Voir la notice du chapitre de livre
@article{LKN_2022_31_31_a0,
author = {Yu. G. Prokhorov},
title = {Fano threefolds},
journal = {Lekcionnye kursy NOC},
pages = {3--154},
year = {2022},
volume = {31},
number = {31},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/LKN_2022_31_31_a0/}
}
Yu. G. Prokhorov. Fano threefolds. Lekcionnye kursy NOC, Fano threefolds, Tome 31 (2022) no. 31, pp. 3-154. http://geodesic.mathdoc.fr/item/LKN_2022_31_31_a0/
[1] V. V. Przhyalkovskii, I. A. Cheltsov, K. A. Shramov, “Giperellipticheskie i trigonalnye mnogoobraziya Fano”, Izv. RAN. Ser. matem., 69:2 (2005), 145–204
[2] Kh. Klemens, Ya. Kollar, S. Mori, Mnogomernaya kompleksnaya geometriya, Mir, M., 1993
[3] I. A. Cheltsov, K. A. Shramov, “Log-kanonicheskie porogi neosobykh trekhmernykh mnogoobrazii Fano”, Uspekhi matem. nauk, 63:5(383) (2008), 73–180 | Zbl
[4] F. Griffits, Dzh. Kharris, Printsipy algebraicheskoi geometrii, v. 1, 2, Mir, M., 1982
[5] B. G. Moishezon, “Teorema Kastelnuovo–Enrikvesa o styagivanii dlya proizvolnoi razmernosti”, Izv. AN SSSR. Ser. matem., 33:5 (1969), 974–1025 ; http://mi.mathnet.ru/izv2189 | Zbl
[6] V. V. Shokurov, “Teorema Netera–Enrikvesa o kanonicheskikh krivykh”, Matem. sb., 86(128):3(11) (1971), 367–408 ; http://mi.mathnet.ru/msb3299 | Zbl
[7] Yu. I. Manin, Kubicheskie formy: Algebra, geometriya, arifmetika, Nauka, M., 1972
[8] V. A. Iskovskikh, “Trekhmernye mnogoobraziya Fano. {I}”, Izv. AN SSSR. Ser. matem., 41:3 (1977), 516–562 ; http://mi.mathnet.ru/izv1823 | Zbl
[9] F. A. Bogomolov, “Golomorfnye tenzory i vektornye rassloeniya na proektivnykh mnogoobraziyakh”, Izv. AN SSSR. Ser. matem., 42:6 (1978), 1227–1287
[10] V. A. Iskovskikh, “Trekhmernye mnogoobraziya Fano. {II}”, Izv. AN SSSR. Ser. matem., 42:3 (1978), 506–549 ; http://mi.mathnet.ru/izv1778 | Zbl
[11] V. A. Iskovskikh, “Antikanonicheskie modeli trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat., 12 (1979), 59–157 ; http://mi.mathnet.ru/intd34 | Zbl
[12] V. V. Shokurov, “Suschestvovanie pryamoi na mnogoobraziyakh Fano”, Izv. AN SSSR. Ser. mat., 43:4 (1979), 922–964 ; http://mi.mathnet.ru/izv1741 | Zbl
[13] V. A. Iskovskikh, “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat., 12 (1979), 159–236 | Zbl
[14] V. V. Shokurov, “Gladkost obschego antikanonicheskogo divizora na mnogoobrazii Fano”, Izv. AN SSSR. Ser. mat., 43:2 (1979), 430–441 ; http://mi.mathnet.ru/izv1720 | Zbl
