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Izv. RAN. Ser. Mat., 2000, Volume 64, Issue 3, Pages 131–150 (Mi izv291)  

This article is cited in 19 scientific papers (total in 19 papers)

Birationally rigid Fano fibrations

A. V. Pukhlikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We prove the birational superrigidity of a general Fano fibration $\pi\colon V\to\mathbf P^1$ whose fibre is a Fano hypersurface $W_M\subset\mathbf P^M$ of index 1. If the fibration is sufficiently twisted over the base $\mathbf P^1$, then $V$ has no other structure of a fibration into rationally connected varieties. We also formulate and discuss conjectures on birational rigidity for a large class of Fano varieties and Fano fibrations over a base of arbitrary dimension.

DOI: https://doi.org/10.4213/im291

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English version:
Izvestiya: Mathematics, 2000, 64:3, 563–581

Bibliographic databases:

MSC: 14E07, 14J45
Received: 03.11.1998

Citation: A. V. Pukhlikov, “Birationally rigid Fano fibrations”, Izv. RAN. Ser. Mat., 64:3 (2000), 131–150; Izv. Math., 64:3 (2000), 563–581

Citation in format AMSBIB
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\paper Birationally rigid Fano fibrations
\jour Izv. RAN. Ser. Mat.
\yr 2000
\vol 64
\issue 3
\pages 131--150
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\vol 64
\issue 3
\pages 563--581
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    This publication is cited in the following articles:
    1. A. V. Pukhlikov, “Fiber-wise birational correspondences”, Math. Notes, 68:1 (2000), 103–112  mathnet  crossref  mathscinet  zmath  isi  elib
    2. 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
    3. I. V. Sobolev, “On a series of birationally rigid varieties with a pencil of Fano hypersurfaces”, Sb. Math., 192:10 (2001), 1543–1551  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. V. Pukhlikov, “Birational Automorphisms of Algebraic Varieties with a Pencil of Fano Double Covers”, Funct. Anal. Appl., 36:1 (2002), 74–75  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. 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
    6. A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269  mathnet  crossref  crossref  mathscinet  zmath
    7. A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. 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
    9. 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
    10. A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. Pukhlikov A.V., “Birationally rigid fano direct products”, Dokl. Math., 71:2 (2005), 198–200  mathnet  mathscinet  zmath  isi  elib
    12. 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
    13. 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
    14. 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
    15. A. V. Pukhlikov, “Explicit examples of birationally rigid Fano varieties”, Mosc. Math. J., 7:3 (2007), 543–560  mathnet  crossref  mathscinet  zmath
    16. 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
    17. 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
    18. 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
    19. A. V. Pukhlikov, “Birationally rigid Fano fibre spaces. II”, Izv. Math., 79:4 (2015), 809–837  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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