Abstract
We study log canonical thresholds (also called global log canonical threshold or $\alpha $-invariant) of $\mathbb R$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds; that is, given a natural number $d$ and a positive real number $\epsilon $, the set of Fano varieties of dimension $d$ with $\epsilon $-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which, in particular, answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case.
-
[Alexeev]
V. Alexeev, "Boundedness and $K^2$ for log surfaces," Internat. J. Math., vol. 5, iss. 6, pp. 779-810, 1994.
@ARTICLE{Alexeev,
author = {Alexeev, Valery},
title = {Boundedness and {$K^2$} for log surfaces},
journal = {Internat. J. Math.},
fjournal = {International Journal of Mathematics},
volume = {5},
year = {1994},
number = {6},
pages = {779--810},
issn = {0129-167X},
mrclass = {14J10 (14J25)},
mrnumber = {1298994},
mrreviewer = {Mark Gross},
doi = {10.1142/S0129167X94000395},
url = {https://doi.org/10.1142/S0129167X94000395},
zblnumber = {0838.14028},
} -
[Alexeev-Brion] V. A. Alexeev and M. Brion, "Boundedness of spherical Fano varieties," in The Fano Conference, Univ. Torino, Turin, 2004, pp. 69-80.
@INCOLLECTION{Alexeev-Brion,
author = {Alexeev, Valery A. and Brion, Michel},
title = {Boundedness of spherical {F}ano varieties},
booktitle = {The {F}ano {C}onference},
pages = {69--80},
publisher = {Univ. Torino, Turin},
year = {2004},
mrclass = {14J45 (14M17 14M25)},
mrnumber = {2112568},
mrreviewer = {Franz Pauer},
zblnumber = {1087.14028},
} -
[Ambro]
F. Ambro, "Variation of log canonical thresholds in linear systems," Int. Math. Res. Not. IMRN, iss. 14, pp. 4418-4448, 2016.
@ARTICLE{Ambro,
author = {Ambro, Florin},
title = {Variation of log canonical thresholds in linear systems},
journal = {Int. Math. Res. Not. IMRN},
fjournal = {International Mathematics Research Notices. IMRN},
year = {2016},
number = {14},
pages = {4418--4448},
issn = {1073-7928},
mrclass = {14E30 (14J45 14M25)},
mrnumber = {3556423},
mrreviewer = {Alex Massarenti},
doi = {10.1093/imrn/rnv284},
url = {https://doi.org/10.1093/imrn/rnv284},
zblnumber = {1404.14020},
} -
[B-lcy-fibs] C. Birkar, Log Calabi-Yau fibrations, 2018.
@MISC{B-lcy-fibs,
author = {Birkar, Caucher},
title = {Log {C}alabi-{Y}au fibrations},
year = {2018},
arxiv = {1811.10709},
zblnumber = {},
} -
[B-compl]
C. Birkar, "Anti-pluricanonical systems on Fano varieties," Ann. of Math. (2), vol. 190, iss. 2, pp. 345-463, 2019.
@ARTICLE{B-compl,
author = {Birkar, Caucher},
title = {Anti-pluricanonical systems on {F}ano varieties},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {190},
year = {2019},
number = {2},
pages = {345--463},
issn = {0003-486X},
mrclass = {14J45 (14C20 14E05 14E30)},
mrnumber = {3997127},
doi = {10.4007/annals.2019.190.2.1},
url = {https://doi.org/10.4007/annals.2019.190.2.1},
zblnumber = {07107180},
} -
[BCHM]
C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, "Existence of minimal models for varieties of log general type," J. Amer. Math. Soc., vol. 23, iss. 2, pp. 405-468, 2010.
@ARTICLE{BCHM,
author = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James},
title = {Existence of minimal models for varieties of log general type},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {23},
year = {2010},
number = {2},
pages = {405--468},
issn = {0894-0347},
mrclass = {14E30 (14E05)},
mrnumber = {2601039},
mrreviewer = {Mark Gross},
doi = {10.1090/S0894-0347-09-00649-3},
url = {https://doi.org/10.1090/S0894-0347-09-00649-3},
zblnumber = {1210.14019},
} -
[A-Borisov]
A. Borisov, "Boundedness of Fano threefolds with log-terminal singularities of given index," J. Math. Sci. Univ. Tokyo, vol. 8, iss. 2, pp. 329-342, 2001.