[15] N. P. Gushel, “O mnogoobraziyakh Fano roda $6$”, Izv. AN SSSR. Ser. mat., 46:3 (1982), 1159–1174 ; http://mi.mathnet.ru/izv1701
[16] N. P. Gushel, “O mnogoobraziyakh Fano roda $8$”, Uspekhi Mat. Nauk., 38:1 (1983), 163–164 ; http://mi.mathnet.ru/umn2834 | Zbl
[17] S. L. Tregub, “Konstruktsiya biratsionalnogo izomorfizma trekhmernoi kubiki i mnogoobraziya Fano pervogo roda s $g=8$, svyazannaya s normalnoi ratsionalnoi krivoi stepeni 4”, Vestn. MGU. Ser. mat., mekh., 1985, no. 6, 99–101 | Zbl
[18] I. R. Shafarevich, Osnovy algebraicheskoi geometrii, t. I, II, 2 izd., Nauka, M., 1988
[19] V. A. Iskovskikh, Lektsii po trekhmernym algebraicheskim mnogoobraziyam. Mnogoobraziya Fano, Izd-vo MGU, 1988
[20] V. A. Iskovskikh, “Dvoinaya proektsiya iz pryamoi na trekhmernykh mnogoobraziyakh Fano pervogo roda”, Matem. sb., 180:2 (1989), 260–278 ; http://mi.mathnet.ru/msb1609
[21] Yu. G. Prokhorov, “Ob ekzoticheskikh mnogoobraziyakh Fano”, Vestnik MGU. Ser. 1, mat. mekh., 1990, no. 3, 34–37 | Zbl
[22] Yu. G. Prokhorov, “Gruppy avtomorfizmov mnogoobrazii Fano”, Uspekhi mat. nauk., 45:3 (1990), 195–196 | Zbl
[23] V. V. Shokurov, “Trekhmernye logperestroiki”, Izv. RAN. Ser. matem., 56:1 (1992), 105–203 | Zbl
[24] N. P. Gushel, “O trekhmernykh mnogoobraziyakh Fano roda $8$”, Algebra i analiz, 4:1 (1992), 120–134 ; http://mi.mathnet.ru/aa302
[25] Yu. G. Prokhorov, “O suschestvovanii khoroshikh divizorov na mnogoobraziyakh Fano koindeksa $3$”, Tr. MIAN, 208 (1995), 266–277 | Zbl
[26] Yu. G. Prokhorov, Osobennosti algebraicheskikh mnogoobrazii, MTsNMO, M., 2009, 128 pp.
[27] Yu. G. Prokhorov, Ratsionalnye poverkhnosti, Lekts. kursy NOTs, 24, MIAN, M., 2015 ; http://mi.mathnet.ru/eng/book1590
[28] Yu. G. Prokhorov, “O chisle osobykh tochek trekhmernykh terminalnykh faktorialnykh mnogoobrazii Fano”, Matematicheskie zametki, 101:6 (2017), 949–954 | Zbl
[29] Yu. G. Prokhorov, “Ekvivariantnaya programma minimalnykh modelei”, Uspekhi mat. nauk, 76:3 (2021), 93–182 ; http://mi.mathnet.ru/rm9990 | Zbl
[30] D. Mamford, Lektsii o krivykh na algebraicheskoi poverkhnosti, Biblioteka sbornika Matematika, Mir, M., 1968
[31] N. Burbaki, Kommutativnaya algebra. Elementy matematiki, Mir, M., 1971
[32] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981
[33] T. Springer, Teoriya invariantov, Matematika – Novoe v zarubezhnoi nauke, 24, Mir, M., 1981
[34] Mamford D., Krasnaya kniga o mnogoobraziyakh i skhemakh, MTsMNO, M., 2007, 296 pp.
