Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Slopes of vector bundles on projective curves and applications to tight closure problems

Author: Holger Brenner
Journal: Trans. Amer. Math. Soc. 356 (2004), 371-392
MSC (2000): Primary 13A35, 14H60
Published electronically: August 25, 2003
MathSciNet review: 2020037
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous $R_+$-primary ideal in a two-dimensional normal standard-graded algebra $R$ in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.

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

  • 1. A. Alzati, M. Bertolini, G. M. Besana, Numerical criteria for very ampleness of divisors on projective bundles over an elliptic curve, Can. J. Math. 48 (6) (1996), 1121-1137. MR 97k:14032
  • 2. C. M. Barton, Tensor products of ample vector bundles in characteristic $p$, Am. J. Math. 93 (1971), 429-438. MR 44:6713
  • 3. H. Brenner, Tight closure and projective bundles, J. Algebra 265 (2003), 45-78.
  • 4. H. Brenner, Tight closure and plus closure for cones over elliptic curves, submitted.
  • 5. F. Campana and H. Flenner, A characterization of ample vector bundles on a curve, Math. Ann. 287 (1990), 571-575. MR 91f:14027
  • 6. D. Gieseker, p-ample bundles and their chern classes, Nagoya Math. J. 43 (1971), 91-116. MR 45:5139
  • 7. A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique II, Inst. Hautes Études Sci. Publ. Math. 8 (1961). MR 36:177b
  • 8. A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique III, Inst. Hautes Études Sci. Publ. Math. 11 (1961). MR 36:177c
  • 9. G. Harder, M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215-248. MR 51:509
  • 10. R. Hartshorne, Ample vector bundles, Publ. Math. I.H.E.S. 29 (1966), 63-94. MR 33:1313
  • 11. R. Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73-89. MR 45:1929
  • 12. R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Springer-Verlag, Berlin-Heidelberg-New York, 1970. MR 44:211
  • 13. R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. MR 57:3116
  • 14. M. Hochster, Solid closure, Contemp. Math. 159 (1994), 103-172. MR 95a:13011
  • 15. C. Huneke, Tight Closure and Its Applications, AMS, 1996. MR 96m:13001
  • 16. C. Huneke, Tight Closure, Parameter Ideals, and Geometry, in Six Lectures on Commutative Algebra, Birkhäuser, Basel, 1998. MR 99j:13001
  • 17. C. Huneke, K. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127-152. MR 98e:13007
  • 18. D. Huybrechts, M. Lehn, The Geometry of Moduli Spaces of Sheaves, Viehweg, Braunschweig, 1997. MR 98g:14012
  • 19. P. Ionescu and M. Toma, On very ample vector bundles on curves, Int. Jour. of Math. 8, No. 5 (1997), 633-643. MR 98h:14036
  • 20. H. Lange, Zur Klassifikation von Regelmannigfaltigkeiten, Math. Ann. 262 (1983), 447-459. MR 85b:14019
  • 21. R. Lazarsfeld, Positivity in Algebraic Geometry (Preliminary Draft), 2001.
  • 22. 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 89k:14022
  • 23. S. Mukai, F. Sakai, Maximal subbundles of vector bundles on a curve, Manuscripta math. 52 (1985), 251-256. MR 86k:14013
  • 24. C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Birkhäuser, Boston, Basel, Stuttgart, 1980. MR 81b:14001
  • 25. C. S. Seshadri, Fibrés vectorielle sur les courbes algébrique, Asterisque 96, 1982. MR 85b:14023
  • 26. K. E. Smith, Tight closure in graded rings, J. Math. Kyoto Univ 37, No. 1 (1997), 35-53. MR 98e:13009
  • 27. X. Sun, Remarks on semistability of $G$-bundles in positive characteristic, Comp. Math. 194 (1999), 41-52. MR 2001e:14032
  • 28. H. Tango, On the behaviour of extensions of vector bundles under the Frobenius map, Nagoya Math. J. 48 (1972), 73-89. MR 47:3401
  • 29. A. Vraciu, $*$-independence and special tight closure, J. Algebra 249, 2 (2002), 544-565. MR 2003d:13003

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13A35, 14H60

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

Additional Information

Holger Brenner
Affiliation: Mathematische Fakultät, Ruhr-Universität Bochum, 44780 Bochum, Germany

Received by editor(s): May 21, 2002
Received by editor(s) in revised form: February 19, 2003
Published electronically: August 25, 2003
Article copyright: © Copyright 2003 American Mathematical Society