@ARTICLE{A-Borisov,
author = {Borisov, Alexandr},
title = {Boundedness of {F}ano threefolds with log-terminal singularities of given index},
journal = {J. Math. Sci. Univ. Tokyo},
fjournal = {The Univ. of Tokyo. Journal of Mathematical Sciences},
volume = {8},
year = {2001},
number = {2},
pages = {329--342},
issn = {1340-5705},
mrclass = {14J45 (14E05 14E30 14J30)},
mrnumber = {1837167},
mrreviewer = {Yuri G. Prokhorov},
url = {https://www.ms.u-tokyo.ac.jp/journal/abstract/jms080208.html},
zblnumber = {1073.14539},
} -
[A-L-Borisov]
A. A. Borisov and L. A. Borisov, "Singular toric Fano varieties," Sb. Math., vol. 75, iss. 1, pp. 277-283, 1993.
@ARTICLE{A-L-Borisov,
author = {Borisov, A. A. and Borisov, L. A.},
title = {Singular toric {F}ano varieties},
journal = {Sb. Math.},
fjournal = {Russian Academy of Sciences. Sbornik Mathematics},
volume = {75},
year = {1993},
number = {1},
pages = {277--283},
issn = {0368-8666},
mrclass = {14J45 (14M25)},
mrnumber = {1166957},
mrreviewer = {Yuri G. Prokhorov},
doi = {10.1070/SM1993v075n01ABEH003385},
url = {https://doi.org/10.1070/SM1993v075n01ABEH003385},
zblnumber = {0786.14028},
} -
[Campana]
F. Campana, "Une version géométrique généralisée du théorème du produit de Nadel," Bull. Soc. Math. France, vol. 119, iss. 4, pp. 479-493, 1991.
@ARTICLE{Campana,
author = {Campana, F.},
title = {Une version géométrique généralisée du théorème du produit de {N}adel},
journal = {Bull. Soc. Math. France},
fjournal = {Bulletin de la Société Mathématique de France},
volume = {119},
year = {1991},
number = {4},
pages = {479--493},
issn = {0037-9484},
mrclass = {14J45 (14C20)},
mrnumber = {1136848},
mrreviewer = {Vasile Br\^ınz\u{a}nescu},
doi = {10.24033/bsmf.2176},
url = {https://doi.org/10.24033/bsmf.2176},
zblnumber = {0783.14023},
} -
[cheltsov-shramov]
I. A. Chelcprimetsov and K. A. Shramov, "Log-canonical thresholds for nonsingular Fano threefolds," Uspekhi Mat. Nauk, vol. 63, iss. 5(383), pp. 73-180, 2008.
@ARTICLE{cheltsov-shramov,
author = {Chel{\cprime}tsov, I. A. and Shramov, K. A.},
title = {Log-canonical thresholds for nonsingular {F}ano threefolds},
journal = {Uspekhi Mat. Nauk},
fjournal = {Uspekhi Matematicheskikh Nauk},
volume = {63},
year = {2008},
number = {5(383)},
pages = {73--180},
issn = {0042-1316},
mrclass = {14J45 (14J30 32Q20)},
mrnumber = {2484031},
mrreviewer = {Alexandr V. Pukhlikov},
doi = {10.1070/RM2008v063n05ABEH004561},
url = {https://doi.org/10.1070/RM2008v063n05ABEH004561},
zblnumber = {1167.14024},
} -
[toric-cox-etal]
D. A. Cox, J. B. Little, and H. K. Schenck, Toric Varieties, Amer. Math. Soc., Providence, RI, 2011, vol. 124.
@BOOK{toric-cox-etal,
author = {Cox, David A. and Little, John B. and Schenck, Henry K.},
title = {Toric Varieties},
series = {Grad. Stud. in Math.},
volume = {124},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2011},
pages = {xxiv+841},
isbn = {978-0-8218-4819-7},
mrclass = {14M25 (05A15 05E45 52B12)},
mrnumber = {2810322},
mrreviewer = {Ivan Arzhantsev},
doi = {10.1090/gsm/124},
url = {https://doi.org/10.1090/gsm/124},
zblnumber = {1223.14001},
} -
[HMX2]
C. D. Hacon, J. McKernan, and C. Xu, "ACC for log canonical thresholds," Ann. of Math. (2), vol. 180, iss. 2, pp. 523-571, 2014.
