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Mat. Sb. (N.S.), 1988, Volume 135(177), Number 4, Pages 472–496 (Mi msb1719)  

This article is cited in 35 scientific papers (total in 35 papers)

Birational automorphisms of a three-dimensional quartic with a quadratic singularity

A. V. Pukhlikov

Abstract: The author studies birational automorphisms of a hypersurface of degree 4 in the four-dimensional projective space over an algebraically closed field having a unique ordinary double point of general type. It is shown that the group of birational automorphisms of a quartic is a semidirect product of a normal subgroup of finite index freely generated by 25 birational involutions and the (finite) group of biregular (projective) automorphisms of the quartic. Such a quartic is not rational.
The proof is based on the techniques of maximal singularities due to V. A. Iskovskih and Yu. I. Manin.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 63:2, 457–482

Bibliographic databases:

UDC: 513.6
MSC: 14E05, 14E35, 14J30, 14M20
Received: 02.12.1986

Citation: A. V. Pukhlikov, “Birational automorphisms of a three-dimensional quartic with a quadratic singularity”, Mat. Sb. (N.S.), 135(177):4 (1988), 472–496; Math. USSR-Sb., 63:2 (1989), 457–482

Citation in format AMSBIB
\by A.~V.~Pukhlikov
\paper Birational automorphisms of a three-dimensional quartic with a quadratic singularity
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 135(177)
\issue 4
\pages 472--496
\jour Math. USSR-Sb.
\yr 1989
\vol 63
\issue 2
\pages 457--482

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    This publication is cited in the following articles:
    1. Manin Y., “Notes on the Arithmetic of Fano Threefolds”, Compos. Math., 85:1 (1993), 37–55  mathscinet  zmath  isi
    2. 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
    3. M. M. Grinenko, “Birational automorphisms of a three-dimensional double cone”, Sb. Math., 189:7 (1998), 991–1007  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. A. V. Pukhlikov, “Birationally rigid Fano double hypersurfaces”, Sb. Math., 191:6 (2000), 883–908  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Grinenko M., “Birational Correspondences of a Double Cone”, Algebra, ed. Bahturin Y., Walter de Gruyter & Co, 2000, 125–133  mathscinet  zmath  isi
    6. A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces with isolated singularities”, Sb. Math., 193:3 (2002), 445–471  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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. I. A. Cheltsov, “Anticanonical models of three-dimensional Fano varieties of degree 4”, Sb. Math., 194:4 (2003), 617–640  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. I. A. Cheltsov, “Regularization of Birational Automorphisms”, Math. Notes, 76:2 (2004), 264–275  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. “Birational geometry of terminal quartic 3-folds, I”, American Journal of Mathematics, 126:4 (2004), 739–761  crossref  mathscinet  zmath  isi
    12. A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. 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
    14. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. I. A. Cheltsov, “Local inequalities and birational superrigidity of Fano varieties”, Izv. Math., 70:3 (2006), 605–639  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. Cheltsov I., “Nonrational nodal quartic threefolds”, Pacific Journal of Mathematics, 226:1 (2006), 65  crossref  mathscinet  zmath  isi  elib
    17. Cheltsov I. Park J., “Weighted Fano Threefold Hypersurfaces”, J. Reine Angew. Math., 600 (2006), 81–116  crossref  mathscinet  zmath  isi  elib
    18. I. A. Cheltsov, “Factoriality of nodal three-dimensional varieties and connectedness of the locus of log canonical singularities”, Sb. Math., 197:3 (2006), 387–414  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    19. 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
    20. 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
    21. Cheltsov I., “On nodal sextic fivefold”, Mathematische Nachrichten, 280:12 (2007), 1344  crossref  mathscinet  zmath  isi  elib
    22. K. A. Shramov, “$\mathbb Q$-factorial quartic threefolds”, Sb. Math., 198:8 (2007), 1165–1174  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    23. A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130  mathnet  crossref  mathscinet  elib
    24. 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
    25. 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
    26. 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
    27. M. M. Grinenko, “Three-dimensional quartics containing a plane”, Sb. Math., 202:2 (2011), 207–241  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. Hausen J., Herppich E., Suess H., “Multigraded Factorial Rings and Fano Varieties with Torus Action”, Doc Math, 16 (2011)  mathscinet  zmath  isi
    29. J.W.. Cutrone, N.A.. Marshburn, “Towards the classification of weak Fano threefolds with ρ = 2”, centr.eur.j.math, 11:9 (2013), 1552  crossref  mathscinet  zmath
    30. 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
    31. Suzuki F., “Birational rigidity of complete intersections”, Math. Z., 285:1-2 (2017), 479–492  crossref  mathscinet  zmath  isi  scopus
    32. de Fernex T., “Birational Rigidity of Singular Fano Hypersurfaces”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 17:3 (2017), 911–929  mathscinet  zmath  isi
    33. A. V. Pukhlikov, “Automorphisms of certain affine complements in projective space”, Sb. Math., 209:2 (2018), 276–289  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    34. Pukhlikov A.V., “Birationally Rigid Complete Intersections With a Singular Point of High Multiplicity”, Proc. Edinb. Math. Soc., 62:1 (2019), 221–239  crossref  mathscinet  zmath  isi  scopus
    35. Yuri G. Prokhorov, “Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I”, Proc. Steklov Inst. Math., 307 (2019), 210–231  mathnet  crossref  crossref  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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