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Fano Hypersurfaces and their Birational Geometry

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Automorphisms in Birational and Affine Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 79))

Abstract

We survey some results on the nonrationality and birational rigidity of certain hypersurfaces of Fano type. The focus is on hypersurfaces of Fano index one, but hypersurfaces of higher index are also discussed.

2010 Mathematics Subject Classification. Primary: 14E08; Secondary: 14J45, 14E05, 14B05, 14N30.

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Notes

  1. 1.

    This definition can be generalized to all Mori fiber spaces, see [5].

  2. 2.

    Mori fiber spaces are the output of the minimal model program for projective manifolds of negative Kodaira dimension. It is natural to motivate the notion of birational rigidity also from this point of view: a Mori fiber space is birationally rigid (resp., superrigid) if, within its own birational class, it is the unique answer of the program up to birational (resp., biregular) automorphisms preserving the fibration.

  3. 3.

    The hypothesis in Manin’s theorem that κ be perfect can be removed, cf. [22].

  4. 4.

    For a comparison, one should notice how similar the arguments are. We decided to use the same exact wording when the argument is the same so that the differences will stand out.

  5. 5.

    A partial solution to Problem 12 in the special case \(g(N) = N - 1\) has been recently announced in [32].

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Acknowledgements

The research was partially supported by NSF CAREER grant DMS-0847059 and a Simons Fellowship.

The author would like to thank the referee for useful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT, 48112-0090, USA

    Tommaso de Fernex

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  1. Tommaso de Fernex
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Correspondence to Tommaso de Fernex .

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Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom

    Ivan Cheltsov

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Ciro Ciliberto

  3. Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany

    Hubert Flenner

  4. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    James McKernan

  5. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Yuri G. Prokhorov

  6. Institute Fourier, UMR 5582, CNRS-UJF, University of Grenoble I, Saint Martin D’Hères, France

    Mikhail Zaidenberg

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de Fernex, T. (2014). Fano Hypersurfaces and their Birational Geometry. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_6

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