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Mat. Sb., 2000, Volume 191, Number 9, Pages 139–160 (Mi msb511)  

This article is cited in 21 scientific papers (total in 21 papers)

On a smooth quintic 4-fold

I. A. Cheltsov


Abstract: The birational geometry of an arbitrary smooth quintic 4-fold is studied using the properties of log pairs. As a result, a new proof of its birational rigidity is given and all birational maps of a smooth quintic 4-fold into fibrations with general fibre of Kodaira dimension zero are described.
In the Addendum similar results are obtained for all smooth hypersurfaces of degree $n$ in $\mathbb P^n$ in the case of $n$ equal to 6, 7, or 8.

DOI: https://doi.org/10.4213/sm511

Full text: PDF file (312 kB)
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English version:
Sbornik: Mathematics, 2000, 191:9, 1399–1419

Bibliographic databases:

UDC: 513.6
MSC: 14J35, 14E05
Received: 10.06.1999 and 19.01.2000

Citation: I. A. Cheltsov, “On a smooth quintic 4-fold”, Mat. Sb., 191:9 (2000), 139–160; Sb. Math., 191:9 (2000), 1399–1419

Citation in format AMSBIB
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\paper On a smooth quintic 4-fold
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. A. Iskovskikh, “Birational rigidity of Fano hypersurfaces in the framework of Mori theory”, Russian Math. Surveys, 56:2 (2001), 207–291  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. I. A. Cheltsov, “Log canonical thresholds on hypersurfaces”, Sb. Math., 192:8 (2001), 1241–1257  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. I. A. Cheltsov, J. Park, “Total log canonical thresholds and generalized Eckardt points”, Sb. Math., 193:5 (2002), 779–789  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269  mathnet  crossref  crossref  mathscinet  zmath
    5. Pukhlikov, AV, “Birational geometry of higher-dimensional Fano hypersurfaces”, Doklady Mathematics, 66:1 (2002), 22  mathnet  mathscinet  zmath  isi  elib
    6. 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
    7. 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
    8. 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
    9. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. I. A. Cheltsov, L. Votslav, “Nonrational Complete Intersections”, Proc. Steklov Inst. Math., 246 (2004), 303–307  mathnet  mathscinet  zmath
    11. I. A. Cheltsov, “Double space with double line”, Sb. Math., 195:10 (2004), 1503–1544  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. 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
    13. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. 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
    15. 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
    16. Cheltsov, I, “On nodal sextic fivefold”, Mathematische Nachrichten, 280:12 (2007), 1344  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    17. Pukhlikov, AV, “Birational Geometry of Algebraic Varieties with a Pencil of Fano Cyclic Covers”, Pure and Applied Mathematics Quarterly, 5:2 (2009), 641  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    18. de Fernex T., “Birationally Rigid Hypersurfaces”, Invent. Math., 192:3 (2013), 533–566  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    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. 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
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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