Equivariant minimal model program
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 3, pp. 461-542 Cet article a éte moissonné depuis la source Math-Net.Ru

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The purpose of the survey is to systematize a vast amount of information about the minimal model program for varieties with group actions. We discuss the basic methods of the theory and give sketches of the proofs of some principal results. Bibliography: 243 titles.
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Yu. G. Prokhorov. Equivariant minimal model program. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 3, pp. 461-542. http://geodesic.mathdoc.fr/item/RM_2021_76_3_a2/

[1] S. S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, Pure Appl. Math., 24, Academic Press, New York–London, 1966, ix+291 pp. | MR | Zbl

[2] D. Abramovich, Jianhua Wang, “Equivariant resolution of singularities in characteristic $0$”, Math. Res. Lett., 4:2-3 (1997), 427–433 | DOI | MR | Zbl

[3] H. Ahmadinezhad, “On conjugacy classes of the Klein simple group in Cremona group”, Glasg. Math. J., 59:2 (2017), 395–400 | DOI | MR | Zbl

[4] H. Ahmadinezhad, I. Cheltsov, J. Park, C. Shramov, Double Veronese cones with 28 nodes, 2019, 47 pp., arXiv: 1910.10533

[5] H. Ahmadinezhad, M. Fedorchuk, I. Krylov, Stability of fibrations over one-dimensional bases, 2020 (v1 – 2019), 34 pp., arXiv: 1912.08779

[6] V. Alexeev, “General elephants of $\mathbb{Q}$-Fano 3-folds”, Compositio Math., 91:1 (1994), 91–116 | MR | Zbl

[7] F. Ambro, “Quasi-log varieties”, Biratsionalnaya geometriya: lineinye sistemy i konechno porozhdennye algebry, Sbornik statei, Trudy MIAN, 240, Nauka, MAIK “Nauka/Interperiodika”, M., 2003, 220–239 | MR | Zbl

[8] A. A. Avilov, “Suschestvovanie standartnykh modelei rassloenii na koniki nad algebraicheski nezamknutymi polyami”, Matem. sb., 205:12 (2014), 3–16 | DOI | MR | Zbl

[9] A. A. Avilov, “Avtomorfizmy osobykh trekhmernykh kubicheskikh giperpoverkhnostei i gruppa Kremony”, Matem. zametki, 100:3 (2016), 461–464 | DOI | MR | Zbl

[10] A. A. Avilov, “Avtomorfizmy trekhmernykh mnogoobrazii, predstavimykh v vide peresecheniya dvukh kvadrik”, Matem. sb., 207:3 (2016), 3–18 | DOI | MR | Zbl

[11] A. Avilov, “Automorphisms of singular three-dimensional cubic hypersurfaces”, Eur. J. Math., 4:3 (2018), 761–777 | DOI | MR | Zbl

[12] A. A. Avilov, “Biregulyarnaya i biratsionalnaya geometriya dvoinykh nakrytii proektivnogo prostranstva s vetvleniem v kvartike s 15 obyknovennymi dvoinymi tochkami”, Izv. RAN. Ser. matem., 83:3 (2019), 5–14 | DOI | MR | Zbl

[13] H. F. Baker, A locus with 25920 linear self-transformations, Cambridge Tracts in Math. and Math. Phys., 39, Cambridge Univ. Press, Cambridge; The Macmillan Company, New York, 1946, xi+107 pp. | MR | Zbl

[14] T. Bandman, Yu. G. Zarhin, “Jordan groups, conic bundles and abelian varieties”, Algebr. Geom., 4:2 (2017), 229–246 | DOI | MR | Zbl

[15] T. Bandman, Yu. G. Zarhin, “Jordan properties of automorphism groups of certain open algebraic varieties”, Transform. Groups, 24:3 (2019), 721–739 | DOI | MR | Zbl

[16] T. Bandman and Yu. G. Zarhin, “Bimeromorphic automorphism groups of certain $\mathbb{P}^1$-bundles”, Eur. J. Math., 7 (2021), 641–670 | DOI | MR

[17] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), 4, Springer-Verlag, Berlin, 1984, x+304 pp. | DOI | MR | Zbl

[18] L. Bayle, “Classification des variétés complexes projectives de dimension trois dont une section hyperplane générale est une surface d'Enriques”, J. Reine Angew. Math., 1994:449 (1994), 9–63 | DOI | MR | Zbl

[19] A. Beauville, “Non-rationality of the symmetric sextic Fano threefold”, Geometry and arithmetic, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, 57–60 | DOI | MR | Zbl

[20] E. Bierstone, P. D. Milman, “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant”, Invent. Math., 128:2 (1997), 207–302 | DOI | MR | Zbl

[21] E. Bierstone, P. D. Milman, “Functoriality in resolution of singularities”, Publ. Res. Inst. Math. Sci., 44:2 (2008), 609–639 | DOI | MR | Zbl

[22] C. Birkar, “Existence of log canonical flips and a special LMMP”, Publ. Math. Inst. Hautes Études Sci., 115 (2012), 325–368 | DOI | MR | Zbl

[23] C. Birkar, “Existence of flips and minimal models for 3-folds in char $p$”, Ann. Sci. Éc. Norm. Supér. (4), 49:1 (2016), 169–212 | DOI | MR | Zbl

[24] C. Birkar, “Singularities of linear systems and boundedness of Fano varieties”, Ann. of Math. (2), 193:2 (2021), 347–405 | DOI | MR | Zbl

[25] C. Birkar, P. Cascini, C. D. Hacon, J. McKernan, “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc., 23:2 (2010), 405–468 | DOI | MR | Zbl

[26] C. Birkar, J. Waldron, “Existence of {M}ori fibre spaces for 3-folds in $\operatorname{char}p$”, Adv. Math., 313 (2017), 62–101 | DOI | MR | Zbl

[27] J. Blanc, “Linearisation of finite Abelian subgroups of the Cremona group of the plane”, Groups Geom. Dyn., 3:2 (2009), 215–266 | DOI | MR | Zbl

[28] J. Blanc, “Algebraic structures of groups of birational transformations”, Algebraic groups: structure and actions, Proc. Sympos. Pure Math., 94, Amer. Math. Soc., Providence, RI, 2017, 17–30 | MR | Zbl