[35] D. Aizenbad, Kommutativnaya algebra s pritselom na algebraicheskuyu geometriyu, MTsNMO, 2017
[36] I. R. Shafarevich, B. G. Averbukh, Yu. K. Vainberg i dr., “Algebraicheskie poverkhnosti”, Trudy matem. in-ta im. V. A. Steklova RAN, 75 (1965), 1–215
[37] D. N. Akhiezer, “Lie group actions in complex analysis”, Aspects of Mathematics, E27, Friedr. Vieweg Sohn, Braunschweig, 1995 | DOI
[38] Ambro F., “Ladders on Fano varieties”, J. Math. Sci. (New York), 94:1 (1999), 1126–1135 | DOI | Zbl
[39] M. Andreatta, J. A. Wiśniewski, “A view on contractions of higher dimensional varieties”, Algebraic geometry, Proceedings of the Summer Research Institute (Santa Cruz, CA, USA, July 9–29, 1995), American Mathematical Society, Providence, RI, 1997, 153–183 | Zbl
[40] M. Andreatta, J. A. Wiśniewski, “Contractions of smooth varieties. {II}. Computations and applications”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1:2 (1998), 343–360
[41] M. Andreatta, J. A. Wiśniewski, “On contractions of smooth varieties”, J. Algebraic Geom., 7:2 (1998), 253–312 | Zbl
[42] M. F. Atiyah, “On analytic surfaces with double points”, Proc. R. Soc. Lond., Ser. A, 247 (1958), 237–244 | DOI | Zbl
[43] A. Beauville, “Variétés de {P}rym et jacobiennes intermédiaires”, Ann. Sci. École Norm. Sup. (4), 10:3 (1977), 309–391 | DOI | Zbl
[44] C. Birkar, “Singularities of linear systems and boundedness of Fano varieties”, Ann. of Math. (2), 193:2 (2021), 347–405 ; https://doi.org/10.4007/annals.2021.193.2.1 | DOI | Zbl
[45] F. Campana, “On twistor spaces of the class ${\mathcal C}$”, J. Differ. Geom., 33:2 (1991), 541–549 | DOI | Zbl
[46] F. Campana, “Connexité rationnelle des variétés de Fano”, Ann. Sci. École Norm. Sup. (4), 25:5 (1992), 539–545 | DOI | Zbl
[47] C. H. Clemens, P. A. Griffiths, The intermediate Jacobian of the cubic threefold, 95 (1972), 281–356 | Zbl
[48] S. D. Cutkosky, “On Fano $3$-folds”, Manuscripta Math., 64:2 (1989), 189–204 | DOI | Zbl
[49] O. Debarre, A. Iliev, L. Manivel, “On nodal prime Fano threefolds of degree 10”, Sci. China Math., 54:8 (2011), 1591–1609 ; http://dx.doi.org/10.1007/s11425-011-4182-0 | DOI | Zbl
[50] O. Debarre, A. Kuznetsov, “Gushel–Mukai varieties: classification and birationalities”, Algebr. Geom., 5:1 (2018), 15–76 ; http://content.algebraicgeometry.nl/2018-1/2018-1-002.pdf | Zbl
[51] I. Dolgachev, “Weighted projective varieties”, Group actions and vector fields (Vancouver, BC, 1981), Lecture Notes in Math., 956, Springer, Berlin, 1982, 34–71 | DOI
[52] I. V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012 | Zbl
[53] G. Fano, “Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulli”, Atti Congresso Bologna, 4 (1931), 115–121 | Zbl
[54] G. Fano, “Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche”, Comment. Math. Helv., 14 (1942), 202–211 | DOI | Zbl
[55] Flips and abundance for algebraic threefolds, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211, ed. J. Kollár, Société Mathématique de France, Paris, 1992, 258 pp. | Zbl
[56] T. Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, 155, Cambridge University Press, Cambridge, 1990 | DOI | Zbl
[57] F. Hidaka, K. Watanabe, “Normal Gorenstein surfaces with ample anti-canonical divisor”, Tokyo J. Math., 4:2 (1981), 319–330 | DOI | Zbl
[58] D. Huybrechts, Lectures on $K$3 surfaces, Cambridge University Press, Cambridge, 2016 ; http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf | Zbl
[59] V. A. Iskovskikh, Yu. Prokhorov, Fano varieties, Algebraic geometry {V}, Encyclopaedia Math. Sci., 47, Springer, Berlin, 1999 | Zbl