@ARTICLE{HMX2,
author = {Hacon, Christopher D. and McKernan, James and Xu, Chenyang},
title = {A{CC} for log canonical thresholds},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {180},
year = {2014},
number = {2},
pages = {523--571},
issn = {0003-486X},
mrclass = {14E05 (14C20 14E30)},
mrnumber = {3224718},
mrreviewer = {Alexandr V. Pukhlikov},
doi = {10.4007/annals.2014.180.2.3},
url = {https://doi.org/10.4007/annals.2014.180.2.3},
zblnumber = {1320.14023},
} -
[HX]
C. D. Hacon and C. Xu, "Boundedness of log Calabi-Yau pairs of Fano type," Math. Res. Lett., vol. 22, iss. 6, pp. 1699-1716, 2015.
@ARTICLE{HX,
author = {Hacon, Christopher D. and Xu, Chenyang},
title = {Boundedness of log {C}alabi-{Y}au pairs of {F}ano type},
journal = {Math. Res. Lett.},
fjournal = {Mathematical Research Letters},
volume = {22},
year = {2015},
number = {6},
pages = {1699--1716},
issn = {1073-2780},
mrclass = {14E05},
mrnumber = {3507257},
mrreviewer = {Lei Zhang},
doi = {10.4310/MRL.2015.v22.n6.a8},
url = {https://doi.org/10.4310/MRL.2015.v22.n6.a8},
zblnumber = {1362.14018},
} -
[Isk-Prokh] V. A. Iskovskikh and Y. G. Prokhorov, "Fano varieties," in Algebraic Geometry, V, Springer, Berlin, 1999, vol. 47, pp. 1-247.
@INCOLLECTION{Isk-Prokh,
author = {Iskovskikh, V. A. and Prokhorov, Yu. G.},
title = {Fano varieties},
booktitle = {Algebraic Geometry, {V}},
series = {Encyclopaedia Math. Sci.},
volume = {47},
pages = {1--247},
publisher = {Springer, Berlin},
year = {1999},
mrclass = {14J45 (14E07 14F22 14K30)},
mrnumber = {1668579},
mrreviewer = {Takao Fujita},
zblnumber = {0912.14013},
} -
[Jiang-3]
C. Jiang, "Boundedness of $\Bbb Q$-Fano varieties with degrees and alpha-invariants bounded from below," Ann. Sci. Éc. Norm. Supér. (4), vol. 53, iss. 5, pp. 1235-1248, 2020.
@ARTICLE{Jiang-3,
author = {Jiang, Chen},
title = {Boundedness of {$\Bbb Q$}-{F}ano varieties with degrees and alpha-invariants bounded from below},
journal = {Ann. Sci. \'{E}c. Norm. Supér. (4)},
fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
volume = {53},
year = {2020},
number = {5},
pages = {1235--1248},
issn = {0012-9593},
mrclass = {14J45 (14L24)},
mrnumber = {4174851},
doi = {},
url = {},
zblnumber = {},
} -
[Jiang-2] C. Jiang, Boundedness of anti-canonical volumes of singular log Fano threefolds, 2014.
@MISC{Jiang-2,
author = {Jiang, Chen},
title = {Boundedness of anti-canonical volumes of singular log {F}ano threefolds},
note = {to appear in \emph{Comm. Anal. Geom.}},
year = {2014},
arxiv = {1411.6728v2},
zblnumber = {},
} -
[Jiang]
C. Jiang, "On birational boundedness of Fano fibrations," Amer. J. Math., vol. 140, iss. 5, pp. 1253-1276, 2018.
@ARTICLE{Jiang,
author = {Jiang, Chen},
title = {On birational boundedness of {F}ano fibrations},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {140},
year = {2018},
number = {5},
pages = {1253--1276},
issn = {0002-9327},
mrclass = {14E05 (14D06 14J45)},
mrnumber = {3862064},
mrreviewer = {Alex Massarenti},
doi = {10.1353/ajm.2018.0030},
url = {https://doi.org/10.1353/ajm.2018.0030},
zblnumber = {1404.14021},
} -
[kawakita]
