A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243

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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

English version PDF (302 kB) Citations (36) Russian version article

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DOI: https://doi.org/10.1070/IM2002v066n06ABEH000413

Abstract: We prove that a smooth Fano hypersurface

$V=V_M\subset\mathbb P^M$ ,

$M\geqslant 6$ , 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:

de Fernex T., “Birational Rigidity and K-Stability of Fano Hypersurfaces With Ordinary Double Points”, Rend. Circ. Mat. Palermo, 2022
A. V. Pukhlikov, “Effective results in the theory of birational rigidity”, Russian Math. Surveys, 77:2 (2022), 301–354
Nathan Chen, David Stapleton, “Higher index Fano varieties with finitely many birational automorphisms”, Compositio Math., 158:11 (2022), 2033
A. V. Pukhlikov, “Birational geometry of singular Fano double spaces of index two”, Sb. Math., 212:4 (2021), 551–566
Suzuki F., “Birational Superrigidity and K-Stability of Projectively Normal Fano Manifolds of Index One”, Mich. Math. J., 70:4 (2021), 779–792
Zhuang Z., “Birational Superrigidity Is Not a Locally Closed Property”, Sel. Math.-New Ser., 26:1 (2020), UNSP 11
Pukhlikov V A., “Birational Geometry of Singular Fano Hypersurfaces of Index Two”, Manuscr. Math., 161:1-2 (2020), 161–203
Ziquan Zhuang, “Birational superrigidity and K-stability of Fano complete intersections of index 1”, Duke Math. J., 169:12 (2020)
Stibitz Ch., Zhuang Z., “K-Stability of Birationally Superrigid Fano Varieties”, Compos. Math., 155:9 (2019), 1845–1852
Kollar J., “Algebraic Hypersurfaces”, Bull. Amer. Math. Soc., 56:4 (2019), 543–568
János Kollár, Lecture Notes of the Unione Matematica Italiana, 26, Birational Geometry of Hypersurfaces, 2019, 129
Thomas Eckl, Aleksandr Pukhlikov, “Effective Birational Rigidity of Fano Double Hypersurfaces”, Arnold Math J., 4:3-4 (2018), 505
Suzuki F., “Birational rigidity of complete intersections”, Math. Z., 285:1-2 (2017), 479–492
de Fernex T., “Birational Rigidity of Singular Fano Hypersurfaces”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 17:3 (2017), 911–929
Aleksandr V. Pukhlikov, “The $4n^2$ 4 n 2 -Inequality for Complete Intersection Singularities”, Arnold Math J., 3:2 (2017), 187
Pukhlikov A.V., “Birational geometry of Fano hypersurfaces of index two”, Math. Ann., 366:1-2 (2016), 721–782
A. V. Pukhlikov, “Birational geometry of higher-dimensional Fano varieties”, Proc. Steklov Inst. Math., 288, suppl. 2 (2015), S1–S150
Jelonek Z., Lenarcik T., “Automorphisms of Affine Smooth Varieties”, Proc. Amer. Math. Soc., 142:4 (2014), 1157–1163
Demailly J.-P., Hoang Hiep Pham, “A Sharp Lower Bound For the Log Canonical Threshold”, Acta Math., 212:1 (2014), 1–9
Tommaso de Fernex, Springer Proceedings in Mathematics & Statistics, 79, Automorphisms in Birational and Affine Geometry, 2014, 103

Citing articles in Google Scholar: Russian citations, English citations
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Известия Российской академии наук. Серия математическая

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Abstract page:	518
Russian version PDF:	197
English version PDF:	30
References:	100
First page:	2

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