[29] J. Blanc, I. Cheltsov, A. Duncan, Yu. Prokhorov, Finite quasisimple groups acting on rationally connected threefolds, 2018, 41 pp., arXiv: 1809.09226

[30] J. Blanc, A. Fanelli, R. Terpereau, Connected algebraic groups acting on $3$-dimensional Mori fibrations, 2021 (v1 – 2019), 83 pp., arXiv: 1912.11364

[31] J. Blanc, J.-P. Furter, “Topologies and structures of the Cremona groups”, Ann. of Math. (2), 178:3 (2013), 1173–1198 | DOI | MR | Zbl

[32] J. Blanc, S. Lamy, S. Zimmermann, Quotients of higher dimensional Cremona groups, 2019, 80 pp., arXiv: 1901.04145

[33] J. Blanc, E. Yasinsky, “Quotients of groups of birational transformations of cubic Del Pezzo fibrations”, J. Éc. Polytech. Math., 7 (2020), 1089–1112 ; (2020 (v1 – 2019)), 26 pp., arXiv: 1907.04696 | DOI | MR | Zbl

[34] H. F. Blichfeldt, Finite collineation groups, Univ. Chicago Press, Chicago, IL, 1917, xi+194 pp.

[35] F. A. Bogomolov, “Golomorfnye tenzory i vektornye rassloeniya na proektivnykh mnogoobraziyakh”, Izv. AN SSSR. Ser. matem., 42:6 (1978), 1227–1287 ; F. A. Bogomolov, “Holomorphic tensors and vector bundles on projective varieties”, Math. USSR-Izv., 13:3 (1979), 499–555 | MR | Zbl | DOI

[36] G. Brown et al., Graded Ring Database http://www.grdb.co.uk

[37] F. Campana, “Connexité rationnelle des variétés de Fano”, Ann. Sci. École Norm. Sup. (4), 25:5 (1992), 539–545 | DOI | MR | Zbl

[38] F. Campana, H. Flenner, “Projective threefolds containing a smooth rational surface with ample normal bundle”, J. Reine Angew. Math., 1993:440 (1993), 77–98 | DOI | MR | Zbl

[39] F. Campana, J. Winkelmann, “Rational connectedness and order of non-degenerate meromorphic maps from $\mathbb C^n$”, Eur. J. Math., 2:1 (2016), 87–95 | DOI | MR | Zbl

[40] J. Caravantes, “Low codimension Fano–Enriques threefolds”, Note Mat., 28:2 (2008), 117–147 | DOI | MR | Zbl

[41] I. Cheltsov, A. Dubouloz, T. Kishimoto, Toric $G$-solid Fano threefolds, 2020, 29 pp., arXiv: 2007.14197

[42] I. Cheltsov, V. Przyjalkowski, C. Shramov, “Quartic double solids with icosahedral symmetry”, Eur. J. Math., 2:1 (2016), 96–119 | DOI | MR | Zbl

[43] I. Cheltsov, C. Shramov, “Three embeddings of the Klein simple group into the Cremona group of rank three”, Transform. Groups, 17:2 (2012), 303–350 | DOI | MR | Zbl

[44] I. Cheltsov, C. Shramov, “Five embeddings of one simple group”, Trans. Amer. Math. Soc., 366:3 (2014), 1289–1331 | DOI | MR | Zbl

[45] I. Cheltsov, C. Shramov, Cremona groups and the icosahedron, Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2016, xxi+504 pp. | DOI | MR | Zbl

[46] I. Cheltsov, C. Shramov, “Two rational nodal quartic $3$-folds”, Q. J. Math., 67:4 (2016), 573–601 | DOI | MR | Zbl

[47] C. Chevalley, “Invariants of finite groups generated by reflections”, Amer. J. Math., 77:4 (1955), 778–782 | DOI | MR | Zbl

[48] C. H. Clemens, P. A. Griffiths, “The intermediate Jacobian of the cubic threefold”, Ann. of Math. (2), 95:2 (1972), 281–356 | DOI | MR | Zbl

[49] A. Conte, J. P. Murre, “Algebraic varieties of dimension three whose hyperplane sections are Enriques surfaces”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 12:1 (1985), 43–80 | MR | Zbl

[50] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, With comput. assistance from J. G. Thackray, Oxford Univ. Press, Eynsham, 1985, xxxiv+252 pp. | MR | Zbl

[51] A. Corti, “Factoring birational maps of threefolds after Sarkisov”, J. Algebraic Geom., 4:2 (1995), 223–254 | MR | Zbl

[52] A. Corti, “Del Pezzo surfaces over Dedekind schemes”, Ann. of Math. (2), 144:3 (1996), 641–683 | DOI | MR | Zbl

[53] A. Corti (ed.), Flips for $3$-folds and $4$-folds, Oxford Lecture Ser. Math. Appl., 35, Oxford Univ. Press, Oxford, 2007, x+189 pp. | DOI | MR | Zbl

[54] V. Cossart, O. Piltant, “Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings”, J. Algebra, 320:3 (2008), 1051–1082 | DOI | MR | Zbl

[55] V. Cossart, O. Piltant, “Resolution of singularities of threefolds in positive characteristic. II”, J. Algebra, 321:7 (2009), 1836–1976 | DOI | MR | Zbl

[56] Ch. Kertis, I. Rainer, Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Nauka, M., 1969, 668 pp. ; C. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Pure Appl. Math., XI, Interscience Publishers, a division of John Wiley Sons, New York–London, 1962, xiv+685 pp. | MR | MR | Zbl

[57] S. Cutkosky, “Elementary contractions of Gorenstein threefolds”, Math. Ann., 280:3 (1988), 521–525 | DOI | MR | Zbl

[58] V. I. Danilov, “Biratsionalnaya geometriya toricheskikh mnogoobrazii”, Izv. AN SSSR. Ser. matem., 46:5 (1982), 971–982 ; V. I. Danilov, “Birational geometry of toric 3-folds”, Math. USSR-Izv., 21:2 (1983), 269–280 | MR | Zbl | DOI

[59] O. Das, C. Hacon, The log minimal model program for Kähler $3$-folds, 2020, 50 pp., arXiv: 2009.05924

