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Mat. Sb., 2000, Volume 191, Number 6, Pages 101–126 (Mi msb485)  

This article is cited in 27 scientific papers (total in 27 papers)

Birationally rigid Fano double hypersurfaces

A. V. Pukhlikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: A general Fano double hypersurface $V$ of index 1 $(\sigma\colon V\to Q_m\subset \mathbb P^{M+1}$ is a double cover branched over a smooth divisor $W=W^*_{2l}\subset\mathbb P^{M+1}$, here $m+l=M+1\geqslant 5)$ is proved to be birationally superrigid; in particular, such a hypersurface admits no non-trivial structures of a fibration into uniruled varieties, and it is non-rational. Its groups of birational and biregular automorphisms coincide.


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English version:
Sbornik: Mathematics, 2000, 191:6, 883–908

Bibliographic databases:

UDC: 513.6
MSC: 14J45
Received: 28.06.1999

Citation: A. V. Pukhlikov, “Birationally rigid Fano double hypersurfaces”, Mat. Sb., 191:6 (2000), 101–126; Sb. Math., 191:6 (2000), 883–908

Citation in format AMSBIB
\by A.~V.~Pukhlikov
\paper Birationally rigid Fano double hypersurfaces
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 6
\pages 101--126
\jour Sb. Math.
\yr 2000
\vol 191
\issue 6
\pages 883--908

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    This publication is cited in the following articles:
    1. Foord D., “Birationally Rigid Fano Cyclic Covers Over a Hypersurface Containing a Singular Point”, Eur. J. Math.  crossref  isi
    2. Pukhlikov, AV, “Birationally rigid Fano complete intersections”, Journal fur Die Reine und Angewandte Mathematik, 541 (2001), 55  crossref  mathscinet  zmath  isi  elib
    3. 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
    4. 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
    5. Pukhlikov, AV, “Birational geometry of higher-dimensional Fano hypersurfaces”, Doklady Mathematics, 66:1 (2002), 22  mathnet  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. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  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. 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. 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
    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. Cheltsov, I, “Double cubics and double quartics”, Mathematische Zeitschrift, 253:1 (2006), 75  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    15. Ivan Cheltsov, “On nodal sextic fivefold”, Math Nachr, 280:12 (2007), 1344  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    16. 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
    17. A. V. Pukhlikov, “Explicit examples of birationally rigid Fano varieties”, Mosc. Math. J., 7:3 (2007), 543–560  mathnet  crossref  mathscinet  zmath
    18. Castravet, AM, “Examples of fano varieties of index one that are not birationally rigid”, Proceedings of the American Mathematical Society, 135:12 (2007), 3783  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    19. A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130  mathnet  crossref  mathscinet  elib
    20. A. V. Pukhlikov, “Birational geometry of Fano double covers”, Sb. Math., 199:8 (2008), 1225–1250  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    21. 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
    22. A. V. Pukhlikov, “Singularities of Algebraic Subvarieties and Problems of Birational Geometry”, Proc. Steklov Inst. Math., 267 (2009), 235–247  mathnet  crossref  mathscinet  zmath  isi  elib
    23. Odaka Yu. Okada T., “Birational Superrigidity and Slope Stability of Fano Manifolds”, Math. Z., 275:3-4 (2013), 1109–1119  crossref  mathscinet  zmath  isi  scopus  scopus
    24. 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
    25. E. Johnstone, “Birationally Rigid Singular Double Quadrics and Double Cubics”, Math. Notes, 102:4 (2017), 508–515  mathnet  crossref  crossref  mathscinet  isi  elib
    26. Evans D. Pukhlikov A., “Birationally Rigid Complete Intersections of Codimension Two”, Bull. Korean. Math. Soc., 54:5 (2017), 1627–1654  crossref  mathscinet  zmath  isi  scopus
    27. Zhuang Z., “Birational Superrigidity Is Not a Locally Closed Property”, Sel. Math.-New Ser., 26:1 (2020), UNSP 11  crossref  mathscinet  isi
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