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

Brenner, Holger

doi:10.1090/S0002-9947-03-03391-9

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

Trans. Amer. Math. Soc. 356 (2004), 371-392

DOI: https://doi.org/10.1090/S0002-9947-03-03391-9

Published electronically: August 25, 2003

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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

Alberto Alzati, Marina Bertolini, and Gian Mario Besana, Numerical criteria for very ampleness of divisors on projective bundles over an elliptic curve, Canad. J. Math. 48 (1996), no. 6, 1121–1137. MR 1426895, DOI 10.4153/CJM-1996-058-1
Charles M. Barton, Tensor products of ample vector bundles in characteristic $p$, Amer. J. Math. 93 (1971), 429–438. MR 289525, DOI 10.2307/2373385
H. Brenner, Tight closure and projective bundles, J. Algebra 265 (2003), 45–78.
H. Brenner, Tight closure and plus closure for cones over elliptic curves, submitted.
F. Campana and H. Flenner, A characterization of ample vector bundles on a curve, Math. Ann. 287 (1990), no. 4, 571–575. MR 1066815, DOI 10.1007/BF01446914
David Gieseker, $p$-ample bundles and their Chern classes, Nagoya Math. J. 43 (1971), 91–116. MR 296078, DOI 10.1017/S0027763000014380
A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248. MR 364254, DOI 10.1007/BF01357141
Robin Hartshorne, Ample vector bundles, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 63–94. MR 193092
Robin Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73–89. MR 292847, DOI 10.1017/S0027763000014379
Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977, DOI 10.1007/BFb0067839
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Melvin Hochster, Solid closure, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 103–172. MR 1266182, DOI 10.1090/conm/159/01508
Craig Huneke, Tight closure and its applications, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With an appendix by Melvin Hochster. MR 1377268, DOI 10.1016/0167-4889(95)00136-0
Craig Huneke, Tight closure, parameter ideals, and geometry, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 187–239. MR 1648666
Craig Huneke and Karen E. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127–152. MR 1437301
Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1450870, DOI 10.1007/978-3-663-11624-0
Paltin Ionescu and Matei Toma, On very ample vector bundles on curves, Internat. J. Math. 8 (1997), no. 5, 633–643. MR 1468354, DOI 10.1142/S0129167X97000330
Herbert Lange, Zur Klassifikation von Regelmannigfaltigkeiten, Math. Ann. 262 (1983), no. 4, 447–459 (German). MR 696517, DOI 10.1007/BF01456060
R. Lazarsfeld, Positivity in Algebraic Geometry (Preliminary Draft), 2001.
Yoichi Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449–476. MR 946247, DOI 10.2969/aspm/01010449
Shigeru Mukai and Fumio Sakai, Maximal subbundles of vector bundles on a curve, Manuscripta Math. 52 (1985), no. 1-3, 251–256. MR 790801, DOI 10.1007/BF01171494
Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
C. S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque, vol. 96, Société Mathématique de France, Paris, 1982 (French). Notes written by J.-M. Drezet from a course at the École Normale Supérieure, June 1980. MR 699278
Karen E. Smith, Tight closure in graded rings, J. Math. Kyoto Univ. 37 (1997), no. 1, 35–53. MR 1447362, DOI 10.1215/kjm/1250518397
Xiaotao Sun, Remarks on semistability of $G$-bundles in positive characteristic, Compositio Math. 119 (1999), no. 1, 41–52. MR 1711507, DOI 10.1023/A:1001512029096
Hiroshi Tango, On the behavior of extensions of vector bundles under the Frobenius map, Nagoya Math. J. 48 (1972), 73–89. MR 314851, DOI 10.1017/S0027763000015099
Adela Vraciu, $\ast$-independence and special tight closure, J. Algebra 249 (2002), no. 2, 544–565. MR 1901172, DOI 10.1006/jabr.2001.9074