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Nef vector bundles on a projective space with first Chern class three

  • Masahiro OhnoEmail author
Article
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Abstract

We classify nef vector bundles on a projective space with first Chern class three over an algebraically closed field of characteristic zero; we see, in particular, that these nef vector bundles are globally generated if the second Chern class is less than eight, and that there exist nef but non-globally generated vector bundles with second Chern class eight and nine on a projective plane.

Keywords

Nef vector bundles Fano bundles Spectral sequences 

Mathematics Subject Classification

14J60 14F05 

Notes

References

  1. 1.
    Anghel, C., Manolache, N.: Globally generated vector bundles on \(\mathbb{P}^n\) with \(c_1=3\). Math. Nachr. 286(14–15), 1407–1423 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bănică, C.: Smooth reflexive sheaves. Rev. Roum. Math. Pures Appl. 36(9–10), 571–593 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Beilinson, A.A.: Coherent sheaves on \({P}^n\) and problems in linear algebra. Funktsional. Anal. i Prilozhen. 12(3), 214–216 (1978)CrossRefGoogle Scholar
  4. 4.
    Bondal, A.I.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989)MathSciNetGoogle Scholar
  5. 5.
    Chiodera, L., Ellia, P.: Rank two globally generated vector bundles with \(c_1\le 5\). Rend. Istit. Mat. Univ. Trieste 44, 413–422 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fulton, W.: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2, second edn. Springer, Berlin (1998)Google Scholar
  7. 7.
    Kawamata, Y.: A generalization of Kodaira–Ramanujam’s vanishing theorem. Math. Ann. 261(1), 43–46 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kumar, N.M., Peterson, C., Rao, A.P.: Standard vector bundle deformations on \(\mathbb{P}^n\). In: Cutkosky, S.D., Eddidin, D., Qin, Z., Zhang, Q. (eds.) Vector Bundles and Representation Theory (Columbia, MO, 2002), no. 322 in Contemporary Mathematics, pp. 151–163. American Mathematical Society, Providence (2003)CrossRefGoogle Scholar
  9. 9.
    Langer, A.: Fano 4-folds with scroll structure. Nagoya Math. J. 150, 135–176 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lazarsfeld, R.: Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 49. Springer, Berlin (2004)Google Scholar
  11. 11.
    Ohno, M.: Nef vector bundles on a projective space or a hyperquadric with the first Chern class small. (2014) arXiv:1409.4191
  12. 12.
    Ohno, M.: Nef vector bundles on a projective space with first Chern class 3 and second Chern class 8. Matematiche (Catania) 72(2), 69–81 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ohno, M., Terakawa, H.: A spectral sequence and nef vector bundles of the first Chern class two on hyperquadrics. Ann. Univ. Ferrara Sez. VII Sci. Mat. 60(2), 397–406 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces, Progress in Mathematics, vol. 3. Birkhäuser, Boston (1980)CrossRefzbMATHGoogle Scholar
  15. 15.
    Peternell, T., Szurek, M., Wiśniewski, J.A.: Numerically effective vector bundles with small Chern classes. In: Hulek, K., Peternell, T., Schneider, M., Schreyer, F.O. (eds.) Complex Algebraic Varieties, Proceedings, Bayreuth, 1990, no. 1507 in Lecture Notes in Mathematics, pp. 145–156. Springer, Berlin (1992)Google Scholar
  16. 16.
    Sato, E.: Uniform vector bundles on a projective space. J. Math. Soc. Jpn. 28(1), 123–132 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sierra, J.C., Ugaglia, L.: Globally generated vector bundles on projective spaces II. J. Pure Appl. Algebra 218(1), 174–180 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Szurek, M., Wiśniewski, J.A.: On Fano manifolds, which are \(\mathbb{P}^k\)-bundles over \(\mathbb{P}^2\). Nagoya Math. J. 120, 89–101 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tango, H.: An example of indecomposable vector bundle of rank \(n-1\) on \(\mathbf{P}^n\). J. Math. Kyoto Univ. 16(1), 137–141 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Trautmann, G.: Darstellung von Vektorraumbündeln über \(\mathbf{C}\setminus \{0\}\). Arch. Math. (Basel) 24, 303–313 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vetter, U.: Zu einem Satz von G. Trautmann über den Rang gewisser kohärenter analytischer Moduln. Arch. Math. (Basel) 24, 158–161 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Viehweg, E.: Vanishing theorems. J. Reine Angew. Math. 335, 1–8 (1982)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of Informatics and EngineeringThe University of Electro-CommunicationsChofu-shiJapan

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