[60] O. Das, J. Waldron, On the log minimal model program for $3$-folds over imperfect fields of characteristic $p>5$, 2019, 42 pp., arXiv: 1911.04394

[61] O. Debarre, A. Kuznetsov, “Gushel–Mukai varieties: classification and birationalities”, Algebr. Geom., 5:1 (2018), 15–76 | DOI | MR | Zbl

[62] M. Demazure, “Sous-groupes algébriques de rang maximum du groupe de Cremona”, Ann. Sci. École Norm. Sup. (4), 3:4 (1970), 507–588 | DOI | MR | Zbl

[63] J. D. Dixon, B. Mortimer, Permutation groups, Grad. Texts in Math., 163, Springer-Verlag, New York, 1996, xii+346 pp. | DOI | MR | Zbl

[64] I. V. Dolgachev, “On elements of order $p^s$ in the plane Cremona group over a field of characteristic $p$”, Mnogomernaya algebraicheskaya geometriya, Sbornik statei. Posvyaschaetsya pamyati chlena-korrespondenta RAN Vasiliya Alekseevicha Iskovskikh, Trudy MIAN, 264, MAIK “Nauka/Interperiodika”, M., 2009, 55–62 | MR | Zbl

[65] I. V. Dolgachev, “Finite subgroups of the plane Cremona group”, Algebraic geometry in East Asia – Seoul 2008, Adv. Stud. Pure Math., 60, Math. Soc. Japan, Tokyo, 2010, 1–49 | DOI | MR | Zbl

[66] I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge Univ. Press, Cambridge, 2012, xii+639 pp. | DOI | MR | Zbl

[67] I. Dolgachev, A. Duncan, “Automorphisms of cubic surfaces in positive characteristic”, Izv. RAN. Ser. matem., 83:3 (2019), 15–92 | DOI | MR | Zbl

[68] I. V. Dolgachev, V. A. Iskovskikh, “Finite subgroups of the plane Cremona group”, Algebra, arithmetic, and geometry, In honor of Yu. I. Manin, v. I, Progr. Math., 269, Birkhäuser Boston, Boston, MA, 2009, 443–548 | DOI | MR | Zbl

[69] A. Dubouloz, S. Lamy, “Automorphisms of open surfaces with irreducible boundary”, Osaka J. Math., 52:3 (2015), 747–791 | MR | Zbl

[70] A. H. Durfee, “Fifteen characterizations of rational double points and simple critical points”, Enseign. Math. (2), 25:1-2 (1979), 131–163 | MR | Zbl

[71] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995, xvi+785 pp. | DOI | MR | Zbl

[72] F. Enriques, G. Fano, “Sui gruppi continui di trasformazioni Cremoniane dello spazio”, Ann. Mat. Pura Appl. (2), 26 (1898), 59–98 | DOI | Zbl

[73] G. Fano, “Sulle varietà algebriche a tre dimensioni le cui sezioni iperpiane sono superficie di genere zero e bigenere uno”, Mem. Mat. Sci. Fis. Natur. Soc. Ital. Sci. (3), 24 (1938), 41–66 | Zbl

[74] W. Feit, “The current situation in the theory of finite simple groups”, Actes du Congrès international des mathématiciens (Nice, 1970), v. 1, Gauthier-Villars, Paris, 1971, 55–93 | MR | Zbl

[75] A. R. Fletcher, “Contributions to Riemann–Roch on projective $3$-folds with only canonical singularities and applications”, Algebraic geometry – Bowdoin 1985, Part 1 (Brunswick, ME, 1985), Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, 221–231 | DOI | MR | Zbl

[76] A. Fujiki, “Closedness of the Douady spaces of compact Kähler spaces”, Publ. Res. Inst. Math. Sci., 14:1 (1978/79), 1–52 | DOI | MR | Zbl

[77] O. Fujino, “Applications of Kawamata's positivity theorem”, Proc. Japan Acad. Ser. A Math. Sci., 75:6 (1999), 75–79 | DOI | MR | Zbl

[78] O. Fujino, “Fundamental theorems for the log minimal model program”, Publ. Res. Inst. Math. Sci., 47:3 (2011), 727–789 | DOI | MR | Zbl

[79] T. Fujita, “On the structure of polarized manifolds with total deficiency one. I”, J. Math. Soc. Japan, 32:4 (1980), 709–725 ; II, 33:3 (1981), 415–434 ; III, 36:1 (1984), 75–89 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[80] T. Fujita, “Projective varieties of $\Delta$-genus one”, Algebraic and topological theories (Kinosaki, 1984), Kinokuniya, Tokyo, 1986, 149–175 | MR | Zbl

[81] T. Fujita, “On singular Del Pezzo varieties”, Algebraic geometry (L'Aquila, 1988), Lecture Notes in Math., 1417, Springer, Berlin, 1990, 117–128 | DOI | MR | Zbl

[82] T. Graber, J. Harris, J. Starr, “Families of rationally connected varieties”, J. Amer. Math. Soc., 16:1 (2003), 57–67 | DOI | MR | Zbl

[83] D. Greb, S. Kebekus, S. J. Kovács, T. Peternell, “Differential forms on log canonical spaces”, Publ. Math. Inst. Hautes Études Sci., 114 (2011), 87–169 | DOI | MR | Zbl

[84] C. D. Hacon, J. McKernan, “The Sarkisov program”, J. Algebraic Geom., 22:2 (2013), 389–405 | DOI | MR | Zbl

[85] C. Hacon, J. Witaszek, The minimal model program for threefolds in characteristic five, 2019, 36 pp., arXiv: 1911.12895

[86] C. Hacon, J. Witaszek, On the relative minimal model program for threefolds in low characteristics, 2020 (v1 – 2019), 17 pp., arXiv: 1909.12872

[87] C. D. Hacon, Chenyang Xu, “Existence of log canonical closures”, Invent. Math., 192:1 (2013), 161–195 | DOI | MR | Zbl

[88] C. D. Hacon, Chenyang Xu, “On the three dimensional minimal model program in positive characteristic”, J. Amer. Math. Soc., 28:3 (2015), 711–744 | DOI | MR | Zbl

