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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 6, Pages 153–186 (Mi izv671)  

This article is cited in 40 scientific papers (total in 40 papers)

Birational geometry of Fano direct products

A. V. Pukhlikov

Abstract: We prove the birational superrigidity of direct products $V=F_1\times…\times F_K$ of primitive Fano varieties of the following two types: either $F_i\subset\mathbb P^M$ is a general hypersurface of degree $M$, $M\geqslant 6$, or $F_i\stackrel{\sigma}{\to}{\mathbb P}^M$ is a general double space of index 1, $M\geqslant 3$. In particular, every structure of a rationally connected fibre space on $V$ is given by the projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Kollár and the technique of hypertangent divisors.

DOI: https://doi.org/10.4213/im671

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English version:
Izvestiya: Mathematics, 2005, 69:6, 1225–1255

Bibliographic databases:

UDC: 512.7
MSC: 14E05, 14J45
Received: 26.01.2005

Citation: A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. RAN. Ser. Mat., 69:6 (2005), 153–186; Izv. Math., 69:6 (2005), 1225–1255

Citation in format AMSBIB
\by A.~V.~Pukhlikov
\paper Birational geometry of Fano direct products
\jour Izv. RAN. Ser. Mat.
\yr 2005
\vol 69
\issue 6
\pages 153--186
\jour Izv. Math.
\yr 2005
\vol 69
\issue 6
\pages 1225--1255

