Birational geometry of higher-dimensional Fano varieties

  • A. V. PukhlikovEmail author


STEKLOV Institute Exceptional Divisor Maximal Singularity Fano Variety BIRATIONAL Geometry 
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  1. 1.
    A.V. Pukhlikov, “Birational automorphisms of Fano hypersurfaces,” Invent. Math. 134(2), 401–426 (1998).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    A.V. Pukhlikov, “Birationally rigid Fano complete intersections,” J. Reine Angew. Math. 541, 55–79 (2001).zbMATHMathSciNetGoogle Scholar
  3. 3.
    A.V. Pukhlikov, “Birationally rigid iterated Fano double covers,” Izv.: Math. 67(3), 555–596 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A.V. Pukhlikov, “Birationally rigid Fano hypersurfaces with isolated singularities,” Sb.: Math. 193(3), 445–471 (2002).zbMATHMathSciNetGoogle Scholar
  5. 5.
    A.V. Pukhlikov, “Birational geometry of Fano double spaces of index two,” Izv.: Math. 74(5), 925–991 (2010).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    M. Noether, “Über Flächen welche Schaaren rationaler Curven besitzen,” Math. Ann. 3(2), 161–227 (1871)CrossRefMathSciNetGoogle Scholar
  7. 7.
    G. Fano, “Sopra alcune varieta algebriche a tre dimensioni aventi tutti i generi nulli,” Torino Atti 43, 973–984 (1908).Google Scholar
  8. 8.
    G. Fano, “Osservazioni sopra alcune varieta non razionali aventi tutti i generi nulli,” Atti Acc. Torino 50, 1067–1072 (1915).zbMATHGoogle Scholar
  9. 9.
    G. Fano, “Nuove ricerche sulle varieta algebriche a tre dimensioni a curve-sezioni canoniche,” Comment. Pontif. Acad. Sci. 11, 635–720 (1947).zbMATHMathSciNetGoogle Scholar
  10. 10.
    V.A. Iskovskikh, Yu. I. Manin, “Three-dimensional quartics and counterexamples to the Lüroth problem,,” Math. USSR-Sb. 15(1), 141–166 (1971).CrossRefGoogle Scholar
  11. 11.
    V.A. Iskovskikh, “Birational automorphisms of three-dimensional algebraic varieties,” J. Soviet Math. 13(6), 815–868 (1980).CrossRefzbMATHGoogle Scholar
  12. 12.
    V.A. Iskovskikh, A.V. Pukhlikov, “Birational automorphisms of multi-dimensional algebraic varieties,” J.Math. Sci. 82(4), 3528–3613 (1996).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    A.V. Pukhlikov, “Birational automorphisms of a double space and double quadric,”Math. USSR-Izv. 32(1), 233–243 (1989).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    A.V. Pukhlikov, “Birationally rigid Fano double hypersurfaces,” Sb.: Math. 191(6), 883–908 (2000).MathSciNetGoogle Scholar
  15. 15.
    A.V. Pukhlikov, “Birational geometry of algebraic varieties with a pencil of Fano cyclic covers,” Pure Appl. Math. Q. 5(2), 641–700 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Yu. I. Manin, “Rational surfaces over perfect fields,” Publ. Math. IHES 30, 55–97 (1966).CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Yu. I. Manin, “Rational surfaces over perfect fields. II,” Math. USSR-Sb. 1(2), 141–168 (1967).CrossRefGoogle Scholar
  18. 18.
    J. Kollár, Flips and Abundance for Algebraic Threefolds, Asterisque 211, (Soc. Math. France, Paris, 1992).Google Scholar
  19. 19.
    V.G. Sarkisov, Birational Maps of Standard ℚ-Fano Fibrations, Preprint (Kurchatov Institute of Atomic Energy, 1989).Google Scholar
  20. 20.
    V.A. Iskovskikh, “Birational rigidity of Fano hypersurfaces in the framework of Mori theory,” Russ. Math. Surveys 56(2), 207–291 (2001).CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    I. A. Chel’tsov, “Birationally rigid Fano varieties,” Russ. Math. Surveys 60(5), 875–965 (2005).CrossRefMathSciNetGoogle Scholar
  22. 22.
    M. Reid, Birational Geometry of 3-Folds According to Sarkisov, Warwick Preprint (1991).Google Scholar
  23. 23.
    A. Corti, “Factoring birationalmaps of three-folds after Sarkisov,” J. Algebraic Geom. 4(2), 223–254 (1995).zbMATHMathSciNetGoogle Scholar
  24. 24.
