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Uspekhi Mat. Nauk, 2001, Volume 56, Issue 2(338), Pages 3–86 (Mi umn382)  

This article is cited in 25 scientific papers (total in 26 papers)

Birational rigidity of Fano hypersurfaces in the framework of Mori theory

V. A. Iskovskikh

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This survey reflects the contemporary state of Mori theory and its log version. The main stress is on applications of the theory of log pairs to the birational geometry of varieties of negative Kodaira dimension (as is known, they are close to rational varieties; however, it is also known that many varieties of negative Kodaira dimension are birationally rigid, which is peculiar to a more general class than that of rational varieties), namely, to the Sarkisov program of factorizing birational maps of Mori models that are Mori fibre spaces under the above restrictions. In particular, we present a new proof of the birational rigidity of a non-singular three-dimensional quartic (the Iskovskikh–Manin theorem, which claims that such a quartic is not rational) and of another anticanonical hypersurface in a weighted projective space (from the Corti–Pukhlikov–Reid list). We also present Chel'tsov's results on the birational rigidity of smooth hypersurfaces of degree $N$ in $\mathbb P^N$ for $4\leqslant N\leqslant 8$; the proofs use the Shokurov connectedness theorem.

DOI: https://doi.org/10.4213/rm382

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English version:
Russian Mathematical Surveys, 2001, 56:2, 207–291

Bibliographic databases:

UDC: 512.76
MSC: Primary 14E30, 14E05, 14E15, 14J30; Secondary 14E07, 53C24
Received: 13.02.2001

Citation: V. A. Iskovskikh, “Birational rigidity of Fano hypersurfaces in the framework of Mori theory”, Uspekhi Mat. Nauk, 56:2(338) (2001), 3–86; Russian Math. Surveys, 56:2 (2001), 207–291

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    This publication is cited in the following articles:
    1. A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269  mathnet  crossref  crossref  mathscinet  zmath
    2. Pukhlikov A.V., “Birational geometry of higher-dimensional Fano hypersurfaces”, Dokl. Math., 66:1 (2002), 22–24  mathnet  mathscinet  zmath  isi  elib
    3. Shokurov V.V., “Letters of a bi-rationalist IV. geometry of log flips”, Volume in Memory of Paolo Francia, Algebraic Geometry, 2002, 313–328  mathscinet  zmath  isi
    4. de Fernex T., Ein L., Mustaţă M., “Bounds for log canonical thresholds with applications to birational rigidity”, Math. Res. Lett., 10:2-3 (2003), 219–236  mathscinet  zmath  isi  elib
    5. A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. I. A. Cheltsov, “Anticanonical models of three-dimensional Fano varieties of degree 4”, Sb. Math., 194:4 (2003), 617–640  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. V. A. Iskovskikh, V. V. Shokurov, “Birational models and flips”, Russian Math. Surveys, 60:1 (2005), 27–94  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. 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
    10. A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130  mathnet  crossref  mathscinet  elib
    11. Proc. Steklov Inst. Math., 264 (2009), 96–101  mathnet  crossref  mathscinet  isi  elib
    12. 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
    13. Brown G., Ryder D., “Elliptic fibrations on cubic surfaces”, J. Pure Appl. Algebra, 214:4 (2010), 410–421  crossref  mathscinet  zmath  isi  scopus
    14. Cheltsov I., Karzhemanov I., “Halphen pencils on quartic threefolds”, Adv. Math., 223:2 (2010), 594–618  crossref  mathscinet  zmath  isi  elib  scopus
    15. 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
    16. 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
    17. Kishimoto T. Prokhorov Yu. Zaidenberg M., “Group Actions on Affine Cones”, Affine Algebraic Geometry: the Russell Festschrift, CRM Proceedings & Lecture Notes, 54, ed. Daigle D. Ganong R. Koras M., Amer Mathematical Soc, 2011, 123–163  crossref  mathscinet  zmath  isi
    18. de Fernex T., “Birationally Rigid Hypersurfaces”, Invent. Math., 192:3 (2013), 533–566  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    19. 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  scopus
    20. I. A. Cheltsov, “Two local inequalities”, Izv. Math., 78:2 (2014), 375–426  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. Pukhlikov A.V., “Birationally Rigid Fano Complete Intersections. II”, J. Reine Angew. Math., 688 (2014), 209–218  crossref  mathscinet  zmath  isi  scopus  scopus
    22. 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
    23. Cheltsov I. Park J., “Birational Rigidity and Main Theorem”, Mem. Am. Math. Soc., 246:1167 (2017), 1+  mathscinet  isi
    24. Karzhemanov I., “On Endomorphisms of Hypersurfaces”, Kodai. Math. J., 40:3 (2017), 615–624  crossref  mathscinet  zmath  isi  scopus
    25. Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surveys, 73:3 (2018), 375–456  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    26. Kollar J., “Algebraic Hypersurfaces”, Bull. Amer. Math. Soc., 56:4 (2019), 543–568  crossref  isi
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