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    This publication is cited in the following articles:
    1. Pukhlikov V A., “Canonical and Log Canonical Thresholds of Multiple Projective Spaces”, Eur. J. Math.  crossref  isi
    2. Kim I.-K. Okada T. Won J., “Alpha Invariants of Birationally Bi-Rigid Fano 3-Folds i”, Eur. J. Math.  crossref  mathscinet  isi
    3. Pukhlikov A.V., “Birational geometry of algebraic varieties with a pencil of Fano complete intersections”, Manuscripta Math., 121:4 (2006), 491–526  crossref  mathscinet  zmath  isi  elib  scopus
    4. 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
    5. A. V. Pukhlikov, “Explicit examples of birationally rigid Fano varieties”, Mosc. Math. J., 7:3 (2007), 543–560  mathnet  crossref  mathscinet  zmath
    6. 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
    7. A. V. Pukhlikov, “Birational geometry of Fano double covers”, Sb. Math., 199:8 (2008), 1225–1250  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. I. A. Cheltsov, K. A. Shramov, “Log canonical thresholds of smooth Fano threefolds”, Russian Math. Surveys, 63:5 (2008), 859–958  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Cheltsov I., “Log canonical thresholds of del Pezzo surfaces”, Geom. Funct. Anal., 18:4 (2008), 1118–1144  crossref  mathscinet  zmath  isi  scopus
    10. A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130  mathnet  crossref  mathscinet  elib
    11. Cheltsov I., “Fano varieties with many selfmaps”, Adv. Math., 217:1 (2008), 97–124  crossref  mathscinet  zmath  isi  scopus
    12. 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
    13. I. A. Cheltsov, K. A. Shramov, “Extremal Metrics on del Pezzo Threefolds”, Proc. Steklov Inst. Math., 264 (2009), 30–44  mathnet  crossref  mathscinet  isi  elib  elib
    14. A. V. Pukhlikov, “Birational Geometry of Singular Fano Varieties”, Proc. Steklov Inst. Math., 264 (2009), 159–177  mathnet  crossref  mathscinet  isi  elib  elib
    15. I. A. Cheltsov, “Log canonical thresholds of three-dimensional Fano hypersurfaces”, Izv. Math., 73:4 (2009), 727–795  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. Pukhlikov A.V., “Birational geometry of algebraic varieties with a pencil of Fano cyclic covers”, Pure Appl. Math. Q., 5:2, Special Issue: In honor of Friedrich Hirzebruch, Part 1 (2009), 641–700  crossref  mathscinet  zmath  isi  elib  scopus
    17. 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
    18. A. V. Pukhlikov, “Existence of the Kähler–Einstein Metric on Certain Fano Complete Intersections”, Math. Notes, 88:4 (2010), 552–558  mathnet  crossref  crossref  mathscinet  isi
    19. 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
    20. 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
    21. Park J., Won J., “Log-canonical thresholds on del Pezzo surfaces of degrees $\ge 2$”, Nagoya Math. J., 200 (2010), 1–26  crossref  mathscinet  zmath  isi  elib  scopus
    22. Park Jihun, Won Joonyeong, “Log canonical thresholds on Gorenstein canonical del Pezzo surfaces”, Proc. Edin. Math. Soc. (2), 54:1 (2011), 187–219  crossref  zmath  isi  scopus
    23. Ivan Cheltsov, Dimitra Kosta, “Computing α-Invariants of Singular del Pezzo Surfaces”, J Geom Anal, 2012  crossref  mathscinet  isi  scopus
    24. Ivan Cheltsov, Jihun Park, Joonyeong Won, “Log canonical thresholds of certain Fano hypersurfaces”, Math. Z, 2013  crossref  mathscinet  isi  scopus
    25. I. A. Cheltsov, “Two local inequalities”, Izv. Math., 78:2 (2014), 375–426  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. 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
    27. Thomas Eckl, Aleksandr Pukhlikov, “On the global log canonical threshold of Fano complete intersections”, European Journal of Mathematics, 2015  crossref  mathscinet  scopus
    28. A. V. Pukhlikov, “Birationally rigid Fano fibre spaces. II”, Izv. Math., 79:4 (2015), 809–837  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    29. A. V. Pukhlikov, “Birational geometry of algebraic varieties fibred into Fano double spaces”, Izv. Math., 81:3 (2017), 618–644  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    30. Kim I.-K., Okada T., Won J., “Alpha Invariants of Birationally Rigid Fano Three-Folds”, Int. Math. Res. Notices, 2018, no. 9, 2745–2800  crossref  mathscinet  isi
    31. Krylov I., “Birational Geometry of Del Pezzo Fibrations With Terminal Quotient Singularities”, J. Lond. Math. Soc.-Second Ser., 97:2 (2018), 222–246  crossref  mathscinet  zmath  isi  scopus
    32. Krylov I., “Rationally Connected Non-Fano Type Varieties”, Eur. J. Math., 4:1, 1, SI (2018), 335–355  crossref  mathscinet  zmath  isi  scopus
    33. Pukhlikov A.V., “Canonical and Log Canonical Thresholds of Fano Complete Intersections”, Eur. J. Math., 4:1, 1, SI (2018), 381–398  crossref  mathscinet  zmath  isi  scopus
    34. Cheltsov I. Parka J. Shramov C., “Alpha-Invariants and Purely Log Terminal Blow-Ups”, Eur. J. Math., 4:3, 2, SI (2018), 845–858  crossref  mathscinet  isi  scopus
    35. Cheltsov I. Dubouloz A. Park J., “Super-Rigid Affine Fano Varieties”, Compos. Math., 154:11 (2018), 2462–2484  crossref  mathscinet  zmath  isi
    36. Cheltsov I.A., Rubinstein Ya.A., Zhang K., “Basis Log Canonical Thresholds, Local Intersection Estimates, and Asymptotically Log Del Pezzo Surfaces”, Sel. Math.-New Ser., 25:2 (2019), UNSP 34  crossref  mathscinet  isi  scopus
    37. Fujita K., “K-Stability of Fano Manifolds With Not Small Alpha Invariants”, J. Inst. Math. Jussieu, 18:3 (2019), 519–530  crossref  isi
    38. Pukhlikov V A., “Rationally Connected Rational Double Covers of Primitive Fano Varieties”, Epijournal Geom. Algebr., 4 (2020), 18  mathscinet  isi
    39. A. V. Pukhlikov, “Birational geometry of singular Fano double spaces of index two”, Sb. Math., 212:4 (2021), 551–566  mathnet  crossref  crossref  isi
    40. A. V. Pukhlikov, “Remarks on Galois Rational Coverings”, Math. Notes, 110:2 (2021), 242–247  mathnet  crossref  crossref  isi
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