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Izv. RAN. Ser. Mat., 1998, Volume 62, Issue 1, Pages 123–164 (Mi izv188)  

This article is cited in 54 scientific papers (total in 55 papers)

Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces

A. V. Pukhlikov

Institute of Systems Analysis, Russian Academy of Sciences

Abstract: In this paper it is proved that there is only one pencil of rational surfaces on a smooth three-dimensional variety $V$ fibred into Del Pezzo surfaces of degree 1, 2 or 3 by a morphism $\pi\colon V\to\mathbb P^1$ (Iskovskikh's conjecture), provided that the class of 1-cycles $(MK^2_V-f)$ (where $K_V$ is the canonical class and $f$ is the class of a line in a fibre) is not effective for any $M\in\mathbb Z$. This condition is satisfied if $V$ is sufficiently “twisted” over the base of the pencil. This implies that these varieties admit no conic bundle structures, and, in particular, that they are nonrational. Certain higher-dimensional generalizations of this theorem are considered: similar statements are true for varieties with a pencil of thee-dimensional quartics of general position, and analogous results are obtained in these cases.

DOI: https://doi.org/10.4213/im188

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English version:
Izvestiya: Mathematics, 1998, 62:1, 115–155

Bibliographic databases:

MSC: 14J50, 14E15
Received: 29.05.1996

Citation: A. V. Pukhlikov, “Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces”, Izv. RAN. Ser. Mat., 62:1 (1998), 123–164; Izv. Math., 62:1 (1998), 115–155