M. Kawakita, "Inversion of adjunction on log canonicity," Invent. Math., vol. 167, iss. 1, pp. 129-133, 2007.
@ARTICLE{kawakita,
author = {Kawakita, Masayuki},
title = {Inversion of adjunction on log canonicity},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {167},
year = {2007},
number = {1},
pages = {129--133},
issn = {0020-9910},
mrclass = {14E30 (14N30)},
mrnumber = {2264806},
mrreviewer = {Carla Novelli},
doi = {10.1007/s00222-006-0008-z},
url = {https://doi.org/10.1007/s00222-006-0008-z},
zblnumber = {1114.14009},
} -
[kawamata-bnd-ext-ray]
Y. Kawamata, "On the length of an extremal rational curve," Invent. Math., vol. 105, iss. 3, pp. 609-611, 1991.
@ARTICLE{kawamata-bnd-ext-ray,
author = {Kawamata, Yujiro},
title = {On the length of an extremal rational curve},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {105},
year = {1991},
number = {3},
pages = {609--611},
issn = {0020-9910},
mrclass = {14E30 (14E99)},
mrnumber = {1117153},
mrreviewer = {Peter Nielsen},
doi = {10.1007/BF01232281},
url = {https://doi.org/10.1007/BF01232281},
zblnumber = {0751.14007},
} -
[kawamata-term-3-folds]
Y. Kawamata, "Boundedness of $\Bbb Q$-Fano threefolds," in Proceedings of the International Conference on Algebra, Part 3, 1992, pp. 439-445.
@INPROCEEDINGS{kawamata-term-3-folds,
author = {Kawamata, Yujiro},
title = {Boundedness of {$\bold Q$}-{F}ano threefolds},
booktitle = {Proceedings of the {I}nternational {C}onference on {A}lgebra, {P}art 3},
venue = {{N}ovosibirsk, 1989},
series = {Contemp. Math.},
volume = {131.3},
pages = {439--445},
publisher = {Amer. Math. Soc., Providence, RI},
year = {1992},
mrclass = {14J45 (14E30)},
mrnumber = {1175897},
mrreviewer = {Jaros\l aw A. Wi\'{s}niewski},
doi = {10.1090/conm/131.3/1175897},
url = {https://doi.org/10.1090/conm/131.3/1175897},
zblnumber = {0785.14024},
} -
[Kedlaya]
K. S. Kedlaya, "More étale covers of affine spaces in positive characteristic," J. Algebraic Geom., vol. 14, iss. 1, pp. 187-192, 2005.
@ARTICLE{Kedlaya,
author = {Kedlaya, Kiran S.},
title = {More étale covers of affine spaces in positive characteristic},
journal = {J. Algebraic Geom.},
fjournal = {Journal of Algebraic Geometry},
volume = {14},
year = {2005},
number = {1},
pages = {187--192},
issn = {1056-3911},
mrclass = {14E20 (13A35)},
mrnumber = {2092132},
mrreviewer = {Gordon Heier},
doi = {10.1090/S1056-3911-04-00381-9},
url = {https://doi.org/10.1090/S1056-3911-04-00381-9},
zblnumber = {1065.14020},
} -
[Kollar-toroidal]
J. Kollár, "Partial resolution by toroidal blow-ups," Tunis. J. Math., vol. 1, iss. 1, pp. 3-12, 2019.
@ARTICLE{Kollar-toroidal,
author = {Koll\'{a}r, J\'{a}nos},
title = {Partial resolution by toroidal blow-ups},
journal = {Tunis. J. Math.},
fjournal = {Tunisian Journal of Mathematics},
volume = {1},
year = {2019},
number = {1},
pages = {3--12},
issn = {2576-7658},
mrclass = {14E30 (14E15)},
mrnumber = {3907731},
mrreviewer = {Yu-Chao Tu},
doi = {10.2140/tunis.2019.1.3},
url = {https://doi.org/10.2140/tunis.2019.1.3},
zblnumber = {1408.14060},
} -
[Kollar-flip-abundance] Flips and Abundance for Algebraic Threefolds, Kollár, J., Ed., Soc. Math. France, Paris, 1992, vol. 211.
