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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

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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
\by A.~V.~Pukhlikov
\paper Birational automorphisms of algebraic threefolds with a~pencil of Del Pezzo surfaces
\jour Izv. RAN. Ser. Mat.
\yr 1998
\vol 62
\issue 1
\pages 123--164
\jour Izv. Math.
\yr 1998
\vol 62
\issue 1
\pages 115--155

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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
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    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
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    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
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    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
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    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
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    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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