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Izv. Akad. Nauk SSSR Ser. Mat., 1988, Volume 52, Issue 1, Pages 229–239 (Mi izv1177)  

This article is cited in 23 scientific papers (total in 23 papers)

Birational automorphisms of a double space and double quadric

A. V. Pukhlikov

Abstract: It is proved that the groups of birational automorphisms are the same as the groups of biregular automorphisms for two series of Fano varieties of dimension 4 and higher: double spaces of index 1 and double quadrics of index 1. From this it follows that the varieties are not rational. The proof that the respective automorphism groups are the same extends the method of V.  A. Iskovskikh and Yu. I. Manin (the method of isolating and excluding maximal singularities) to arbitrary dimension.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1989, 32:1, 233–243

Bibliographic databases:

Document Type: Article
UDC: 513.6
MSC: Primary 14E05, 14C20, 14J50; Secondary 14C05, 14J26
Received: 19.06.1986

Citation: A. V. Pukhlikov, “Birational automorphisms of a double space and double quadric”, Izv. Akad. Nauk SSSR Ser. Mat., 52:1 (1988), 229–239; Math. USSR-Izv., 32:1 (1989), 233–243

Citation in format AMSBIB
\by A.~V.~Pukhlikov
\paper Birational automorphisms of a~double space and double quadric
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1988
\vol 52
\issue 1
\pages 229--239
\jour Math. USSR-Izv.
\yr 1989
\vol 32
\issue 1
\pages 233--243

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    This publication is cited in the following articles:
    1. M. M. Grinenko, “Birational automorphisms of a three-dimensional double quadric with an elementary singularity”, Sb. Math., 189:1 (1998), 97–114  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. V. Pukhlikov, “Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces”, Izv. Math., 62:1 (1998), 115–155  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. V. Pukhlikov, “Certain examples of birationally rigid varieties pencil of double quadrics”, J Math Sci, 94:1 (1999), 986  crossref
    4. A. V. Pukhlikov, “Birationally rigid Fano double hypersurfaces”, Sb. Math., 191:6 (2000), 883–908  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. 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
    6. A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. V. Pukhlikov, “Fano direct products as rationally connected fibrations”, J. Math. Sci., 140:2 (2007), 303–320  mathnet  crossref  mathscinet  zmath  elib  elib
    8. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. 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
    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. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. 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
    13. A. V. Pukhlikov, “Explicit examples of birationally rigid Fano varieties”, Mosc. Math. J., 7:3 (2007), 543–560  mathnet  mathscinet  zmath
    14. Cheltsov I., “On nodal sextic fivefold”, Mathematische Nachrichten, 280:12 (2007), 1344–1353  crossref  isi  elib
    15. A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130  mathnet  crossref  mathscinet  elib
    16. I. A. Cheltsov, “Extremal metrics on two Fano varieties”, Sb. Math., 200:1 (2009), 95–132  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. R. J. Mullany, “Fano Double Spaces with a Big Singular Locus”, Math. Notes, 87:3 (2010), 444–448  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. 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
    19. 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
    20. A. V. Pukhlikov, “Birationally rigid complete intersections of quadrics and cubics”, Izv. Math., 77:4 (2013), 795–845  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. 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
    22. Victor V. Przyjalkowski, Constantin A. Shramov, “Double quadrics with large automorphism groups”, Proc. Steklov Inst. Math., 294 (2016), 154–175  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    23. E. Johnstone, “Birationally Rigid Singular Double Quadrics and Double Cubics”, Math. Notes, 102:4 (2017), 508–515  mathnet  crossref  crossref  mathscinet  isi  elib
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