[89] R. Khartskhorn, Algebraicheskaya geometriya, Mir, M., 1981, 600 pp. ; R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, Springer-Verlag, New York–Heidelberg, 1977, xvi+496 pp. | MR | Zbl | DOI | MR | Zbl

[90] K. Hashizume, Y. Nakamura, H. Tanaka, “Minimal model program for log canonical threefolds in positive characteristic”, Math. Res. Lett., 27:4 (2020), 1003–1054 ; (2019 (v1 – 2017)), 38 pp., arXiv: 1711.10706 | DOI | MR | Zbl

[91] A. Höring, T. Peternell, “Mori fibre spaces for Kähler threefolds”, J. Math. Sci. Univ. Tokyo, 22:1 (2015), 219–246 | MR | Zbl

[92] A. Höring, T. Peternell, “Minimal models for Kähler threefolds”, Invent. Math., 203:1 (2016), 217–264 | DOI | MR | Zbl

[93] B. Hunt, The geometry of some special arithmetic quotients, Lecture Notes in Math., 1637, Springer-Verlag, Berlin, 1996, xiv+332 pp. | DOI | MR | Zbl

[94] D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves, Aspects Math., E31, Friedr. Vieweg Sohn, Braunschweig, 1997, xiv+269 pp. | MR | Zbl

[95] A. R. Iano-Fletcher, “Working with weighted complete intersections”, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 101–173 | DOI | MR | Zbl

[96] S. Ishii, Introduction to singularities, Springer, Tokyo, 2014, viii+223 pp. | DOI | MR | Zbl

[97] V. A. Iskovskikh, “Ratsionalnye poverkhnosti s puchkom ratsionalnykh krivykh”, Matem. sb., 74(116):4 (1967), 608–638 | MR | Zbl

[98] V. A. Iskovskikh, “Ratsionalnye poverkhnosti s puchkom ratsionalnykh krivykh i s polozhitelnym kvadratom kanonicheskogo klassa”, Matem. sb., 83(125):1(9) (1970), 90–119 | MR | Zbl

[99] V. A. Iskovskikh, “Trekhmernye mnogoobraziya Fano. I”, Izv. AN SSSR. Ser. matem., 41:3 (1977), 516–562 | MR | Zbl

[100] V. A. Iskovskikh, “Trekhmernye mnogoobraziya Fano. II”, Izv. AN SSSR. Ser. matem., 42:3 (1978), 506–549 | MR | Zbl

[101] V. A. Iskovskikh, “Antikanonicheskie modeli trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem., 12, VINITI, M., 1979, 59–157 ; V. A. Iskovskikh, “Anticanonical models of three-dimensional algebraic varieties”, J. Soviet Math., 13:6 (1980), 745–814 | MR | Zbl | DOI

[102] V. A. Iskovskikh, “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem., 12, VINITI, M., 1979, 159–236 ; V. A. Iskovskikh, “Birational automorphisms of three-dimensional algebraic varieties”, J. Soviet Math., 13:6 (1980), 815–868 | MR | Zbl | DOI

[103] V. A. Iskovskikh, “Minimalnye modeli ratsionalnykh poverkhnostei nad proizvolnymi polyami”, Izv. AN SSSR. Ser. matem., 43:1 (1979), 19–43 ; V. A. Iskovskih, “Minimal models of rational surfaces over arbitrary fields”, Math. USSR-Izv., 14:1 (1980), 17–39 | MR | Zbl | DOI

[104] V. A. Iskovskikh, “Faktorizatsiya biratsionalnykh otobrazhenii ratsionalnykh poverkhnostei s tochki zreniya teorii Mori”, UMN, 51:4(310) (1996), 3–72 | DOI | MR | Zbl

[105] V. A. Iskovskikh, Yu. G. Prokhorov, “Fano varieties”, Algebraic geometry V, Encyclopaedia Math. Sci., 47, Springer, Berlin, 1999, 1–247 | MR | Zbl

[106] V. A. Iskovskikh, V. V. Shokurov, “Biratsionalnye modeli i perestroiki”, UMN, 60:1(361) (2005), 29–98 | DOI | MR | Zbl

[107] P. Jahnke, I. Radloff, “Terminal Fano threefolds and their smoothings”, Math. Z., 269:3-4 (2011), 1129–1136 | DOI | MR | Zbl

[108] M. C. Jordan, “Mémoire sur les équations différentielles linéaires à intégrale algébrique”, J. Reine Angew. Math., 1878:84 (1878), 89–215 | DOI | MR | Zbl

[109] Y. Kachi, “Extremal contractions from $4$-dimensional manifolds to $3$-folds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24:1 (1997), 63–131 | MR | Zbl

[110] A.-S. Kaloghiros, “The defect of Fano $3$-folds”, J. Algebraic Geom., 20:1 (2011), 127–149 ; “Errata”, 21:2 (2012), 397–399 | DOI | MR | Zbl | DOI | MR | Zbl

[111] M. Kawakita, “Divisorial contractions in dimension three which contract divisors to smooth points”, Invent. Math., 145:1 (2001), 105–119 | DOI | MR | Zbl

[112] M. Kawakita, “Divisorial contractions in dimension three which contract divisors to compound $A_1$ points”, Compositio Math., 133:1 (2002), 95–116 | DOI | MR | Zbl

[113] M. Kawakita, “General elephants of three-fold divisorial contractions”, J. Amer. Math. Soc., 16:2 (2003), 331–362 | DOI | MR | Zbl

[114] M. Kawakita, “Three-fold divisorial contractions to singularities of higher indices”, Duke Math. J., 130:1 (2005), 57–126 | DOI | MR | Zbl

[115] Y. Kawamata, “The cone of curves of algebraic varieties”, Ann. of Math. (2), 119:3 (1984), 603–633 | DOI | MR | Zbl

[116] Y. Kawamata, “Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces”, Ann. of Math. (2), 127:1 (1988), 93–163 | DOI | MR | Zbl

[117] Y. Kawamata, “On the length of an extremal rational curve”, Invent. Math., 105:3 (1991), 609–611 | DOI | MR | Zbl

[118] Y. Kawamata, “Boundedness of $\mathbf{Q}$-Fano threefolds”, Proceedings of the international conference on algebra, Part 3 (Novosibirsk, 1989), Contemp. Math., 131, Part 3, Amer. Math. Soc., Providence, RI, 1992, 439–445 | DOI | MR | Zbl