@BOOK{Kollar-flip-abundance, editor = {Koll\'{a}r, J\'{a}nos},
title = {Flips and Abundance for Algebraic Threefolds},
note = {A summer seminar at the University of Utah, Salt Lake City, 1991},
series = {Astérisque},
publisher={Soc. Math. France, Paris},
volume = {211},
pages = {258 pp.},
mrnumber = {1225842},
year = {1992},
zblnumber = {0782.00075},
} -
[KMM-smooth-fano]
J. Kollár, Y. Miyaoka, and S. Mori, "Rational connectedness and boundedness of Fano manifolds," J. Differential Geom., vol. 36, iss. 3, pp. 765-779, 1992.
@ARTICLE{KMM-smooth-fano,
author = {Koll\'{a}r, J\'{a}nos and Miyaoka, Yoichi and Mori, Shigefumi},
title = {Rational connectedness and boundedness of {F}ano manifolds},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {36},
year = {1992},
number = {3},
pages = {765--779},
issn = {0022-040X},
mrclass = {14J45},
mrnumber = {1189503},
mrreviewer = {Yuri G. Prokhorov},
url = {http://projecteuclid.org/euclid.jdg/1214453188},
zblnumber = {0759.14032},
} -
[KMMT-can-3-folds]
J. Kollár, Y. Miyaoka, S. Mori, and H. Takagi, "Boundedness of canonical $\Bbb Q$-Fano 3-folds," Proc. Japan Acad. Ser. A Math. Sci., vol. 76, iss. 5, pp. 73-77, 2000.
@ARTICLE{KMMT-can-3-folds,
author = {Koll\'{a}r, J\'{a}nos and Miyaoka, Yoichi and Mori, Shigefumi and Takagi, Hiromichi},
title = {Boundedness of canonical {$\bold Q$}-{F}ano 3-folds},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
fjournal = {Japan Academy. Proceedings. Series A. Mathematical Sciences},
volume = {76},
year = {2000},
number = {5},
pages = {73--77},
issn = {0386-2194},
mrclass = {14J45 (14E30)},
mrnumber = {1771144},
mrreviewer = {Yuri G. Prokhorov},
url = {http://projecteuclid.org/euclid.pja/1148393517},
zblnumber = {},
} -
[kollar-mori]
J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Univ. Press, Cambridge, 1998, vol. 134.
@BOOK{kollar-mori,
author = {Koll\'{a}r, J\'{a}nos and Mori, Shigefumi},
title = {Birational Geometry of Algebraic Varieties},
series = {Cambridge Tracts in Math.},
volume = {134},
note = {with the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1998},
pages = {viii+254},
isbn = {0-521-63277-3},
mrclass = {14E30},
mrnumber = {1658959},
mrreviewer = {Mark Gross},
doi = {10.1017/CBO9780511662560},
url = {https://doi.org/10.1017/CBO9780511662560},
zblnumber = {},
} -
[Lazar]
R. Lazarsfeld, Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals, Springer-Verlag, Berlin, 2004, vol. 49.
@BOOK{Lazar,
author = {Lazarsfeld, Robert},
title = {Positivity in Algebraic Geometry. {II}. Positivity for Vector Bundles, and Multiplier Ideals},
series = {Ergeb. Mathe. Grenzgeb.},
volume = {49},
publisher = {Springer-Verlag, Berlin},
year = {2004},
pages = {xviii+385},
isbn = {3-540-22534-X},
mrclass = {14-02 (14C20 14F05 14F17)},
mrnumber = {2095472},
mrreviewer = {Mihnea Popa},
doi = {10.1007/978-3-642-18810-7},
url = {https://doi.org/10.1007/978-3-642-18810-7},
zblnumber = {1093.14500},
} -
[Lin]
