There is no Bogomolov type restriction theorem for strong semistability in positive characteristic

Brenner, Holger

doi:10.1090/S0002-9939-05-07843-3

There is no Bogomolov type restriction theorem for strong semistability in positive characteristic
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by Holger Brenner

Proc. Amer. Math. Soc. 133 (2005), 1941-1947

DOI: https://doi.org/10.1090/S0002-9939-05-07843-3

Published electronically: January 31, 2005

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Abstract:

We give an example of a strongly semistable vector bundle of rank two on the projective plane such that there exist smooth curves of arbitrary high degree with the property that the restriction of the bundle to the curve is not strongly semistable anymore. This shows that a Bogomolov type restriction theorem does not hold for strong semistability in positive characteristic.

References

Fedor A. Bogomolov, Stability of vector bundles on surfaces and curves, Einstein metrics and Yang-Mills connections (Sanda, 1990) Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 35–49. MR 1215277
H. Brenner. Computing the tight closure in dimension two. To appear in Mathematics of Computation, 2005.
Holger Brenner, Slopes of vector bundles on projective curves and applications to tight closure problems, Trans. Amer. Math. Soc. 356 (2004), no. 1, 371–392. MR 2020037, DOI 10.1090/S0002-9947-03-03391-9
C. Deninger and A. Werner. Vector bundles and $p$-adic representations I. Preprint, ArXiv, 2003.
Hubert Flenner, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv. 59 (1984), no. 4, 635–650. MR 780080, DOI 10.1007/BF02566370
Melvin Hochster, Tight closure in equal characteristic, big Cohen-Macaulay algebras, and solid closure, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 173–196. MR 1266183, DOI 10.1090/conm/159/01507
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
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
Herbert Lange and Ulrich Stuhler, Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1977), no. 1, 73–83 (German). MR 472827, DOI 10.1007/BF01215129
Adrian Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), no. 1, 251–276. MR 2051393, DOI 10.4007/annals.2004.159.251
V. B. Mehta and A. Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1981/82), no. 3, 213–224. MR 649194, DOI 10.1007/BF01450677
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