  • 8. R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.MR 0463157 (57:3116)
  • 9. M. Hochster, Solid closure, Contemp. Math. 159 (1994), 103-172.MR 1266182 (95a:13011)
  • 10. M. Hochster, C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31-116.MR 1017784 (91g:13010)
  • 11. C. Huneke, Tight Closure and Its Applications, AMS, 1996. MR 1377268 (96m:13001)
  • 12. C. Huneke, Tight Closure, Parameter Ideals, and Geometry, in Six Lectures on Commutative Algebra, Birkhäuser, Basel, 1998.MR 1648666 (99j:13001)
  • 13. C. Huneke, K. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127-152.MR 1437301 (98e:13007)
  • 14. D. Huybrechts, M. Lehn, The Geometry of Moduli Spaces of Sheaves, Viehweg, Braunschweig, 1997.MR 1450870 (98g:14012)
  • 15. R. Lazarsfeld, Positivity in Algebraic Geometry (Preliminary Draft), 2001.
  • 16. G. J. Leuschke, Appendix: Some examples in tight closure, Trends in Commutative Algebra, MSRI publications, 51, 2004.
  • 17. L. Migliorini, Some observations on cohomologically $p$-ample bundles, Annali di Matematica pura ed applicata (IV), Vol CLXIV, 89-102, (1993). MR 1243950 (94k:14012)
  • 18. Y. Miyaoka, The Chern class and Kodaira dimension of a minimal variety, in Algebraic Geometry, Sendai 1985, Adv. Stud. Pure Math. 10, 1987, 449-476.MR 0946247 (89k:14022)
  • 19. M. Katzman, Gröbner bases and the Frobenius map, preprint, http://www.sheffield.
  • 20. A. Singh, A computation of tight closure in diagonal hypersurfaces, J. Algebra 203, No. 2 (1998), 579-589.MR 1622811 (2000d:13007)
  • 21. K. E. Smith, Tight closure in graded rings, J. Math. Kyoto Univ 37, No. 1 (1997), 35-53.MR 1447362 (98e:13009)
  • 22. K. E. Smith, An introduction to tight closure, in Geometric and combinatorial aspects of commutative algebra, Lect. Notes Pure Appl. Math. 217 (2001), 353-377.MR 1824242 (2002e:13011)
  • 23. S. Sullivant, Tight closure of monomial ideals in Fermat rings, preprint, 2002.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 13A35, 14H60

Retrieve articles in all journals with MSC (2000): 13A35, 14H60

Additional Information

Holger Brenner
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom

Received by editor(s): March 10, 2003
Received by editor(s) in revised form: April 11, 2004
Published electronically: January 27, 2005
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society