Advertisement

Manuscripta Mathematica

, Volume 142, Issue 1–2, pp 1–34 | Cite as

Exceptional sequences on rational \({\mathbb{C}^{*}}\) -surfaces

  • Andreas HocheneggerEmail author
  • Nathan Owen Ilten
Article

Abstract

Inspired by Bondal’s conjecture, we study the behavior of exceptional sequences of line bundles on rational \({\mathbb{C}^{*}}\) -surfaces under homogeneous degenerations. In particular, we provide a sufficient criterion for such a sequence to remain exceptional under a given degeneration. We apply our results to show that, for toric surfaces of Picard rank 3 or 4, all full exceptional sequences of line bundles may be constructed via augmentation. We also discuss how our techniques may be used to construct noncommutative deformations of derived categories.

Mathematics Subject Classification (2010)

Primary: 14M25 14F05 Secondary: 14D06 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bondal A.I.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989)MathSciNetGoogle Scholar
  2. 2.
    Flenner, H., Kaliman, S., Zaidenberg, M.: Completions of \({\mathbb{C}^{*}}\) -surfaces. In: Affine Algebraic Geometry, pp. 149–201. Osaka University Press, Osaka. Also arXiv:math/0511282 (2007)Google Scholar
  3. 3.
    Fulton W.: Introduction to Toric Varieties, volume 131 of Annals of Mathematics Studies. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton (1993)Google Scholar
  4. 4.
    Hille, L., Perling, M.: Exceptional sequences of invertible sheaves on rational surfaces. Compos. Math. 147(4), 1230–1280. Also arXiv:0810.1938 (2011)Google Scholar
  5. 5.
    Hochenegger, A., Ilten, N.O.: Families of invariant divisors on rational complexity-one T-varieties. arXiv:0906.4292v3 [math.AG] (2011)Google Scholar
  6. 6.
    Hochenegger, A.: Exceptional sequences of line bundles and spherical twists—a toric example. Beträge Algebra Geom. arXiv:1108.3734v1 (2011, to appear)Google Scholar
  7. 7.
    Hochenegger, A.: Exzeptionelle Folgen in der torischen Geometrie (Exceptional Sequences in Toric Geometry). PhD thesis, Freie Universität, Berlin. http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000022947 (2011)
  8. 8.
    Huybrechts D.: Fourier-Mukai Transforms in Algebraic Geometry. Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford (2006)CrossRefGoogle Scholar
  9. 9.
    Ilten, N.O., Süß, H.: Polarized complexity-one T-varieties. Mich. Math. J. 60(3), 561–578. Also arXiv:0910.5919 (2011)Google Scholar
  10. 10.
    Ilten, N.O., Vollmert, R.: Deformations of rational T-varieties. J. Algebraic Geom. 21(3), 531–562. Also arXiv:0903.1293 (2012)Google Scholar
  11. 11.
    Liendo, A., Süß, H.: Normal singularities with torus actions. Tohoku Math. J. arXiv:1005.2462v2 (2010, to appear)Google Scholar
  12. 12.
    Manin, Y.I.: Cubic Forms, volume 4 of North-Holland Mathematical Library, 2nd edn. North-Holland Publishing Co., Amsterdam (Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel) (1986)Google Scholar
  13. 13.
    Orlik P., Wagreich P.: Algebraic surfaces with k *-action. Acta Math. 138(1–2), 43–81 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Perling, M.: Examples for exceptional sequences of invertible sheaves on rational surfaces. In: Geometric Methods in Representation Theory II, Seminaire et Congress 25, pp. 369–389. Also arXiv:0904.0529v1 (2010)Google Scholar
  15. 15.
    Petersen, L., Süß, H.: Torus invariant divisors. Israel J. Math. 182(1), 481–504. Also arXiv:0811.0517 (2011)Google Scholar
  16. 16.
    Rudakov, A.N.: Helices and Vector Bundles, volume 148 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (Seminaire Rudakov, Translated from the Russian by A. D. King, P. Kobak and A. Maciocia) (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnCologneGermany
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations