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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 5(365), Pages 71–160 (Mi umn1643)  

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

Birationally rigid Fano varieties

I. A. Cheltsov

Steklov Mathematical Institute, Russian Academy of Sciences

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.


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English version:
Russian Mathematical Surveys, 2005, 60:5, 875–965

Bibliographic databases:

UDC: 512.76
MSC: Primary 14J45, 14E05, 14E30; Secondary 14G22, 14J30, 14E07, 14M20, 14M10
Received: 23.06.2005

Citation: I. A. Cheltsov, “Birationally rigid Fano varieties”, Uspekhi Mat. Nauk, 60:5(365) (2005), 71–160; Russian Math. Surveys, 60:5 (2005), 875–965

Citation in format AMSBIB
\by I.~A.~Cheltsov
\paper Birationally rigid Fano varieties
\jour Uspekhi Mat. Nauk
\yr 2005
\vol 60
\issue 5(365)
\pages 71--160
\jour Russian Math. Surveys
\yr 2005
\vol 60
\issue 5
\pages 875--965

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    This publication is cited in the following articles:
    1. M. M. Grinenko, “Fibrations into del Pezzo surfaces”, Russian Math. Surveys, 61:2 (2006), 255–300  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. I. A. Cheltsov, “Elliptic structures on weighted three-dimensional Fano hypersurfaces”, Izv. Math., 71:4 (2007), 765–862  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. V. Pukhlikov, “Birationally rigid varieties. I. Fano varieties”, Russian Math. Surveys, 62:5 (2007), 857–942  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. I. A. Cheltsov, K. A. Shramov, “Log canonical thresholds of smooth Fano threefolds”, Russian Math. Surveys, 63:5 (2008), 859–958  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. K. A. Shramov, “Birational Rigidity and $\mathbb Q$-Factoriality of a Singular Double Cover of a Quadric Branched over a Divisor of Degree 4”, Math. Notes, 84:2 (2008), 280–289  mathnet  crossref  crossref  mathscinet  isi  elib
    6. A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130  mathnet  crossref  mathscinet  elib
    7. A. V. Pukhlikov, “Birational geometry of Fano double covers”, Sb. Math., 199:8 (2008), 1225–1250  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. Cheltsov I., “Fano varieties with many selfmaps”, Adv. Math., 217:1 (2008), 97–124  crossref  mathscinet  zmath  isi
    9. I. A. Cheltsov, “Extremal metrics on two Fano varieties”, Sb. Math., 200:1 (2009), 95–132  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Proc. Steklov Inst. Math., 264 (2009), 96–101  mathnet  crossref  mathscinet  isi  elib
    11. I. A. Cheltsov, “Log canonical thresholds of three-dimensional Fano hypersurfaces”, Izv. Math., 73:4 (2009), 727–795  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Cheltsov I., “On singular cubic surfaces”, Asian J. Math., 13:2 (2009), 191–214  crossref  mathscinet  zmath  isi
    13. Kishimoto T., Prokhorov Yu., Zaidenberg M., “Group Actions on Affine Cones”, Affine Algebraic Geometry: the Russell Festschrift, CRM Proceedings & Lecture Notes, 54, eds. Daigle D., Ganong R., Koras M., Amer Mathematical Soc, 2011, 123–163  crossref  mathscinet  zmath  isi
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    15. Jean-Pierre Demailly, Hoàn.H.iệp Phạm, “A sharp lower bound for the log canonical threshold”, Acta Math, 212:1 (2014), 1  crossref  mathscinet  zmath  isi
    16. Cheltsov I., Shramov C., “Five Embeddings of One Simple Group”, Trans. Am. Math. Soc., 366:3 (2014), 1289–1331  crossref  mathscinet  zmath  isi
    17. I. A. Cheltsov, “Two local inequalities”, Izv. Math., 78:2 (2014), 375–426  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Pukhlikov A.V., “Birationally Rigid Fano Complete Intersections. II”, J. Reine Angew. Math., 688 (2014), 209–218  crossref  mathscinet  zmath  isi
    19. 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
    20. Pan I., Simis A., “Cremona Maps of de Jonquieres Type”, 67, no. 4, 2015, 923–941  crossref  mathscinet  zmath  isi
    21. Dubouloz A., Kishimoto T., “Log-Uniruled Affine Varieties Without Cylinder-Like Open Subsets”, 143, no. 2, 2015, 383–401  mathscinet  zmath  isi
    22. Beauville A., “The Luroth Problem”, Rationality Problems in Algebraic Geometry, Lect. Notes Math., Lecture Notes in Mathematics, 2172, eds. Pardini R., Pirola G., Springer International Publishing Ag, 2016, 1–27  crossref  mathscinet  isi
    23. E. Johnstone, “Birationally Rigid Singular Double Quadrics and Double Cubics”, Math. Notes, 102:4 (2017), 508–515  mathnet  crossref  crossref  mathscinet  isi  elib
    24. Pirutka A., “Varieties That Are Not Stably Rational, Zero-Cycles and Unramified Cohomology”, Algebraic Geometry: Salt Lake City 2015, Pt 2, Proceedings of Symposia in Pure Mathematics, 97, no. 2, eds. DeFernex T., Hassett B., Mustata M., Olsson M., Popa M., Thomas R., Amer Mathematical Soc, 2018, 459–483  crossref  isi
    25. Bohning Ch., von Bothmer H.-Ch.G., Sosna P., “On the Dynamical Degrees of Reflections on Cubic Fourfolds”, Int. Math. Res. Notices, 2018, no. 14, 4480–4512  crossref  mathscinet  isi
    26. Krylov I., “Birational Geometry of Del Pezzo Fibrations With Terminal Quotient Singularities”, J. Lond. Math. Soc.-Second Ser., 97:2 (2018), 222–246  crossref  mathscinet  zmath  isi  scopus
    27. Krylov I., “Rationally Connected Non-Fano Type Varieties”, Eur. J. Math., 4:1, 1, SI (2018), 335–355  crossref  mathscinet  zmath  isi  scopus
    28. Avilov A., “Automorphisms of Singular Three-Dimensional Cubic Hypersurfaces”, Eur. J. Math., 4:3, 2, SI (2018), 761–777  crossref  mathscinet  isi  scopus
    29. Cheltsov I., Dubouloz A., Park J., “Super-Rigid Affine Fano Varieties”, Compos. Math., 154:11 (2018), 2462–2484  crossref  mathscinet  zmath  isi
    30. Fontanari C., Martinelli D., “A Remark on Rationally Connected Varieties and Mori Dream Spaces”, Proc. Edinb. Math. Soc., 62:1 (2019), 259–263  crossref  isi  scopus
    31. 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  crossref  isi
    32. Kollar J., “Algebraic Hypersurfaces”, Bull. Amer. Math. Soc., 56:4 (2019), 543–568  crossref  isi
    33. Cheltsov I., Shramov C., “Finite Collineation Groups and Birational Rigidity”, Sel. Math.-New Ser., 25:5 (2019), UNSP 71  crossref  isi
    34. Zhuang Z., “Birational Superrigidity Is Not a Locally Closed Property”, Sel. Math.-New Ser., 26:1 (2020), UNSP 11  crossref  isi
    35. 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  crossref  isi
    36. Shramov C., “Automorphisms of Cubic Surfaces Without Points”, Int. J. Math., 31:11 (2020), 2050083  crossref  mathscinet  isi
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