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Okada T., “On Birational Rigidity of Singular Del Pezzo Fibrations of Degree 1”, J. Lond. Math. Soc.-Second Ser.  crossref  isi
    2. M. M. Grinenko, “Birational automorphisms of a three-dimensional double cone”, Sb. Math., 189:7 (1998), 991–1007  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. A. V. Pukhlikov, “Birationally rigid varieties with a sheaf of Fano hypersurfaces”, Russian Math. Surveys, 54:1 (1999), 260–261  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. V. Pukhlikov, “Birationally rigid Fano fibrations”, Izv. Math., 64:3 (2000), 563–581  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. M. M. Grinenko, “Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2”, Sb. Math., 191:5 (2000), 633–653  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. A. V. Pukhlikov, “Birationally rigid Fano double hypersurfaces”, Sb. Math., 191:6 (2000), 883–908  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. V. Pukhlikov, “Birational automorphisms of algebraic varieties with a pencil of double quadrics”, Math. Notes, 67:2 (2000), 192–199  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. A. V. Pukhlikov, “Fiber-wise birational correspondences”, Math. Notes, 68:1 (2000), 103–112  mathnet  crossref  mathscinet  zmath  isi  elib
    9. I. V. Sobolev, “On the birational rigidity of a series of multidimensional Fano varieties”, Math. Notes, 68:3 (2000), 392–396  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. M. M. Grinenko, “On Fibrations into del Pezzo Surfaces”, Math. Notes, 69:4 (2001), 499–513  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    12. I. V. Sobolev, “Birational automorphisms of a class of varieties fibred into cubic surfaces”, Izv. Math., 66:1 (2002), 201–222  mathnet  crossref  crossref  mathscinet  zmath  elib
    13. A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269  mathnet  crossref  crossref  mathscinet  zmath
    14. M. M. Grinenko, “Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2. II”, Sb. Math., 194:5 (2003), 669–695  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. M. M. Grinenko, “On a double cone over a Veronese surface”, Izv. Math., 67:3 (2003), 421–438  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    17. Nakashima T., “Stable rank two bundles on Del-Pezzo fibrations of degree 1 or 2”, Arch. Math. (Basel), 81:3 (2003), 258–265  crossref  mathscinet  zmath  isi  scopus
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    20. I. A. Cheltsov, “The method of degeneracy and the irrationality of three-dimensional varieties with a pencil of del Pezzo surfaces”, Russian Math. Surveys, 59:4 (2004), 792–793  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. A. V. Pukhlikov, “Birationally rigid varieties with a pencil of double Fano covers. I”, Sb. Math., 195:7 (2004), 1039–1071  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    22. I. A. Cheltsov, “Double space with double line”, Sb. Math., 195:10 (2004), 1503–1544  mathnet  crossref  crossref  mathscinet  zmath  isi
    23. A. V. Pukhlikov, “Birationally rigid varieties with a pencil of Fano double covers. II”, Sb. Math., 195:11 (2004), 1665–1702  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    24. M. M. Grinenko, “Mori Structures on a Fano Threefold of Index 2 and Degree 1”, Proc. Steklov Inst. Math., 246 (2004), 103–128  mathnet  mathscinet  zmath
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    27. V. V. Przyjalkowski, I. A. Cheltsov, K. A. Shramov, “Hyperelliptic and trigonal Fano threefolds”, Izv. Math., 69:2 (2005), 365–421  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    28. A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    29. 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
    30. M. M. Grinenko, “Fibrations into del Pezzo surfaces”, Russian Math. Surveys, 61:2 (2006), 255–300  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    31. A. V. Pukhlikov, “Birationally rigid varieties with a pencil of Fano double covers. III”, Sb. Math., 197:3 (2006), 335–368  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    32. K. A. Shramov, “On the rationality of non-singular threefolds with a pencil of Del Pezzo surfaces of degree 4”, Sb. Math., 197:1 (2006), 127–137  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    33. Cheltsov I., “Nonrational nodal quartic threefolds”, Pacific J. Math., 226:1 (2006), 65–81  crossref  mathscinet  zmath  isi  elib  scopus
    34. 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
    35. A. V. Pukhlikov, “Explicit examples of birationally rigid Fano varieties”, Mosc. Math. J., 7:3 (2007), 543–560  mathnet  crossref  mathscinet  zmath
    36. A. V. Pukhlikov, “Birational geometry of Fano double covers”, Sb. Math., 199:8 (2008), 1225–1250  mathnet  crossref  crossref  mathscinet  isi  elib  elib
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    38. 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
    39. 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
    40. 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
    41. 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
    42. M. M. Grinenko, “Three-dimensional quartics containing a plane”, Sb. Math., 202:2 (2011), 207–241  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    45. I. A. Cheltsov, “Two local inequalities”, Izv. Math., 78:2 (2014), 375–426  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    46. 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
    47. A. V. Pukhlikov, “Birationally rigid Fano fibre spaces. II”, Izv. Math., 79:4 (2015), 809–837  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    48. Ahmadinezhad H., “On pliability of del Pezzo fibrations and Cox rings”, J. Reine Angew. Math., 723 (2017), 101–125  crossref  mathscinet  zmath  isi
    49. Dubouloz A., Kishimoto T., “Explicit Biregular/Birational Geometry of Affine Threefolds: Completions of a(3) Into Del Pezzo Fibrations and Mori Conic Bundles”, Algebraic Varieties and Automorphism Groups, Advanced Studies in Pure Mathematics, 75, eds. Masuda K., Kishimoto T., Kojima H., Miyanishi M., Zaidenberg M., Math Soc Japan, 2017, 49–71  mathscinet  zmath  isi
    50. Dubouloz A., Kishimoto T., “Cylinders in Del Pezzo Fibrations”, Isr. J. Math., 225:2 (2018), 797–815  crossref  mathscinet  zmath  isi  scopus
    51. Krylov I., “Birational Geometry of Del Pezzo Fibrations With Terminal Quotient Singularities”, J. Lond. Math. Soc.-Second Ser., 97:2 (2018), 222–246  crossref  mathscinet  zmath  isi  scopus
    52. Krylov I., “Rationally Connected Non-Fano Type Varieties”, Eur. J. Math., 4:1, 1, SI (2018), 335–355  crossref  mathscinet  zmath  isi  scopus
    53. Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surveys, 73:3 (2018), 375–456  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    54. Krylov I., Okada T., “Stable Rationality of Del Pezzo Fibrations of Low Degree Over Projective Spaces”, Int. Math. Res. Notices, 2020:23 (2020), 9075–9119  crossref  isi
    55. Karzhemanov I., “On the Cut-and-Paste Property of Algebraic Varieties”, Bull. Sci. Math., 166 (2021), 102920  crossref  isi
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