Bollettino dell'Unione Matematica Italiana

, Volume 11, Issue 1, pp 69–74 | Cite as

Why should a birational geometer care about Bridgeland stability conditions?

Article
  • 36 Downloads

Abstract

In this survey we borrow from Coskun and Huizenga an example of application of Bridgeland stability conditions to birational geometry and we rephrase it without assuming any previous knowledge about derived categories.

Keywords

Stability condition Ample cone Moduli space Coherent sheaf 

Mathematics Subject Classification

14J60 14E30 

Notes

Acknowledgments

This project started during the visit of D. Martinelli at the Department of Mathematics at the University of Trento. She wishes to thank this institute for the warm hospitality. C. Fontanari and D. Martinelli would like to thank Arend Bayer for useful comments and suggestions.

References

  1. 1.
    Arcara, D., Bertram, A., Coskun, I., Huizenga, J.: The minimal model program for the Hilbert scheme of points on \(\mathbb{P}^2\) and Bridgeland stability. Adv. Math. 235, 580–626 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bayer, A.: A tour to stability conditions on derived categories (2010). http://www.maths.ed.ac.uk/~abayer/dc-lecture-notes.pdf
  3. 3.
    Bayer, A., Macrì, E.: Projectivity and birational geometry of Bridgeland moduli spaces. J. Am. Math. Soc. 27(3), 707–752 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bayer, A., Macrì, E.: MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones. Lagrangian Fibr. Invent. Math. 198(3), 505–590 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166(2), 317–345 (2007)Google Scholar
  6. 6.
    Bridgeland, T.: Stability conditions on K3 surfaces. Duke Math. J. 141(2), 241–291 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Coskun, I., Huizenga, J.: The ample cone of moduli spaces of sheaves on the plane. Algebr. Geom. 3(1), 106–136 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Huybrechts, D.: Introduction to stability conditions. Moduli spaces, 179–229. In: London Math. Soc. Lecture Note Ser., vol. 411. Cambridge Univ. Press, Cambridge (2014)Google Scholar
  9. 9.
    Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. In: Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig (1997)Google Scholar
  10. 10.
    Le Potier, J.: Lectures on vector bundles. In: Cambridge Studies in Advanced Mathematics, vol. 54. Cambridge University Press, Cambridge (1997)Google Scholar
  11. 11.
    Li, C., Zhao, X.: The MMP for deformations of \(Hilb^n \mathbb{P}^2\) (2013). arXiv:1312.1748v1
  12. 12.
    Li, C., Zhao, X.: Birational models of moduli spaces of coherent sheaves on the projective plane (2016). arXiv:1603.05035v1

Copyright information

© Unione Matematica Italiana 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di TrentoPovoItaly
  2. 2.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations