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Birational rigidity of complete intersections

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Abstract

We prove that every smooth complete intersection \(X=X_{d_{1}, \ldots , d_{s}}\subset \mathbb {P}^{\sum _{i=1}^{s}d_{i}}\) defined by s hypersurfaces of degree \(d_{1}, \ldots , d_{s}\) is birationally superrigid if \(5s +1\le \frac{2(\sum _{i=1}^{s}d_{i}+1)}{\sqrt{\prod _{i=1}^{s}d_{i}}}\). In particular, X is non-rational and \({{\mathrm{Bir}}}(X)={{\mathrm{Aut}}}(X)\). We also prove birational superrigidity of singular complete intersections with similar numerical condition. These extend the results proved by Tommaso de Fernex.

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Acknowledgments

The author wishes to express his gratitude to his supervisor Professor Yujiro Kawamata for his encouragement and valuable advice. The author is grateful to Professor Tommaso de Fernex for sending his drafts of the papers [13, 14] and for helpful suggestions. The author wishes to thank Akihiro Kanemitsu, Chen Jiang, Pu Cao and Yusuke Nakamura for careful reading of the manuscript and helpful suggestions. This paper is an extension of the master thesis of the author at University of Tokyo. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan.

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  1. Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, 153-9814, Japan

    Fumiaki Suzuki

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Suzuki, F. Birational rigidity of complete intersections. Math. Z. 285, 479–492 (2017). https://doi.org/10.1007/s00209-016-1717-7

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