J. Lin, "Birational unboundedness of $\Bbb Q$-Fano threefolds," Int. Math. Res. Not., iss. 6, pp. 301-312, 2003.
@ARTICLE{Lin,
author = {Lin, Jiayuan},
title = {Birational unboundedness of {$\Bbb Q$}-{F}ano threefolds},
journal = {Int. Math. Res. Not.},
fjournal = {International Mathematics Research Notices},
year = {2003},
number = {6},
pages = {301--312},
issn = {1073-7928},
mrclass = {14J45 (14E30 14J30)},
mrnumber = {1939018},
mrreviewer = {Takao Fujita},
doi = {10.1155/S1073792803108070},
url = {https://doi.org/10.1155/S1073792803108070},
zblnumber = {1019.14005},
} -
[MP]
J. McKernan and Y. Prokhorov, "Threefold thresholds," Manuscripta Math., vol. 114, iss. 3, pp. 281-304, 2004.
@ARTICLE{MP,
author = {McKernan, James and Prokhorov, Yuri},
title = {Threefold thresholds},
journal = {Manuscripta Math.},
fjournal = {Manuscripta Mathematica},
volume = {114},
year = {2004},
number = {3},
pages = {281--304},
issn = {0025-2611},
mrclass = {14E30 (14B05 14E05 14J30)},
mrnumber = {2075967},
mrreviewer = {Massimiliano Mella},
doi = {10.1007/s00229-004-0457-x},
url = {https://doi.org/10.1007/s00229-004-0457-x},
zblnumber = {1060.14022},
} -
[Nadel]
A. M. Nadel, "The boundedness of degree of Fano varieties with Picard number one," J. Amer. Math. Soc., vol. 4, iss. 4, pp. 681-692, 1991.
@ARTICLE{Nadel,
author = {Nadel, Alan Michael},
title = {The boundedness of degree of {F}ano varieties with {P}icard number one},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {4},
year = {1991},
number = {4},
pages = {681--692},
issn = {0894-0347},
mrclass = {14J45},
mrnumber = {1115788},
mrreviewer = {Harry D'Souza},
doi = {10.2307/2939285},
url = {https://doi.org/10.2307/2939285},
zblnumber = {0754.14026},
} -
[Nikulin-3]
V. V. Nikulin, "Del Pezzo surfaces with log-terminal singularities. III," Izv. Akad. Nauk SSSR Ser. Mat., vol. 53, iss. 6, pp. 1316-1334, 1338, 1989.
@ARTICLE{Nikulin-3,
author = {Nikulin, V. V.},
title = {Del {P}ezzo surfaces with log-terminal singularities. {III}},
journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
fjournal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
volume = {53},
year = {1989},
number = {6},
pages = {1316--1334, 1338},
note = {translation in {\em Math. USSR-Izv}. \textbf{35} (1990), no. 3, 657--675},
issn = {0373-2436},
mrclass = {14J26 (14J05 14J17)},
mrnumber = {1039966},
mrreviewer = {I. Dolgachev},
doi = {10.1070/IM1990v035n03ABEH000721},
url = {https://doi.org/10.1070/IM1990v035n03ABEH000721},
zblnumber = {0711.14018},
} -
[Nikulin-2]
V. V. Nikulin, "Del Pezzo surfaces with log-terminal singularities. II," Izv. Akad. Nauk SSSR Ser. Mat., vol. 52, iss. 5, pp. 1032-1050, 1119, 1988.
@ARTICLE{Nikulin-2,
author = {Nikulin, V. V.},
title = {Del {P}ezzo surfaces with log-terminal singularities. {II}},
journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
fjournal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
volume = {52},
year = {1988},
number = {5},
pages = {1032--1050, 1119},
note = {translation in {\em Math. USSR-Izv}. \textbf{33} (1989), no. 2, 355--372},
issn = {0373-2436},
mrclass = {14J26 (14J05 14J17)},
mrnumber = {0972094},
mrreviewer = {I. Dolgachev},
doi = {10.1070/IM1989v033n02ABEH000836},
url = {https://doi.org/10.1070/IM1989v033n02ABEH000836},
zblnumber = {0704.14030},
} -
[Nikulin]
