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Moduli of Curves pp 27–47Cite as

Higher Dimensional Varieties and their Moduli Spaces

Guanajuato, Mexico

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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.

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  • DOI: 10.1007/978-3-319-59486-6_2
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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 \(K_{X} + \Delta\).

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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.

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Authors and Affiliations

  1. Imperial College London, London, UK

    Paolo Cascini

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  1. Paolo Cascini
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Correspondence to Paolo Cascini .

Editor information

Editors and Affiliations

  1. CIMAT, Guanajuato, Mexico

    Prof. Dr. h.c. Leticia Brambila Paz

  2. Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy

    Prof. Ciro Ciliberto

  3. Instituto Nacional de Matemática Pura, Rio de Janeiro, Brazil

    Prof. Eduardo Esteves

  4. Dipartimento di Matematica, Università di Roma Tre, Roma, Italy

    Dr. Margarida Melo

  5. Collège de France, Paris, France

    Prof. Claire Voisin

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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

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