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Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 6, Pages 159–186 (Mi izv413)  

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

Birationally rigid Fano hypersurfaces

A. V. Pukhlikov

Steklov Mathematical Institute, Russian Academy of Sciences

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.

DOI: https://doi.org/10.4213/im413

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English version:
Izvestiya: Mathematics, 2002, 66:6, 1243–1269

Bibliographic databases:

UDC: 512.9
MSC: 14E05, 14J45
Received: 04.04.2002

Citation: A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. RAN. Ser. Mat., 66:6 (2002), 159–186; Izv. Math., 66:6 (2002), 1243–1269

Citation in format AMSBIB
\by A.~V.~Pukhlikov
\paper Birationally rigid Fano hypersurfaces
\jour Izv. RAN. Ser. Mat.
\yr 2002
\vol 66
\issue 6
\pages 159--186
\jour Izv. Math.
\yr 2002
\vol 66
\issue 6
\pages 1243--1269

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    This publication is cited in the following articles:
    1. I. A. Cheltsov, “Non-rationality of the 4-dimensional smooth complete intersection of a quadric and a quartic not containing planes”, Sb. Math., 194:11 (2003), 1679–1699  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. I. A. Cheltsov, L. Votslav, “Nonrational Complete Intersections”, Proc. Steklov Inst. Math., 246 (2004), 303–307  mathnet  mathscinet  zmath
    5. A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. Pukhlikov A.V., “Birational geometry of algebraic varieties with a pencil of Fano complete intersections”, Manuscripta Math., 121:4 (2006), 491–526  crossref  mathscinet  zmath  isi  elib  scopus
    8. I. A. Cheltsov, “Local inequalities and birational superrigidity of Fano varieties”, Izv. Math., 70:3 (2006), 605–639  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. Cheltsov I., “On nodal sextic fivefold”, Math. Nachr., 280:12 (2007), 1344–1353  crossref  mathscinet  zmath  isi  elib  scopus
    10. 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
    11. Pukhlikov A.V., “Birational geometry of algebraic varieties with a pencil of Fano cyclic covers”, Pure Appl. Math. Q., 5:2, Special Issue: In honor of Friedrich Hirzebruch, Part 1 (2009), 641–700  crossref  mathscinet  zmath  isi  elib  scopus
    12. 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
    13. 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
    14. A. V. Pukhlikov, “$K$-Trivial Structures on Fano Complete Intersections”, Math. Notes, 91:4 (2012), 568–574  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    15. de Fernex T., “Birationally Rigid Hypersurfaces”, Invent. Math., 192:3 (2013), 533–566  crossref  mathscinet  zmath  adsnasa  isi  scopus
    16. 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
    17. Jelonek Z., Lenarcik T., “Automorphisms of Affine Smooth Varieties”, Proc. Amer. Math. Soc., 142:4 (2014), 1157–1163  crossref  mathscinet  zmath  isi  scopus
    18. Demailly J.-P., Hoang Hiep Pham, “A Sharp Lower Bound For the Log Canonical Threshold”, Acta Math., 212:1 (2014), 1–9  crossref  mathscinet  zmath  isi  scopus
    19. 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
    20. Suzuki F., “Birational rigidity of complete intersections”, Math. Z., 285:1-2 (2017), 479–492  crossref  mathscinet  zmath  isi  scopus
    21. de Fernex T., “Birational Rigidity of Singular Fano Hypersurfaces”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 17:3 (2017), 911–929  mathscinet  zmath  isi
    22. Stibitz Ch., Zhuang Z., “K-Stability of Birationally Superrigid Fano Varieties”, Compos. Math., 155:9 (2019), 1845–1852  crossref  isi
    23. Kollar J., “Algebraic Hypersurfaces”, Bull. Amer. Math. Soc., 56:4 (2019), 543–568  crossref  isi
    24. Zhuang Z., “Birational Superrigidity Is Not a Locally Closed Property”, Sel. Math.-New Ser., 26:1 (2020), UNSP 11  crossref  isi
    25. Pukhlikov V A., “Birational Geometry of Singular Fano Hypersurfaces of Index Two”, Manuscr. Math., 161:1-2 (2020), 161–203  crossref  isi
    26. A. V. Pukhlikov, “Birational geometry of singular Fano double spaces of index two”, Sb. Math., 212:4 (2021), 551–566  mathnet  crossref  crossref  isi
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