V. V. Nikulin, "Del Pezzo surfaces with log-terminal singularities," Mat. Sb., vol. 180, iss. 2, pp. 226-243, 304, 1989.
@ARTICLE{Nikulin,
author = {Nikulin, V. V.},
title = {Del {P}ezzo surfaces with log-terminal singularities},
journal = {Mat. Sb.},
fjournal = {Matematicheskiĭ Sbornik},
volume = {180},
year = {1989},
number = {2},
pages = {226--243, 304},
issn = {0368-8666},
mrclass = {14J26 (14E30 14J05)},
mrnumber = {0993456},
mrreviewer = {I. Dolgachev},
note = {translation in {\em Math. USSR-Sb.},
\textbf{66} (1990), no. 1, 231--248},
doi = {10.1070/SM1990v066n01ABEH001314},
url = {https://doi.org/10.1070/SM1990v066n01ABEH001314},
zblnumber = {0674.14024},
} -
[Prokhorov-plt-blowups]
Y. G. Prokhorov, "Blow-ups of canonical singularities," in Algebra, de Gruyter, Berlin, 2000, pp. 301-317.
@INCOLLECTION{Prokhorov-plt-blowups,
author = {Prokhorov, Yu. G.},
title = {Blow-ups of canonical singularities},
booktitle = {Algebra},
venue = {{M}oscow, 1998},
pages = {301--317},
publisher = {de Gruyter, Berlin},
year = {2000},
mrclass = {14E30 (14B05 14E05 14J30)},
mrnumber = {1754677},
mrreviewer = {Massimiliano Mella},
doi = {10.1515/9783110805697},
url = {https://doi.org/10.1515/9783110805697},
zblnumber = {1003.14005},
} -
[PSh-II]
Y. G. Prokhorov and V. V. Shokurov, "Towards the second main theorem on complements," J. Algebraic Geom., vol. 18, iss. 1, pp. 151-199, 2009.
@ARTICLE{PSh-II,
author = {Prokhorov, Yu. G. and Shokurov, V. V.},
title = {Towards the second main theorem on complements},
journal = {J. Algebraic Geom.},
fjournal = {Journal of Algebraic Geometry},
volume = {18},
year = {2009},
number = {1},
pages = {151--199},
issn = {1056-3911},
mrclass = {14C20 (14J45)},
mrnumber = {2448282},
mrreviewer = {Alexandr V. Pukhlikov},
doi = {10.1090/S1056-3911-08-00498-0},
url = {https://doi.org/10.1090/S1056-3911-08-00498-0},
zblnumber = {1159.14020},
} -
[PSh-I]
Y. G. Prokhorov and V. V. Shokurov, "The first fundamental theorem on complements: from global to local," Izv. Ross. Akad. Nauk Ser. Mat., vol. 65, iss. 6, pp. 99-128, 2001.
@ARTICLE{PSh-I,
author = {Prokhorov, Yu. G. and Shokurov, V. V.},
title = {The first fundamental theorem on complements: from global to local},
journal = {Izv. Ross. Akad. Nauk Ser. Mat.},
fjournal = {Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya},
volume = {65},
year = {2001},
number = {6},
pages = {99--128},
issn = {1607-0046},
mrclass = {14E30 (14J40)},
mrnumber = {1892905},
mrreviewer = {Adrian Langer},
doi = {10.1070/IM2001v065n06ABEH000366},
url = {https://doi.org/10.1070/IM2001v065n06ABEH000366},
zblnumber = {1068.14018},
} -
[Prokhorov-Shramov]
Y. Prokhorov and C. Shramov, "Jordan property for Cremona groups," Amer. J. Math., vol. 138, iss. 2, pp. 403-418, 2016.
@ARTICLE{Prokhorov-Shramov,
author = {Prokhorov, Yuri and Shramov, Constantin},
title = {Jordan property for {C}remona groups},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {138},
year = {2016},
number = {2},
pages = {403--418},
issn = {0002-9327},
mrclass = {14E07},
mrnumber = {3483470},
mrreviewer = {Zinovy Reichstein},
doi = {10.1353/ajm.2016.0017},
url = {https://doi.org/10.1353/ajm.2016.0017},
zblnumber = {1343.14010},
} -
[Prokhorov-Shramov-2]
Y. Prokhorov and C. Shramov, "Jordan property for groups of birational selfmaps," Compos. Math., vol. 150, iss. 12, pp. 2054-2072, 2014.