[119] Y. Kawamata, “Termination of log flips for algebraic $3$-folds”, Internat. J. Math., 3:5 (1992), 653–659 | DOI | MR | Zbl

[120] Y. Kawamata, “The minimal discrepancy coefficients of terminal singularities in dimension three”, Appendix to V. V. Shokurov's paper "$3$-fold log flips", Izv. RAN. Ser. matem., 56:1 (1992), 201–203 ; Russian Acad. Sci. Izv. Math., 40:1 (1993), 193–195 | MR | Zbl | DOI

[121] Y. Kawamata, “Divisorial contractions to $3$-dimensional terminal quotient singularities”, Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin, 1996, 241–246 | DOI | MR | Zbl

[122] 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 | MR | Zbl

[123] Y. Kawamata, “Flops connect minimal models”, Publ. Res. Inst. Math. Sci., 44:2 (2008), 419–423 | DOI | MR | Zbl

[124] 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 | MR | Zbl

[125] S. L. Kleiman, “Toward a numerical theory of ampleness”, Ann. of Math. (2), 84:3 (1966), 293–344 | DOI | MR | Zbl

[126] J. Kollár, “Flops”, Nagoya Math. J., 113 (1989), 15–36 | DOI | MR | Zbl

[127] J. Kollár, “Flips, flops, minimal models, etc.”, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 113–199 | MR | Zbl

[128] J. Kollár (ed.), Flips and abundance for algebraic threefolds, Papers from the 2nd summer seminar on algebraic geometry (Univ. of Utah, Salt Lake City, 1991), Astérisque, 211, Soc. Math. France, Paris, 1992, 258 pp. | MR | Zbl

[129] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996, viii+320 pp. | DOI | MR | Zbl

[130] J. Kollár, “Polynomials with integral coefficients, equivalent to a given polynomial”, Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 17–27 | DOI | MR | Zbl

[131] J. Kollár, “Singularities of pairs”, Algebraic geometry – Santa Cruz 1995, Part 1, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997, 221–287 | DOI | MR | Zbl

[132] J. Kollár, Lectures on resolution of singularities, Ann. of Math. Stud., 166, Princeton Univ. Press, Princeton, NJ, 2007, vi+208 pp. | MR | Zbl

[133] J. Kollár, Singularities of the minimal model program, With the collaboration of S. Kovács, Cambridge Tracts in Math., 200, Cambridge Univ. Press, Cambridge, 2013, x+370 pp. | DOI | MR | Zbl

[134] J. Kollár, S. J. Kovács, “Log canonical singularities are Du Bois”, J. Amer. Math. Soc., 23:3 (2010), 791–813 | DOI | MR | Zbl

[135] J. Kollár, Y. Miyaoka, S. Mori, H. Takagi, “Boundedness of canonical $\mathbf{Q}$-Fano 3-folds”, Proc. Japan Acad. Ser. A Math. Sci., 76:5 (2000), 73–77 | DOI | MR | Zbl

[136] J. Kollár, S. Mori, “Classification of three-dimensional flips”, J. Amer. Math. Soc., 5:3 (1992), 533–703 ; S. Mori, “Errata”, J. Amer. Math. Soc., 20:1 (2007), 269–271 | DOI | MR | Zbl | DOI | MR | Zbl

[137] J. Kollár, S. Mori, Birational geometry of algebraic varieties, With the collaboration of C. H. Clemens and A. Corti, transl. from the 1998 Japan. original, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998, viii+254 pp. | DOI | MR | Zbl

[138] J. Kollár, N. I. Shepherd-Barron, “Threefolds and deformations of surface singularities”, Invent. Math., 91:2 (1988), 299–338 | DOI | MR | Zbl

[139] I. Krylov, “Birational geometry of del {P}ezzo fibrations with terminal quotient singularities”, J. Lond. Math. Soc. (2), 97:2 (2018), 222–246 | DOI | MR | Zbl

[140] I. Krylov, Families of embeddings of the alternating group of rank 5 into the Cremona group, 2020, 15 pp., arXiv: 2005.07354

[141] A. Kuznetsov, Yu. Prokhorov, “Prime Fano threefolds of genus 12 with a $\mathbb G_m$-action”, Épijournal Géom. Algébrique, 2 (2018), 3, 14 pp. | DOI | MR | Zbl

[142] A. Kuznetsov, Yu. Prokhorov, Rationality over non-closed fields of Fano threefolds with higher geometric Picard rank, 2021, 34 pp., arXiv: 2103.02934

[143] A. G. Kuznetsov, Yu. G. Prokhorov, C. A. Shramov, “Hilbert schemes of lines and conics and automorphism groups of Fano threefolds”, Jpn. J. Math., 13:1 (2018), 109–185 | DOI | MR | Zbl

[144] A. A. Kuznetsova, “Konechnye $3$-podgruppy v gruppe Kremony ranga 3”, Matem. zametki, 108:5 (2020), 725–749 | DOI | MR | Zbl

[145] Ching-Jui Lai, “Varieties fibered by good minimal models”, Math. Ann., 350:3 (2011), 533–547 | DOI | MR | Zbl

[146] Hsueh-Yung Lin, Algebraic approximations of compact Kähler threefolds, 2018 (v1 – 2017), 42 pp., arXiv: 1710.01083v2

[147] J. Lipman, “Rational singularities, with applications to algebraic surfaces and unique factorization”, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 195–279 | DOI | MR | Zbl

[148] K. Loginov, “Standard models of degree 1 del Pezzo fibrations”, Mosc. Math. J., 18:4 (2018), 721–737 | DOI | MR | Zbl

[149] Yu. I. Manin, “Ratsionalnye poverkhnosti nad sovershennymi polyami”, Inst. Hautes Études Sci. Publ. Math., 30 (1966), 55–97 ; Yu. I. Manin, “Rational surfaces over perfect fields. I”, Twelve papers on algebra, algebraic geometry and topology, Amer. Math. Soc. Transl. Ser. 2, 84, Amer. Math. Soc., Providence, RI, 1969, 137–186 | DOI | MR | Zbl | DOI | Zbl