[60] P. Jahnke, I. Radloff, “Gorenstein Fano threefolds with base points in the anticanonical system”, Compos. Math., 142:2 (2006), 422–432 | DOI | Zbl
[61] Y. Kachi, “Extremal contractions from $4$-dimensional manifolds to $3$-folds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24:1 (1997), 63–131 ; http://www.numdam.org/item?id=ASNSP_1997_4_24_1_63_0 | Zbl
[62] Y. Kawamata, “Boundedness of $\mathbf{Q}$-Fano threefolds”, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), Contemp. Math., 131, Amer. Math. Soc., Providence, RI, 1992, 439–445 | DOI
[63] Y. Kawamata, “On Fujita's freeness conjecture for $3$-folds and $4$-folds”, Math. Ann., 308:3 (1997), 491–505. | DOI | Zbl
[64] Y. Kawamata, “Subadjunction of log canonical divisors for a subvariety of codimension $2$”, Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math., 207, Amer. Math. Soc., Providence, RI, 1997, 79–88 | DOI | Zbl
[65] Y. Kawamata, “Subadjunction of log canonical divisors. {II}”, Amer. J. Math., 120:5 (1998), 893–899 | DOI | Zbl
[66] Y. Kawamata, K. Matsuda, K. Matsuki, “Introduction to the minimal model problem”, Algebraic geometry (Sendai, 1985), Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, 283–360 | DOI
[67] S. L. Kleiman, “Toward a numerical theory of ampleness”, Ann. of Math. (2), 84 (1966), 293–344 | DOI | Zbl
[68] K. Kodaira, Complex manifolds and deformation of complex structures, Translated from the Japanese by Kazuo Akao, With an appendix by Daisuke Fujiwara, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 283, Springer-Verlag, New York, 1986 | DOI | Zbl
[69] J. Kollár, “Toward moduli of singular varieties”, Compositio Math., 56:3 (1985), 369–398 | Zbl
[70] J. Kollár, “Flops”, Nagoya Math. J., 113 (1989), 15–36 | DOI | Zbl
[71] J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 32, Springer-Verlag, Berlin, 1996
[72] J. Kollár, “Singularities of pairs”, Algebraic geometry–Santa Cruz, Proc. Sympos. Pure Math., 62, Amer. Math. Soc., Providence, RI, 1995, 221–287
[73] J. Kollár, Y. Miyaoka, S. Mori, “Rational connectedness and boundedness of Fano manifolds”, J. Differential Geom., 36:3 (1992), 765–779 | Zbl
[74] J. Kollár, S. Mori, Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998 | Zbl
[75] A. Kuznetsov, Yu. Prokhorov, “Prime {F}ano threefolds of genus $12$ with a $\mathbf G_m$-action”, Épijournal de Géométrie Algébrique, 2 (2018), 4560; https://epiga.episciences.org/4560
[76] A. Kuznetsov, Yu. Prokhorov, Rationality of Fano threefolds over non-closed fields, Arxiv e-print 1911.08949, 2019 (to appear)
[77] A. Kuznetsov, Yu. Prokhorov, “Rationality of Mukai varieties over non-closed fields”, Rationality of Varieties, Progress in Mathematics, 342, eds. Farkas G., van der Geer G., Shen M., Taelman L., Birkhäuser, Cham, 2021, 249–290 ; https://doi.org/10.1007/978-3-030-75421-1_10 | DOI
[78] A. Kuznetsov, Yu. Prokhorov, C. Shramov, “Hilbert schemes of lines and conics and automorphism groups of Fano threefolds”, Japanese J. Math., 13:1 (2018), 109–185 | DOI | Zbl
[79] R. Lazarsfeld, Positivity in algebraic geometry. {II}. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 49, Springer-Verlag, Berlin, 2004 | Zbl
[80] R. Lazarsfeld, “A short course on multiplier ideals”, Analytic and algebraic geometry. Common problems, different methods. Lecture notes from the Park City Mathematics Institute (PCMI) graduate summer school on analytic and algebraic geometry (Park City, UT, USA, Summer 2008), American Mathematical Society (AMS), Providence, RI, 2010, 451–494 ; http://arxiv.org/abs/0901.0651 | Zbl
[81] H. Matsumura, Commutative ring theory, Translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1986 | Zbl
