I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875

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Russian Mathematical Surveys, 2005, Volume 60, Issue 5, Pages 875–965
DOI: https://doi.org/10.1070/RM2005v060n05ABEH003736 (Mi rm1643)

This article is cited in 53 scientific papers (total in 53 papers)

Birationally rigid Fano varieties

I. A. Cheltsov

Steklov Mathematical Institute, Russian Academy of Sciences

English version PDF (1050 kB) Citations (53) Russian version article

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DOI: https://doi.org/10.1070/RM2005v060n05ABEH003736

Abstract: The birational superrigidity and, in particular, the non-rationality of a smooth three-dimensional quartic was proved by V. Iskovskikh and Yu. Manin in 1971, and this led immediately to a counterexample to the three-dimensional Lüroth problem. Since then, birational rigidity and superrigidity have been proved for a broad class of higher-dimensional varieties, among which the Fano varieties occupy the central place. The present paper is a survey of the theory of birationally rigid Fano varieties.

Received: 23.06.2005

Bibliographic databases:

Document Type: Article

UDC: 512.76

MSC: Primary 14J45, 14E05, 14E30; Secondary 14G22, 14J30, 14E07, 14M20, 14M10

Language: English

Original paper language: Russian

Citation: I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/rm1643

https://doi.org/10.1070/RM2005v060n05ABEH003736

https://www.mathnet.ru/eng/rm/v60/i5/p71

This publication is cited in the following 53 articles:

Ivan Cheltsov, Yuri Tschinkel, Zhijia Zhang, “Equivariant geometry of the Segre cubic and the Burkhardt quartic”, Sel. Math. New Ser., 31:1 (2025)
Ivan Cheltsov, Yuri Tschinkel, Zhijia Zhang, “Equivariant geometry of singular cubic threefolds”, Forum of Mathematics, Sigma, 13 (2025)
Benson Farb, “Irrationality of the general smooth quartic 3-fold using intermediate Jacobians”, Advances in Mathematics, 465 (2025), 110160
Charles Favre, Alexandra Kuznetsova, “Families of automorphisms on abelian varieties”, Math. Ann., 2024
A. V. Pukhlikov, “Birationally rigid hypersurfaces with quadratic singularities of low rank”, Sb. Math., 215:6 (2024), 823–840
Aleksandr Pukhlikov, “Birationally rigid Fano-Mori fibre spaces”, Forum of Mathematics, Sigma, 12 (2024)
Ivan Cheltsov, Adrien Dubouloz, Takashi Kishimoto, “Toric G-solid Fano threefolds”, Sel. Math. New Ser., 29:2 (2023)
Mikhail Ovcharenko, “The classification of smooth well-formed Fano weighted complete intersections”, Int. J. Math., 34:11 (2023), 2350064
Andrey Trepalin, “Birational classification of pointless del Pezzo surfaces of degree 8”, Eur. J. Math., 9 (2023), 9
Ivan Cheltsov, “Kummer quartic double solids”, Rend. Circ. Mat. Palermo, II. Ser, 72:3 (2023), 1993
A. V. Pukhlikov, “Effective results in the theory of birational rigidity”, Russian Math. Surveys, 77:2 (2022), 301–354
Birkar C., “Generalised Pairs in Birational Geometry”, EMS Surv. Math. Sci., 8:1-2 (2021), 5–24
Kresch A., Tschinkel Yu., “Brauer Groups of Involution Surface Bundles”, Pure Appl. Math. Q., 17:2, SI (2021), 649–669
Zhuang Z., “Birational Superrigidity Is Not a Locally Closed Property”, Sel. Math.-New Ser., 26:1 (2020), UNSP 11
Cheltsov I., Kuznetsov A., Shramov C., “Coble Fourfold, S-6-Invariant Quartic Threefolds, and Wiman-Edge Sextics”, Algebr. Number Theory, 14:1 (2020), 213–274
Shramov C., “Automorphisms of Cubic Surfaces Without Points”, Int. J. Math., 31:11 (2020), 2050083
Fontanari C., Martinelli D., “A Remark on Rationally Connected Varieties and Mori Dream Spaces”, Proc. Edinb. Math. Soc., 62:1 (2019), 259–263
das Dores L., Mauri M., “G-Birational Superrigidity of Del Pezzo Surfaces of Degree 2 and 3”, Eur. J. Math., 5:3, SI (2019), 798–827
Kollar J., “Algebraic Hypersurfaces”, Bull. Amer. Math. Soc., 56:4 (2019), 543–568
Cheltsov I., Shramov C., “Finite Collineation Groups and Birational Rigidity”, Sel. Math.-New Ser., 25:5 (2019), UNSP 71

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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Abstract page:	883
Russian version PDF:	333
English version PDF:	40
References:	104
First page:	2

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