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Higher Dimensional Varieties and their Moduli Spaces

Guanajuato, Mexico

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Moduli of Curves

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

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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.

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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 .

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Editors and Affiliations

  1. CIMAT, Guanajuato, Mexico

    Leticia Brambila Paz

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

    Ciro Ciliberto

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

    Eduardo Esteves

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

    Margarida Melo

  5. Collège de France, Paris, France

    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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