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Uspekhi Mat. Nauk, 2007, Volume 62, Issue 5(377), Pages 15–106 (Mi umn8505)  

This article is cited in 14 scientific papers (total in 15 papers)

Birationally rigid varieties. I. Fano varieties

A. V. Pukhlikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The theory of birational rigidity of rationally connected varieties generalises the classical rationality problem. This paper gives a survey of the current state of this theory and traces its history from Noether's theorem and the Lüroth problem to the latest results on the birational superrigidity of higher-dimensional Fano varieties. The main components of the method of maximal singularities are considered.

DOI: https://doi.org/10.4213/rm8505

Full text: PDF file (1162 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2007, 62:5, 857–942

Bibliographic databases:

UDC: 512.7
MSC: Primary 14J45, 14E08, 14E05; Secondary 14C20, 14G22, 14M10, 14M20
Received: 03.06.2007

Citation: A. V. Pukhlikov, “Birationally rigid varieties. I. Fano varieties”, Uspekhi Mat. Nauk, 62:5(377) (2007), 15–106; Russian Math. Surveys, 62:5 (2007), 857–942

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. A. V. Pukhlikov, “Birational geometry of Fano double covers”, Sb. Math., 199:8 (2008), 1225–1250  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130  mathnet  crossref  mathscinet  elib
    3. A. V. Pukhlikov, “Birational Geometry of Singular Fano Varieties”, Proc. Steklov Inst. Math., 264 (2009), 159–177  mathnet  crossref  mathscinet  isi  elib  elib
    4. F. A. Bogomolov, Vik. S. Kulikov, Yu. I. Manin, V. V. Nikulin, D. O. Orlov, A. N. Parshin, Yu. G. Prokhorov, A. V. Pukhlikov, M. Reid, I. R. Shafarevich, V. V. Shokurov, “Vasilii Alekseevich Iskovskikh (obituary)”, Russian Math. Surveys, 64:5 (2009), 939–946  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. A. V. Pukhlikov, “Singularities of Algebraic Subvarieties and Problems of Birational Geometry”, Proc. Steklov Inst. Math., 267 (2009), 235–247  mathnet  crossref  mathscinet  zmath  isi  elib
    6. R. J. Mullany, “Fano Double Spaces with a Big Singular Locus”, Math. Notes, 87:3 (2010), 444–448  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. A. V. Pukhlikov, “Birational geometry of Fano double spaces of index two”, Izv. Math., 74:5 (2010), 925–991  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. A. V. Pukhlikov, “Birationally rigid varieties. II. Fano fibre spaces”, Russian Math. Surveys, 65:6 (2010), 1083–1171  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Kaloghiros A.-S., “A classification of terminal quartic 3-folds and applications to rationality questions”, Math. Ann., 354:1 (2012), 263–296  crossref  mathscinet  zmath  isi  elib  scopus
    10. A. V. Pukhlikov, “Birational rigidity of Fano complete intersections”, Dokl. Math, 87:1 (2013), 34  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    11. A. V. Pukhlikov, “Birationally rigid complete intersections of quadrics and cubics”, Izv. Math., 77:4 (2013), 795–845  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Yuji Odaka, Takuzo Okada, “Birational superrigidity and slope stability of Fano manifolds”, Math. Z, 2013  crossref  mathscinet  isi  scopus
    13. Pukhlikov A.V., “Birationally Rigid Fano Complete Intersections. II”, J. Reine Angew. Math., 688 (2014), 209–218  crossref  mathscinet  zmath  isi  scopus
    14. 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
    15. Beauville A., “The Luroth Problem”, Rationality Problems in Algebraic Geometry, Lect. Notes Math., Lecture Notes in Mathematics, 2172, ed. Pardini R. Pirola G., Springer International Publishing Ag, 2016, 1–27  crossref  mathscinet  isi  scopus
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