    A. Corti, M. Mella, “Birational geometry of terminal quartic 3-folds. I,” Amer. J. Math. 126(4), 739–761 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    I. A. Cheltsov, M.M. Grinenko, Birational Rigidity is not an Open Property arXiv: math.AG/0612159 (2006).Google Scholar
  26. 26.
    A.V. Pukhlikov, “Birationally rigid varieties with a pencil of Fano double covers. III,” Sb.: Math. 197(3), 335–368 (2006).zbMATHMathSciNetGoogle Scholar
  27. 27.
    M. Mella, “Birational geometry of quartic 3-folds. II. The importance of being Q-factorial,” Math. Ann. 330(1), 107–126 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    I. A. Chel’tsov, “On a smooth quintic 4-fold,” Mat. Sb. 191(9), 1399–1419 (2000).CrossRefMathSciNetGoogle Scholar
  29. 29.
    I. A. Chel’tsov, “Non-rationality of the 4-dimensional smooth complete intersection of a quadric and a quartic not containing planes,” Sb.: Math. 194(11), 1679–1699 (2003).zbMATHMathSciNetGoogle Scholar
  30. 30.
    I. A. Chel’tsov, “Double space with double line,” Sb.: Math. 195(10), 1503–1544 (2004).zbMATHMathSciNetGoogle Scholar
  31. 31.
    I. A. Chel’tsov, “Conic bundles with big discriminant loci,” Izv.: Math. 68(2), 429–434 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    I. A. Cheltsov, “Birationally rigid del Pezzo fibrations,” Manuscripta Math. 116(4), 385–396 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    A.V. Pukhlikov, “Birational geometry of algebraic varieties with a pencil of Fano complete intersections,” Manuscripta Math. 121, 491–526 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    A.V. Pukhlikov, “K-trivial structures on Fano complete intersections,” Math. Notes 91(4), 568–574 (2012).CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    A.V. Pukhlikov, “A remark on the theorem of V. A. Iskovskikh and Yu. I. Manin on a three-dimensional quartic,” Proc. Steklov Inst. Math. 208, 244–254 (1995).MathSciNetGoogle Scholar
  36. 36.
    A.V. Pukhlikov, “Birationally rigid Fano hypersurfaces,” Izv.: Math. 66(6), 1243–1269 (2002).CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    A.V. Pukhlikov, “Birational isomorphisms of four-dimensional quintics,” Invent. Math. 87(2), 303–329 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    A.V. Pukhlikov, “Birational automorphisms of a three-dimensional quartic with a quadratic singularity,” Math. USSR-Sb. 63(2), 457–482 (1989).CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    A.V. Pukhlikov, “Essentials of the method of maximal singularities” in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281, 73–100 (Cambridge Univ. Press, Cambridge, 2000).CrossRefGoogle Scholar
  40. 40.
    W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, (Springer-Verlag, Berlin, 1984).CrossRefzbMATHGoogle Scholar
  41. 41.
    A. Corti, “Singularities of linear systems and 3-fold birational geometry” in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281, 259–312 (Cambridge Univ. Press, Cambridge, 2000).CrossRefGoogle Scholar
  42. 42.
    V. V. Shokurov, “3-Fold log flips,” Izv. Math. 40(1), 95–202 (1993).CrossRefMathSciNetGoogle Scholar
  43. 43.
    Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetics (Nauka, Moscow, 1972) [in Russian].Google Scholar
  44. 44.
    A. Corti, A. Pukhlikov, M. Reid, “Fano 3-fold hypersurfaces,” in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281 175–258 (Cambridge Univ. Press, Cambridge, 2000).CrossRefGoogle Scholar
  45. 45.
    A.V. Pukhlikov, “Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces,” Izv.: Math. 62(1), 115–155 (1998).CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    I.V. Sobolev, “Birational automorphisms of a class of varieties fibred into cubic surfaces,” Izv.: Math. 66(1), 201–222 (2002).CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    M. M. Grinenko, “Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2,” Sb.: Math. 191(5), 633–653 (2000).MathSciNetGoogle Scholar
  48. 48.
    M. M. Grinenko, “Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2. II,” Sb.: Math. 194(5), 669–695 (2003).zbMATHMathSciNetGoogle Scholar
  49. 49.