[82] T. Matsusaka, “Polarized varieties with a given Hilbert polynomial”, Amer. J. Math., 94 (1972), 1027–1077 | DOI | Zbl
[83] V. B. Mehta, A. Ramanathan, “Semistable sheaves on projective varieties and their restriction to curves”, Math. Ann., 258 (1982), 213–224 | DOI | Zbl
[84] M. Mella, “Existence of good divisors on Mukai varieties”, J. Algebr. Geom., 8:2 (1999), 197–206 | Zbl
[85] B. G. Moishezon, “Über algebraische Homologieklassen auf algebraischen Mannigfaltigkeiten”, Izv. Akad. Nauk SSSR, Ser. Mat., 31 (1967), 225–268 | Zbl
[86] S. Mori, “On a generalization of complete intersections”, J. Math. Kyoto Univ., 15:3 (1975), 619–646 | Zbl
[87] S. Mori, “Threefolds whose canonical bundles are not numerically effective”, Ann. Math. (2), 116 (1982), 133–176 | DOI | Zbl
[88] S. Mori, S. Mukai, “Classification of Fano $3$-folds with $B\sb{2}\geq 2$”, Manuscripta Math., 36:2 (1981/82), 147–162 ; “Erratum”, Manuscripta Math., 110 (2003), 407 | DOI | Zbl | DOI
[89] S. Mori, S. Mukai, “On Fano $3$-folds with $B\sb{2}\geq 2$”, Algebraic varieties and analytic varieties, Tokyo, 1981, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, 101–129 | DOI
[90] S. Mukai, “Curves, $K3$ surfaces and Fano $3$-folds of genus $\leq 10$”, Algebraic geometry and commutative algebra, v. I, Kinokuniya, Tokyo, 1988, 357–377
[91] S. Mukai, “Biregular classification of Fano $3$-folds and Fano manifolds of coindex $3$”, Proc. Nat. Acad. Sci. U.S.A., 86:9 (1989), 3000–3002 | DOI | Zbl
[92] S. Mukai, “Fano $3$-folds”, Complex projective geometry, Trieste, 1989/Bergen, 1989, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992, 255–263
[93] S. Mukai, “Curves and symmetric spaces. {I}”, Amer. J. Math., 117:6 (1995), 1627–1644 | DOI | Zbl
[94] S. Mukai, “New developments in the theory of Fano threefolds: vector bundle method and moduli problems [translation of {S}ūgaku 47 (1995), no. 2, 125–144]”, Sugaku Expositions, 15:2 (2002), 125–150
[95] S. Mukai, H. Umemura, “Minimal rational threefolds”, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., 1016, Berlin, Springer, 490–518 | DOI | Zbl
[96] T. Peternell, J. A. Wiśniewski, “On stability of tangent bundles of Fano manifolds with $b\sb 2=1$”, J. Algebraic Geom., 4:2 (1995), 363–384 | Zbl
[97] Yu. Prokhorov, “Simple finite subgroups of the Cremona group of rank $3$”, J. Algebraic Geom., 21:3 (2012), 563–600 | DOI | Zbl
[98] Yu. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surv., 73:3 (2018), 375–456 | DOI | Zbl
[99] M. Reid, “Minimal models of canonical $3$-folds”, Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, 131–180 | DOI
[100] M. Reid, Projective morphisms according to Kawamata, Preprint. Univ. Warwick, 1983; http://www.maths.warwick.ac.uk/~miles/3folds
[101] B. Saint-Donat, “Projective models of $K3$ surfaces”, Amer. J. Math., 96 (1974), 602–639 | DOI | Zbl
[102] F. O. Schreyer, “Geometry and algebra of prime Fano $3$-folds of genus $12$”, Compositio Math., 127:3 (2001), 297–319 | DOI | Zbl
[103] N. I. Shepherd-Barron, “Fano threefolds in positive characteristic”, Compositio Math., 105:3 (1997), 237–265 | DOI | Zbl
[104] A. Steffens, “On the stability of the tangent bundle of Fano manifolds”, Math. Ann., 304:4 (1996), 635–643 | DOI | Zbl
[105] K. Takeuchi, “Some birational maps of Fano $3$-folds”, Compositio Math., 71:3 (1989), 265–283 | Zbl
[106] K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Notes written in collaboration with P. Cherenack, Lecture Notes in Mathematics, 439, Springer-Verlag, Berlin–New York, 1975 | DOI | Zbl
[107] P. M. H. Wilson, “Fano fourfolds of index greater than one”, J. Reine Angew. Math., 379 (1987), 172–181 | Zbl