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Izvestiya: Mathematics, 2002, Volume 66, Issue 6, Pages 1243–1269
DOI: https://doi.org/10.1070/IM2002v066n06ABEH000413
(Mi im413)
 

This article is cited in 36 scientific papers (total in 36 papers)

Birationally rigid Fano hypersurfaces

A. V. Pukhlikov

Steklov Mathematical Institute, Russian Academy of Sciences
References:
Abstract: We prove that a smooth Fano hypersurface V=VMPM, M6, is birationally superrigid. In particular, it cannot be fibred into uniruled varieties by a non-trivial rational map, and every birational map of V onto a minimal Fano variety of the same dimension is a biregular isomorphism. The proof is based on the method of maximal singularities combined with the connectedness principle of Shokurov and Kollar.
Received: 04.04.2002
Bibliographic databases:
Document Type: Article
UDC: 512.9
MSC: 14E05, 14J45
Language: English
Original paper language: Russian
Citation: A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269
Citation in format AMSBIB
\Bibitem{Puk02}
\by A.~V.~Pukhlikov
\paper Birationally rigid Fano hypersurfaces
\jour Izv. Math.
\yr 2002
\vol 66
\issue 6
\pages 1243--1269
\mathnet{http://mi.mathnet.ru/eng/im413}
\crossref{https://doi.org/10.1070/IM2002v066n06ABEH000413}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1970356}
\zmath{https://zbmath.org/?q=an:1083.14012}
Linking options:
  • https://www.mathnet.ru/eng/im413
  • https://doi.org/10.1070/IM2002v066n06ABEH000413
  • https://www.mathnet.ru/eng/im/v66/i6/p159
  • This publication is cited in the following 36 articles:
    1. de Fernex T., “Birational Rigidity and K-Stability of Fano Hypersurfaces With Ordinary Double Points”, Rend. Circ. Mat. Palermo, 2022  crossref  isi  scopus
    2. A. V. Pukhlikov, “Effective results in the theory of birational rigidity”, Russian Math. Surveys, 77:2 (2022), 301–354  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. Nathan Chen, David Stapleton, “Higher index Fano varieties with finitely many birational automorphisms”, Compositio Math., 158:11 (2022), 2033  crossref
    4. A. V. Pukhlikov, “Birational geometry of singular Fano double spaces of index two”, Sb. Math., 212:4 (2021), 551–566  mathnet  crossref  crossref  zmath  adsnasa  isi
    5. Suzuki F., “Birational Superrigidity and K-Stability of Projectively Normal Fano Manifolds of Index One”, Mich. Math. J., 70:4 (2021), 779–792  crossref  mathscinet  isi
    6. Zhuang Z., “Birational Superrigidity Is Not a Locally Closed Property”, Sel. Math.-New Ser., 26:1 (2020), UNSP 11  crossref  mathscinet  isi
    7. Pukhlikov V A., “Birational Geometry of Singular Fano Hypersurfaces of Index Two”, Manuscr. Math., 161:1-2 (2020), 161–203  crossref  mathscinet  isi
    8. Ziquan Zhuang, “Birational superrigidity and K-stability of Fano complete intersections of index 1”, Duke Math. J., 169:12 (2020)  crossref
    9. Stibitz Ch., Zhuang Z., “K-Stability of Birationally Superrigid Fano Varieties”, Compos. Math., 155:9 (2019), 1845–1852  crossref  mathscinet  isi
    10. Kollar J., “Algebraic Hypersurfaces”, Bull. Amer. Math. Soc., 56:4 (2019), 543–568  crossref  mathscinet  isi
    11. János Kollár, Lecture Notes of the Unione Matematica Italiana, 26, Birational Geometry of Hypersurfaces, 2019, 129  crossref
    12. Thomas Eckl, Aleksandr Pukhlikov, “Effective Birational Rigidity of Fano Double Hypersurfaces”, Arnold Math J., 4:3-4 (2018), 505  crossref
    13. Suzuki F., “Birational rigidity of complete intersections”, Math. Z., 285:1-2 (2017), 479–492  crossref  mathscinet  zmath  isi  scopus
    14. de Fernex T., “Birational Rigidity of Singular Fano Hypersurfaces”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 17:3 (2017), 911–929  mathscinet  zmath  isi
    15. Aleksandr V. Pukhlikov, “The 4n2 4 n 2 -Inequality for Complete Intersection Singularities”, Arnold Math J., 3:2 (2017), 187  crossref
    16. Pukhlikov A.V., “Birational geometry of Fano hypersurfaces of index two”, Math. Ann., 366:1-2 (2016), 721–782  crossref  mathscinet  zmath  isi  elib  scopus
    17. A. V. Pukhlikov, “Birational geometry of higher-dimensional Fano varieties”, Proc. Steklov Inst. Math., 288, suppl. 2 (2015), S1–S150  mathnet  crossref  crossref  isi  elib
    18. Jelonek Z., Lenarcik T., “Automorphisms of Affine Smooth Varieties”, Proc. Amer. Math. Soc., 142:4 (2014), 1157–1163  crossref  mathscinet  zmath  isi  scopus
    19. Demailly J.-P., Hoang Hiep Pham, “A Sharp Lower Bound For the Log Canonical Threshold”, Acta Math., 212:1 (2014), 1–9  crossref  mathscinet  zmath  isi  scopus
    20. Tommaso de Fernex, Springer Proceedings in Mathematics & Statistics, 79, Automorphisms in Birational and Affine Geometry, 2014, 103  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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