[150] Yu. I. Manin, “Ratsionalnye poverkhnosti nad sovershennymi polyami. II”, Matem. sb., 72(114):2 (1967), 161–192 | MR | Zbl

[151] Yu. I. Manin, Kubicheskie formy: algebra, geometriya, arifmetika, Nauka, M., 1972, 304 pp. ; Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic, North-Holland Math. Library, 4, North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., New York, 1974, vii+292 pp. | MR | Zbl | MR | Zbl

[152] D. Markushevich, “Minimal discrepancy for a terminal cDV singularity is 1”, J. Math. Sci. Univ. Tokyo, 3:2 (1996), 445–456 | MR | Zbl

[153] K. Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, New York, 2002, xxiv+478 pp. | DOI | MR | Zbl

[154] H. Matsumura, Commutative ring theory, Transl. from the Japan. by M. Reid, Cambridge Stud. Adv. Math., 8, Cambridge Univ. Press, Cambridge, 1986, xiv+320 pp. | DOI | MR | Zbl

[155] Dzh. Milnor, Osobye tochki kompleksnykh giperpoverkhnostei, Mir, M., 1971, 127 pp. ; J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Stud., 61, Princeton Univ. Press, Princeton, NJ; Univ. of Tokyo Press, Tokyo, 1968, iii+122 pp. | MR | Zbl | DOI | MR | Zbl

[156] Y. Miyaoka, S. Mori, “A numerical criterion for uniruledness”, Ann. of Math. (2), 124:1 (1986), 65–69 | DOI | MR | Zbl

[157] S. Mori, “Threefolds whose canonical bundles are not numerically effective”, Ann. of Math. (2), 116:1 (1982), 133–176 | DOI | MR | Zbl

[158] S. Mori, “On $3$-dimensional terminal singularities”, Nagoya Math. J., 98 (1985), 43–66 | DOI | MR | Zbl

[159] S. Mori, “Flip theorem and the existence of minimal models for $3$-folds”, J. Amer. Math. Soc., 1:1 (1988), 117–253 | DOI | MR | Zbl

[160] S. Mori, S. Mukai, “Classification of Fano $3$-folds with $B_{2}\geqslant 2$”, Manuscripta Math., 36:2 (1981/82), 147–162 ; “Erratum”, 110:3 (2003), 407 | DOI | MR | Zbl | DOI | MR

[161] S. Mori, S. Mukai, “Extremal rays and Fano 3-folds”, The Fano conference, Univ. Torino, Turin, 2004, 37–50 | MR | Zbl

[162] S. Mori, Yu. Prokhorov, “On $\mathbb Q$-conic bundles”, Publ. Res. Inst. Math. Sci., 44:2 (2008), 315–369 | DOI | MR | Zbl

[163] S. Mori, Yu. Prokhorov, “On $\mathbb Q$-conic bundles. II”, Publ. Res. Inst. Math. Sci., 44:3 (2008), 955–971 | DOI | MR | Zbl

[164] S. Mori, Yu. Prokhorov, “On $\mathbb Q$-conic bundles. III”, Publ. Res. Inst. Math. Sci., 45:3 (2009), 787–810 | DOI | MR | Zbl

[165] S. Mori, Yu. G. Prokhorov, “Multiple fibers of del Pezzo fibrations”, Mnogomernaya algebraicheskaya geometriya, Sbornik statei. Posvyaschaetsya pamyati chlena-korrespondenta RAN Vasiliya Alekseevicha Iskovskikh, Trudy MIAN, 264, MAIK “Nauka/Interperiodika”, M., 2009, 137–151 | MR | Zbl

[166] S. Mori, Yu. Prokhorov, “Threefold extremal contractions of type (IA)”, Kyoto J. Math., 51:2 (2011), 393–438 | DOI | MR | Zbl

[167] S. Mori, Yu. Prokhorov, “3-fold extremal contractions of types (IC) and (IIB)”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 231–252 | DOI | MR | Zbl

[168] S. Mori, Yu. G. Prokhorov, “Threefold extremal contractions of type (IIA). I”, Izv. RAN. Ser. matem., 80:5 (2016), 77–102 | DOI | MR | Zbl

[169] S. Mori, Yu. Prokhorov, “Threefold extremal contractions of type (IIA). Part II”, Geometry and physics, v. 2, Oxford Univ. Press, Oxford, 2018, 623–652 | DOI | MR | Zbl

[170] Sh. Mori, Yu. G. Prokhorov, “Trekhmernye ekstremalnye okrestnosti krivoi s odnoi negorenshteinovoi tochkoi”, Izv. RAN. Ser. matem., 83:3 (2019), 158–212 | DOI | MR | Zbl

[171] Sh. Mori, Yu. G. Prokhorov, “Obschii antikanonicheskii element dlya trekhmernykh ekstremalnykh styagivanii s odnomernymi sloyami: isklyuchitelnyi sluchai”, Matem. sb., 212:3 (2021), 88–111 | DOI | MR

[172] D. R. Morrison, G. Stevens, “Terminal quotient singularities in dimensions three and four”, Proc. Amer. Math. Soc., 90:1 (1984), 15–20 | DOI | MR | Zbl

[173] S. Mukai, “Finite groups of automorphisms of K3 surfaces and the Mathieu group”, Invent. Math., 94:1 (1988), 183–221 | DOI | MR | Zbl

[174] S. Mukai, H. Umemura, “Minimal rational threefolds”, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., 1016, Springer, Berlin, 1983, 490–518 | DOI | MR | Zbl

[175] T. Nakano, “On equivariant completions of 3-dimensional homogeneous spaces of $\operatorname{SL}(2,\mathbf{C})$”, Japan. J. Math. (N. S.), 15:2 (1989), 221–273 | DOI | MR | Zbl

[176] T. Nakano, “Projective threefolds on which $\operatorname{SL}(2)$ acts with $2$-dimensional general orbits”, Trans. Amer. Math. Soc., 350:7 (1998), 2903–2924 | DOI | MR | Zbl

[177] N. Nakayama, “The lower semi-continuity of the plurigenera of complex varieties”, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987, 551–590 | DOI | MR | Zbl