@ARTICLE{Prokhorov-Shramov-2,
author = {Prokhorov, Yuri and Shramov, Constantin},
title = {Jordan property for groups of birational selfmaps},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {150},
year = {2014},
number = {12},
pages = {2054--2072},
issn = {0010-437X},
mrclass = {14E07},
mrnumber = {3292293},
mrreviewer = {Anne-Sophie Kaloghiros},
doi = {10.1112/S0010437X14007581},
url = {https://doi.org/10.1112/S0010437X14007581},
zblnumber = {1314.14022},
} -
[Serre]
. J-P. Serre, "A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field," Mosc. Math. J., vol. 9, iss. 1, pp. 193-208, back matter, 2009.
@ARTICLE{Serre,
author = {Serre, {\relax J-P}},
title = {A {M}inkowski-style bound for the orders of the finite subgroups of the {C}remona group of rank 2 over an arbitrary field},
journal = {Mosc. Math. J.},
fjournal = {Moscow Mathematical Journal},
volume = {9},
year = {2009},
number = {1},
pages = {193--208, back matter},
issn = {1609-3321},
mrclass = {14E07 (14J26)},
mrnumber = {2567402},
mrreviewer = {Alberto Calabri},
doi = {10.17323/1609-4514-2009-9-1-183-198},
url = {https://doi.org/10.17323/1609-4514-2009-9-1-183-198},
zblnumber = {1203.14017},
} -
[Tian]
G. Tian, "On a set of polarized Kähler metrics on algebraic manifolds," J. Differential Geom., vol. 32, iss. 1, pp. 99-130, 1990.
@ARTICLE{Tian,
author = {Tian, Gang},
title = {On a set of polarized {K}ähler metrics on algebraic manifolds},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {32},
year = {1990},
number = {1},
pages = {99--130},
issn = {0022-040X},
mrclass = {32L07 (32C17 53C55)},
mrnumber = {1064867},
mrreviewer = {John M. Lee},
doi = {10.4310/jdg/1214445039},
url = {https://doi.org/10.4310/jdg/1214445039},
zblnumber = {0706.53036},
} -
[Viehweg]
E. Viehweg, Quasi-Projective Moduli for Polarized Manifolds, Springer-Verlag, Berlin, 1995, vol. 30.
@BOOK{Viehweg,
author = {Viehweg, Eckart},
title = {Quasi-Projective Moduli for Polarized Manifolds},
series = {Ergeb. Math. Grenzgeb.},
volume = {30},
publisher = {Springer-Verlag, Berlin},
year = {1995},
pages = {viii+320},
isbn = {3-540-59255-5},
mrclass = {14-02 (14D20 14D22)},
mrnumber = {1368632},
mrreviewer = {P. E. Newstead},
doi = {10.1007/978-3-642-79745-3},
url = {https://doi.org/10.1007/978-3-642-79745-3},
zblnumber = {0844.14004},
} -
[Xu-plt]
C. Xu, "Finiteness of algebraic fundamental groups," Compos. Math., vol. 150, iss. 3, pp. 409-414, 2014.
@ARTICLE{Xu-plt,
author = {Xu, Chenyang},
title = {Finiteness of algebraic fundamental groups},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {150},
year = {2014},
number = {3},
pages = {409--414},
issn = {0010-437X},
mrclass = {14J17 (14J45)},
mrnumber = {3187625},
mrreviewer = {Andrzej Kozlowski},
doi = {10.1112/S0010437X13007562},
url = {https://doi.org/10.1112/S0010437X13007562},
zblnumber = {1291.14057},
} -
[Zarhin]
Y. G. Zarhin, "Theta groups and products of abelian and rational varieties," Proc. Edinb. Math. Soc. (2), vol. 57, iss. 1, pp. 299-304, 2014.
@ARTICLE{Zarhin,
author = {Zarhin, Yuri G.},
title = {Theta groups and products of abelian and rational varieties},
journal = {Proc. Edinb. Math. Soc. (2)},
fjournal = {Proceedings of the Edinburgh Mathematical Society. Series II},
volume = {57},
year = {2014},
number = {1},
pages = {299--304},
issn = {0013-0915},
mrclass = {14E07 (14K05)},
mrnumber = {3165026},
mrreviewer = {Vladimir L. Popov},
doi = {10.1017/S0013091513000862},
url = {https://doi.org/10.1017/S0013091513000862},
zblnumber = {1311.14018},
}