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Mat. Sb., 1998, Volume 189, Number 7, Pages 37–52 (Mi msb336)  

This article is cited in 22 scientific papers (total in 22 papers)

Birational automorphisms of a three-dimensional double cone

M. M. Grinenko

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Birational rigidity is proved for a three-dimensional double cone, that is, a double cover of a quadric cone in $\mathbb P^2$ with branching in a smooth section of a quartic, and its non-rationality and the absence of conic bundle structures are also proved. The group of birational automorphisms of this variety is computed.


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English version:
Sbornik: Mathematics, 1998, 189:7, 991–1007

Bibliographic databases:

Document Type: Article
UDC: 512.6
MSC: Primary 14E07; Secondary 14E05, 14E15, 14J50
Received: 05.01.1998

Citation: M. M. Grinenko, “Birational automorphisms of a three-dimensional double cone”, Mat. Sb., 189:7 (1998), 37–52; Sb. Math., 189:7 (1998), 991–1007

Citation in format AMSBIB
\by M.~M.~Grinenko
\paper Birational automorphisms of a~three-dimensional double cone
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 7
\pages 37--52
\jour Sb. Math.
\yr 1998
\vol 189
\issue 7
\pages 991--1007

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    This publication is cited in the following articles:
    1. M. Grinenko, “On the birational rigidity of some pencils of Del Pezzo surfaces”, Journal of Mathematical Sciences (New York), 102:2 (2000), 3933  crossref  mathscinet  zmath
    2. A. V. Pukhlikov, “Birationally rigid Fano fibrations”, Izv. Math., 64:3 (2000), 563–581  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. 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
    4. 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
    5. M. M. Grinenko, “On Fibrations into del Pezzo Surfaces”, Math. Notes, 69:4 (2001), 499–513  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. 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
    7. 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
    8. A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. 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
    11. 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
    12. Corti, A, “Birational geometry of terminal quartic 3-folds, I”, American Journal of Mathematics, 126:4 (2004), 739  crossref  mathscinet  zmath  isi
    13. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. 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
    15. A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. 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
    17. Ivan Cheltsov, “Nonrational nodal quartic threefolds”, Pacific J Math, 226:1 (2006), 65  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    18. K. A. Shramov, “Birational Rigidity and $\mathbb Q$-Factoriality of a Singular Double Cover of a Quadric Branched over a Divisor of Degree 4”, Math. Notes, 84:2 (2008), 280–289  mathnet  crossref  crossref  mathscinet  isi  elib
    19. Pukhlikov, AV, “Birational Geometry of Algebraic Varieties with a Pencil of Fano Cyclic Covers”, Pure and Applied Mathematics Quarterly, 5:2 (2009), 641  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    20. Hausen J., Herppich E., Suess H., “Multigraded Factorial Rings and Fano Varieties with Torus Action”, Doc Math, 16 (2011)  mathscinet  zmath  isi
    21. 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
    22. Cheltsov I., Przyjalkowski V., Shramov C., “Which Quartic Double Solids Are Rational?”, J. Algebr. Geom., 28:2 (2019), 201–243  crossref  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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