[178] N. Nakayama, “The singularity of the canonical model of compact Kähler manifolds”, Math. Ann., 280:3 (1988), 509–512 | DOI | MR | Zbl

[179] Y. Namikawa, “Smoothing Fano 3-folds”, J. Algebraic Geom., 6:2 (1997), 307–324 | MR | Zbl

[180] V. V. Nikulin, “Konechnye gruppy avtomorfizmov kelerovykh poverkhnostei tipa $K_3$”, Tr. MMO, 38, Izd-vo Mosk. un-ta, M., 1979, 75–137 ; V. V. Nikulin, “Finite automorphism groups of Kähler K3 surfaces”, Trans. Moscow Math. Soc., 2 (1980), 71–135 | MR | Zbl

[181] T. Okada, “Nonrational del Pezzo fibrations admitting an action of the Klein simple group”, Eur. J. Math., 2:1 (2016), 319–332 | DOI | MR | Zbl

[182] Yu. M. Polyakova, “Semeistvo kategorii logterminalnykh par i avtomorfizmy poverkhnostei”, Izv. RAN. Ser. matem., 74:3 (2010), 103–156 | DOI | MR | Zbl

[183] V. L. Popov, “On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties”, Affine algebraic geometry. The Russell festschrift (McGill Univ., Montreal, QC, 2009), CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011, 289–311 | DOI | MR | Zbl

[184] V. L. Popov, “Tri syuzheta o gruppakh Kremony”, Izv. RAN. Ser. matem., 83:4 (2019), 194–225 | DOI | MR | Zbl

[185] Yu. G. Prokhorov, “O dopolnyaemosti kanonicheskogo divizora dlya rassloenii Mori na koniki”, Matem. sb., 188:11 (1997), 99–120 | DOI | MR | Zbl

[186] Yu. G. Prokhorov, Osobennosti algebraicheskikh mnogoobrazii, MTsNMO, M., 2009, 128 pp.

[187] Yu. Prokhorov, “$\mathbb Q$-Fano threefolds of large Fano index. I”, Doc. Math., 15 (2010), 843–872 | MR | Zbl

[188] Yu. Prokhorov, “$p$-elementary subgroups of the Cremona group of rank 3”, Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, 327–338 | DOI | MR | Zbl

[189] Yu. Prokhorov, “Simple finite subgroups of the Cremona group of rank 3”, J. Algebraic Geom., 21:3 (2012), 563–600 | DOI | MR | Zbl

[190] Yu. Prokhorov, “$G$-Fano threefolds. I”, Adv. Geom., 13:3 (2013), 389–418 | DOI | MR | Zbl

[191] Yu. Prokhorov, “$G$-Fano threefolds. II”, Adv. Geom., 13:3 (2013), 419–434 | DOI | MR | Zbl

[192] Yu. Prokhorov, “2-elementary subgroups of the space Cremona group”, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., 79, Springer, Cham, 2014, 215–229 | DOI | MR | Zbl

[193] Yu. G. Prokhorov, “O trekhmernykh $G$-mnogoobraziyakh Fano”, Izv. RAN. Ser. matem., 79:4 (2015), 159–174 | DOI | MR | Zbl

[194] Yu. G. Prokhorov, “Osobye mnogoobraziya Fano roda 12”, Matem. sb., 207:7 (2016), 101–130 | DOI | MR | Zbl

[195] Yu. G. Prokhorov, “O chisle osobykh tochek trekhmernykh terminalnykh faktorialnykh mnogoobrazii Fano”, Matem. zametki, 101:6 (2017), 949–954 | DOI | MR | Zbl

[196] Yu. G. Prokhorov, “Problema ratsionalnosti dlya rassloenii na koniki”, UMN, 73:3(441) (2018), 3–88 | DOI | MR | Zbl

[197] Yu. Prokhorov, C. Shramov, “Jordan property for groups of birational selfmaps”, Compositio Math., 150:12 (2014), 2054–2072 | DOI | MR | Zbl

[198] Yu. Prokhorov, C. Shramov, “Jordan property for Cremona groups”, Amer. J. Math., 138:2 (2016), 403–418 | DOI | MR | Zbl

[199] Yu. Prokhorov, C. Shramov, “Jordan constant for Cremona group of rank 3”, Mosc. Math. J., 17:3 (2017), 457–509 | DOI | MR | Zbl

[200] Yu. Prokhorov, C. Shramov, “Finite groups of birational selfmaps of threefolds”, Math. Res. Lett., 25:3 (2018), 957–972 | DOI | MR | Zbl

[201] Yu. Prokhorov, C. Shramov, “$p$-subgroups in the space Cremona group”, Math. Nachr., 291:8-9 (2018), 1374–1389 | DOI | MR | Zbl

[202] Yu. G. Prokhorov, K. A. Shramov, “Konechnye gruppy bimeromorfnykh avtomorfizmov unilineichatykh trekhmernykh kelerovykh mnogoobrazii”, Izv. RAN. Ser. matem., 84:5 (2020), 169–196 | DOI | MR | Zbl

[203] Yu. Prokhorov, C. Shramov, “Automorphism groups of Inoue and Kodaira surfaces”, Asian J. Math., 24:2 (2020), 355–368 | DOI | MR | Zbl

[204] Yu. Prokhorov, C. Shramov, “Automorphism groups of compact complex surfaces”, Int. Math. Res. Not. IMRN, Publ. online: 2019, rnz124 (to appear) | DOI

[205] V. V. Przhiyalkovskii, I. A. Cheltsov, K. A. Shramov, “Trekhmernye mnogoobraziya Fano s beskonechnymi gruppami avtomorfizmov”, Izv. RAN. Ser. matem., 83:4 (2019), 226–280 | DOI | MR | Zbl

[206] M. Reid, “Canonical 3-folds”, Journées de géométrie algébrique d'Angers, juillet 1979 / Algebraic geometry, Angers, 1979, Sijthoff Noordhoff, Alphen aan den Rijn–Germantown, MD, 1980, 273–310 | MR | Zbl

[207] 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 | MR | Zbl

[208] M. Reid, “Young person's guide to canonical singularities”, Algebraic geometry – Bowdoin 1985, Part 1 (Brunswick, ME, 1985), Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, 345–414 | DOI | MR | Zbl

