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Slopes of vector bundles on projective curves and applications to tight closure problems. (English) Zbl 1041.13002
In J. Algebra 265, 45–78 (2003; Zbl 1099.13010), H. Brenner has showed how to connect the tight closure theory of standard graded rings to the theory of projective bundles. In this paper, the author applies his method to two dimensional standard graded rings over a field. The author investigates the minimal and maximal slope of locally free sheaves on smooth projective curves and shows how to apply these invariants to tight closure problems. He describes degree bounds that force elements to belong to the tight closure of a given ideal; similarly, he gives degree bounds that guarantee that an element is not in the tight closure of an ideal unless it is in the ideal itself. While it is known that examples in tight closure theory are generally hard to compute, these results provide a new and large class of examples in tight closure theory obtained from a coherent point view.

MSC:
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14H60 Vector bundles on curves and their moduli
13A02 Graded rings
14F17 Vanishing theorems in algebraic geometry
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[1] 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. · Zbl 0957.14008 · doi:10.4153/CJM-1996-058-1 · doi.org
[2] Charles M. Barton, Tensor products of ample vector bundles in characteristic \?, Amer. J. Math. 93 (1971), 429 – 438. · Zbl 0221.14011 · doi:10.2307/2373385 · doi.org
[3] H. Brenner, Tight closure and projective bundles, J. Algebra 265 (2003), 45-78. · Zbl 1099.13010
[4] H. Brenner, Tight closure and plus closure for cones over elliptic curves, submitted. · Zbl 1074.13003
[5] F. Campana and H. Flenner, A characterization of ample vector bundles on a curve, Math. Ann. 287 (1990), no. 4, 571 – 575. · Zbl 0728.14033 · doi:10.1007/BF01446914 · doi.org
[6] David Gieseker, \?-ample bundles and their Chern classes, Nagoya Math. J. 43 (1971), 91 – 116. · Zbl 0221.14010
[7] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167.
[8] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167.
[9] 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. · Zbl 0324.14006 · doi:10.1007/BF01357141 · doi.org
[10] Robin Hartshorne, Ample vector bundles, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 63 – 94. · Zbl 0173.49003
[11] Robin Hartshorne, Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73 – 89. · Zbl 0218.14018
[12] 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. · Zbl 0208.48901
[13] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[14] 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. · Zbl 0812.13006 · doi:10.1090/conm/159/01508 · doi.org
[15] 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. · Zbl 0930.13004
[16] 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. · Zbl 0930.13005
[17] Craig Huneke and Karen E. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127 – 152. · Zbl 0913.13003
[18] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. · Zbl 0872.14002
[19] Paltin Ionescu and Matei Toma, On very ample vector bundles on curves, Internat. J. Math. 8 (1997), no. 5, 633 – 643. · Zbl 0899.14011 · doi:10.1142/S0129167X97000330 · doi.org
[20] Herbert Lange, Zur Klassifikation von Regelmannigfaltigkeiten, Math. Ann. 262 (1983), no. 4, 447 – 459 (German). · Zbl 0492.14003 · doi:10.1007/BF01456060 · doi.org
[21] R. Lazarsfeld, Positivity in Algebraic Geometry (Preliminary Draft), 2001.
[22] 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.
[23] Shigeru Mukai and Fumio Sakai, Maximal subbundles of vector bundles on a curve, Manuscripta Math. 52 (1985), no. 1-3, 251 – 256. · Zbl 0572.14008 · doi:10.1007/BF01171494 · doi.org
[24] Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. · Zbl 0438.32016
[25] 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.
[26] Karen E. Smith, Tight closure in graded rings, J. Math. Kyoto Univ. 37 (1997), no. 1, 35 – 53. · Zbl 0902.13005
[27] Xiaotao Sun, Remarks on semistability of \?-bundles in positive characteristic, Compositio Math. 119 (1999), no. 1, 41 – 52. · Zbl 0951.14031 · doi:10.1023/A:1001512029096 · doi.org
[28] Hiroshi Tango, On the behavior of extensions of vector bundles under the Frobenius map, Nagoya Math. J. 48 (1972), 73 – 89. · Zbl 0239.14007
[29] Adela Vraciu, \ast -independence and special tight closure, J. Algebra 249 (2002), no. 2, 544 – 565. · Zbl 1057.13004 · doi:10.1006/jabr.2001.9074 · doi.org
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