    A.V. Pukhlikov, “Birationally rigid varieties with a pencil of Fano double covers. II,” Sb.: Math. 195(11), 1665–1702 (2004).zbMATHMathSciNetGoogle Scholar
  50. 50.
    A.V. Pukhlikov, “Birationally rigid Fano fibrations,” Izv.: Math. 64(3), 563–581 (2000).CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    A.V. Pukhlikov, “Birational geometry of Fano direct products,” Izv.: Math. 69(6), 1225–1255 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    I.V. Sobolev, “On a series of birationally rigid varieties with a pencil of Fano hypersurfaces,” Sb.: Math. 192(10), 1543–1551 (2001).zbMATHMathSciNetGoogle Scholar
  53. 53.
    I. A. Chel’tsov, “Log-canonical thresholds on hypersurfaces,” Sb.: Math. 192(8), 1241–1257 (2001).zbMATHMathSciNetGoogle Scholar
  54. 54.
    T. Graber, J. Harris, J. Starr, “Families of rationally connected varieties,” J. Amer. Math. Soc. 16(1), 57–67 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    F. Call, G. Lyubeznik, “A simple proof of Grothendieck’s theorem on the parafactoriality of local rings” in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, MA, 1992), Contemp. Math. 159, 15–18 (Amer. Math. Soc., Providence, RI, 1994).CrossRefGoogle Scholar
  56. 56.
    J. Kollár, “Singularities of pairs” in Algebraic Geometry — Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Part 1, 221–287 (Amer. Math. Soc., Providence, RI, 1997).CrossRefGoogle Scholar
  57. 57.
    I. A. Chel’tsov, “Local inequalities and birational superrigidity of Fano varieties,” Izv.: Math. 70(3), 605–639 (2006).CrossRefMathSciNetGoogle Scholar
  58. 58.
    I. Cheltsov, “Double cubics and double quartics,” Math. Z. 253(1), 75–86 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    A.V. Pukhlikov, On the 8n 2-inequality arXiv: math.AG/0811.0183 (2008).Google Scholar
  60. 60.
    M. M. Grinenko, “Mori structures on a Fano threefold of index 2 and degree 1,” Proc. Steklov Inst. Math. 246(3), 103–128 (2004).MathSciNetGoogle Scholar
  61. 61.
    S. I. Khashin, “Birational automorphisms of the Veronese double cone of dimension three,” Vestn. Moskov. Univ. Ser. 1. Mat.Mekh., No. 1, 13–16 (1984).Google Scholar
  62. 62.
    V.A. Iskovskih, “Algebraic threefolds with special regard to the problem of rationality,” in Proceedings of the International Congress of Mathematicians (Warsaw, 1983), Vol. 1, 2, 733–746 (PWN, Warsaw, 1984).Google Scholar
  63. 63.
    M. M. Grinenko, “New Mori structures on a double space of index 2,” Russ. Math. Surveys 59(3), 573–574 (2004).CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    M. M. Grinenko, “Fibrations into del Pezzo surfaces,” Russ. Math. Surveys 61(2), 255–300 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    C. H. Clemens, Ph.A. Griffiths, “The intermediate Jacobian of the cubic threefold,” Ann. of Math. (2) 95(2), 281–356 (1972).CrossRefzbMATHMathSciNetGoogle Scholar
  66. 66.
    A. N. Tyurin, “Five lectures on three-dimensional varieties,” Russ.Math. Surveys 27(5), 1–53 (1972).CrossRefGoogle Scholar
  67. 67.
    A. N. Tyurin, “The middle Jacobian of three-dimensional varieties,” J. Soviet Math. 13(6), 707–745 (1979).CrossRefGoogle Scholar
  68. 68.
    J. Kollár, “Nonrational hypersurfaces,” J. Amer. Math. Soc. 8, 241–249 (1995).CrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    A. S. Tikhomirov, “The intermediate Jacobian of the double covering of P3 branched at a quadric,” Math. USSR-Izv. 17(3), 523–566 (1981).CrossRefzbMATHGoogle Scholar
  70. 70.
    A. S. Tikhomirov, “Singularities of the theta divisor of the intermediate Jacobian of a double cover of P3 of index two,” Math. USSR-Izv. 21(2), 355–373 (1983).CrossRefzbMATHGoogle Scholar
  71. 71.
    A. S. Tikhomirov, “Letter to the editors of the journal ‘Izvestiya AN SSSR Seriya Matematicheskaya’,” Math. USSR-Izv. 27(1), 201 (1986).CrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.University of LiverpoolLiverpoolUK

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