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Mat. Sb., 2002, Volume 193, Number 3, Pages 135–160 (Mi msb640)  

This article is cited in 15 scientific papers (total in 15 papers)

Birationally rigid Fano hypersurfaces with isolated singularities

A. V. Pukhlikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: It is proved that a general Fano hypersurface $V=V_M\subset{\mathbb P}^M$ of index 1 with isolated singularities in general position is birationally rigid. Hence it cannot be fibred into uniruled varieties of smaller dimension by a rational map, and each ${\mathbb Q}$-Fano variety $V'$ with Picard number 1 birationally equivalent to $V$ is in fact isomorphic to $V$. In particular, $V$ is non-rational. The group of birational self-maps of $V$ is either {1} or ${\mathbb Z}/2{\mathbb Z}$, depending on whether $V$ has a terminal point of the maximum possible multiplicity $M- 2$. The proof is based on a combination of the method of maximal singularities and the techniques of hypertangent systems with Shokurov's connectedness principle.

DOI: https://doi.org/10.4213/sm640

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English version:
Sbornik: Mathematics, 2002, 193:3, 445–471

Bibliographic databases:

UDC: 513.6
MSC: 14E05, 14J45
Received: 04.09.2001

Citation: A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces with isolated singularities”, Mat. Sb., 193:3 (2002), 135–160; Sb. Math., 193:3 (2002), 445–471

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269  mathnet  crossref  crossref  mathscinet  zmath
    2. A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Ivan Cheltsov, “On nodal sextic fivefold”, Math Nachr, 280:12 (2007), 1344  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. 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
    8. 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
    9. A. V. Pukhlikov, “Birational Geometry of Singular Fano Varieties”, Proc. Steklov Inst. Math., 264 (2009), 159–177  mathnet  crossref  mathscinet  isi  elib  elib
    10. 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
    11. 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
    12. 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
    13. Suzuki F., “Birational rigidity of complete intersections”, Math. Z., 285:1-2 (2017), 479–492  crossref  mathscinet  zmath  isi  scopus
    14. de Fernex T., “Birational Rigidity of Singular Fano Hypersurfaces”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 17:3 (2017), 911–929  mathscinet  zmath  isi
    15. 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
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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