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Birational geometry of hypersurfaces in products of projective spaces

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Abstract

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in \({\mathbb {P}}^m \times {\mathbb {P}}^n\), we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using this, we give new simple examples of non-Mori dream spaces and analogues of Mumford’s example of a strictly nef line bundle which is not ample.

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Acknowledgments

Thanks to Burt Totaro for his advice and encouragement. Also thanks to Laurent Gruson, Antonio Laface, Victor Lozovanu, Christian Peskine, Diane MacLagan and Kenji Oguiso for useful discussions and comments. After this paper was written, we learned that some of the hypersurfaces in Theorem 1.1 were considered by Ito in [10], who shows that they are Mori dream spaces using a different argument.

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  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    John Christian Ottem

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  1. John Christian Ottem
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Ottem, J.C. Birational geometry of hypersurfaces in products of projective spaces. Math. Z. 280, 135–148 (2015). https://doi.org/10.1007/s00209-015-1415-x

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  • DOI: https://doi.org/10.1007/s00209-015-1415-x

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