Skip to main content

Higher Dimensional Varieties and their Moduli Spaces

Guanajuato, Mexico

  • 872 Accesses

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN,volume 21)

Abstract

To explain some of the main ideas of the Minimal Model Program and some of the tools used, we use some basic facts from graph theory. In particular, we describe a directed graph associated to the category of projective varieties. For this reason, we recall some of the basic definitions in graph theory.

Keywords

  • Minimal Model Program
  • Mori Fiber Space
  • Birational Contraction
  • Higher Dimensional Projective Varieties
  • Proper Birational Morphism

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-59486-6_2
  • Chapter length: 21 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-59486-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)

Notes

  1. 1.

    Note that this definition is slightly different than the one given in [2].

  2. 2.

    Note that a priori λ might be an irrational number. On the other hand, Theorem 2.2 holds in a more general context, assuming that \(\Delta\) is a \(\mathbb{R}\)-divisor rather than a \(\mathbb{Q}\)-divisor.

  3. 3.

    The main idea, on why we can do this, relies on the fact that any divisor contracted by the birational contraction X −−→ X′ in contained in the stable base locus of \(K_{X} + \Delta\).

References

  1. C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips. Duke Math. J. 136(1), 173–180 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. C. Birkar, P. Cascini, C.D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Google Scholar 

  3. A. Bruno, K. Matsuki, Log Sarkisov program. Int. J. Math. 8(4), 451–494 (1997)

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. P. Cascini, V. Lazić, The minimal model program revisited, in Contributions to Algebraic Geometry (European Mathematical Society, Zurich, 2012), pp. 169–187

    CrossRef  MATH  Google Scholar 

  5. P. Cascini, V. Lazić, New outlook on the minimal model program, I. Duke Math. J. 161(12), 2415–2467 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. A. Corti, Factoring birational maps of threefolds after Sarkisov. J. Algebr. Geom. 4(2), 223–254 (1995)

    MathSciNet  MATH  Google Scholar 

  7. A. Corti (ed.), Flips for 3-Folds and 4-Folds. Oxford Lecture Series in Mathematics and its Applications, vol. 35 (Oxford University Press, Oxford, 2007)

    Google Scholar 

  8. A. Corti, V. Lazić, New outlook on the minimal model program, II. Math. Ann. 356(2), 617–633 (2013)

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. O. Fujino, S. Mori, A canonical bundle formula. J. Differ. Geom. 56(1), 167–188 (2000)

    MathSciNet  CrossRef  MATH  Google Scholar 

  10. Y. Gongyo, On weak Fano varieties with log canonical singularities. J. Reine Angew. Math. 665, 237–252 (2012)

    MathSciNet  MATH  Google Scholar 

  11. C.D. Hacon, J. McKernan, Existence of minimal models for varieties of log general type II. J. Am. Math. Soc. 23(2), 469–490 (2010)

    Google Scholar 

  12. C. Hacon, J. McKernan, The Sarkisov program. J. Algebr. Geom. 22(2), 389–405 (2013)

    Google Scholar 

  13. C. Hacon, J. McKernan, C. Xu, Acc for log canonical thresholds (2012)

    Google Scholar 

  14. Y. Kawamata, On the length of an extremal rational curve. Invent. Math. 105, 609–611 (1991)

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Y. Kawamata, Flops connect minimal models. Publ. Res. Inst. Math. Sci. 44(2), 419–423 (2008)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998)

    Google Scholar 

  17. J. Kollár et al., Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992

    MATH  Google Scholar 

  18. R. Lazarsfeld, Positivity in Algebraic Geometry. I. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48 (Springer, Berlin, 2004)

    Google Scholar 

  19. B. Lehmann, A cone theorem for NEF curves. J. Algebr. Geom. 21(3), 473–493 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. C.-L. Wang, K-equivalence in birational geometry and characterizations of complex elliptic genera. J. Algebr. Geom. 12(2), 285–306 (2003)

    MathSciNet  CrossRef  MATH  Google Scholar 

Download references

Acknowledgements

These are the notes for the CIMPA-CIMAT-ICTP School “Moduli of Curves” in Guanajuato, México, 22 February–4 March 2016. I would like to thank the organisers and all the participants for the invitation and for giving me the opportunity to present this material at the school. I would also like to thank the referee for reading a preliminary version of these notes and providing many useful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Paolo Cascini .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2017 The Author(s)

About this chapter

Cite this chapter

Cascini, P. (2017). Higher Dimensional Varieties and their Moduli Spaces. In: Brambila Paz, L., Ciliberto, C., Esteves, E., Melo, M., Voisin, C. (eds) Moduli of Curves. Lecture Notes of the Unione Matematica Italiana, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-59486-6_2

Download citation