[209] M. Reid, “Nonnormal del Pezzo surfaces”, Publ. Res. Inst. Math. Sci., 30:5 (1994), 695–727 | DOI | MR | Zbl

[210] M. Rosenlicht, “Some basic theorems on algebraic groups”, Amer. J. Math., 78:2 (1956), 401–443 | DOI | MR | Zbl

[211] B. Saint-Donat, “Projective models of $K$-3 surfaces”, Amer. J. Math., 96:4 (1974), 602–639 | DOI | MR | Zbl

[212] T. Sano, “Classification of non-Gorenstein $\mathbf Q$-Fano $d$-folds of Fano index greater than $d-2$”, Nagoya Math. J., 142 (1996), 133–143 | DOI | MR | Zbl

[213] V. G. Sarkisov, “O strukturakh rassloenii na koniki”, Izv. AN SSSR. Ser. matem., 46:2 (1982), 371–408 ; V. G. Sarkisov, “On conic bundle structures”, Math. USSR-Izv., 20:2 (1983), 355–390 | MR | Zbl | DOI

[214] R. L. E. Schwarzenberger, “Vector bundles on the projective plane”, Proc. London Math. Soc. (3), 11 (1961), 623–640 | DOI | MR | Zbl

[215] 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., 9:1 (2009), 183–198 | DOI | MR | Zbl

[216] J.-P. Serre, “Le groupe de Cremona et ses sous-groupes finis”, Séminaire Bourbaki, Exposés 997–1011, v. 2008/2009, Astérisque, 332, Soc. Math. France, Paris, 2010, Exp. No. 1000, vii, 75–100 | MR | Zbl

[217] J.-P. Serre, Problems for the Edinburgh workshop on Cremona groups, 2010, \par http://mi-ras.ru/~prokhoro/preprints.html

[218] I. R. Shafarevich, “On some infinite-dimensional groups”, Rend. Mat. e Appl. (5), 25:1-2 (1966), 208–212 | MR | Zbl

[219] I. R. Shafarevich, B. G. Averbukh, Yu. R. Vainberg, A. B. Zhizhchenko, Yu. I. Manin, B. G. Moishezon, G. N. Tyurina, A. N. Tyurin, “Algebraicheskie poverkhnosti”, Tr. MIAN SSSR, 75, Nauka, M., 1965, 3–215 | MR | MR | Zbl

[220] G. C. Shephard, J. A. Todd, “Finite unitary reflection groups”, Canad. J. Math., 6 (1954), 274–304 | DOI | MR | Zbl

[221] Kil-Ho Shin, “3-dimensional Fano varieties with canonical singularities”, Tokyo J. Math., 12:2 (1989), 375–385 | DOI | MR | Zbl

[222] V. V. Shokurov, “Teorema o neobraschenii v nul”, Izv. AN SSSR. Ser. matem., 49:3 (1985), 635–651 ; V. V. Shokurov, “The nonvanishing theorem”, Math. USSR-Izv., 26:3 (1986), 591–604 | MR | Zbl | DOI

[223] V. V. Shokurov, “Trekhmernye logperestroiki”, Izv. RAN. Ser. matem., 56:1 (1992), 105–203 ; V. V. Shokurov, “3-fold log flips”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 95–202 | MR | Zbl | DOI

[224] V. V. Shokurov, “3-fold log models”, Algebraic geometry 4, J. Math. Sci. (N. Y.), 81:3 (1996), 2667–2699 | DOI | MR | Zbl

[225] V. V. Shokurov, “Prelimiting flips”, Biratsionalnaya geometriya: lineinye sistemy i konechno porozhdennye algebry, Sbornik statei, Trudy MIAN, 240, Nauka, MAIK “Nauka/Interperiodika”, M., 2003, 82–219 | MR | Zbl

[226] V. V. Shokurov, “Pisma o biratsionalnom. VII. Uporyadochennyi obryv”, Mnogomernaya algebraicheskaya geometriya, Sbornik statei. Posvyaschaetsya pamyati chlena-korrespondenta RAN Vasiliya Alekseevicha Iskovskikh, Trudy MIAN, 264, MAIK “Nauka/Interperiodika”, M., 2009, 184–208 | MR | Zbl

[227] H. Sumihiro, “Equivariant completion”, J. Math. Kyoto Univ., 14 (1974), 1–28 | DOI | MR | Zbl

[228] K. Suzuki, “On Fano indices of $\mathbb{Q}$-Fano 3-folds”, Manuscripta Math., 114:2 (2004), 229–246 | DOI | MR | Zbl

[229] V. I. Tsygankov, “Uravneniya $G$-minimalnykh rassloenii na koniki”, Matem. sb., 202:11 (2011), 103–160 | DOI | MR | Zbl

[230] N. Tziolas, “Terminal $3$-fold divisorial contractions of a surface to a curve. I”, Compositio Math., 139:3 (2003), 239–261 | DOI | MR | Zbl

[231] N. Tziolas, “Three dimensional divisorial extremal neighborhoods”, Math. Ann., 333:2 (2005), 315–354 | DOI | MR | Zbl

[232] N. Tziolas, “Three-fold divisorial extremal neighborhoods over $cE_7$ and $c_6$ compound DuVal singularities”, Internat. J. Math., 21:1 (2010), 1–23 | DOI | MR | Zbl

[233] H. Umemura, “Sur les sous-groupes algébriques primitifs du groupe de Cremona à trois variables”, Nagoya Math. J., 79 (1980), 47–67 | DOI | MR | Zbl

[234] H. Umemura, “Maximal algebraic subgroups of the Cremona group of three variables. Imprimitive algebraic subgroups of exceptional type”, Nagoya Math. J., 87 (1982), 59–78 | DOI | MR | Zbl

[235] H. Umemura, “On the maximal connected algebraic subgroups of the Cremona group. I”, Nagoya Math. J., 88 (1982), 213–246 | DOI | MR | Zbl

[236] H. Umemura, “On the maximal connected algebraic subgroups of the Cremona group. II”, Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., 6, North-Holland, Amsterdam, 1985, 349–436 | DOI | MR | Zbl

[237] H. Umemura, “Minimal rational threefolds. II”, Nagoya Math. J., 110 (1988), 15–80 | DOI | MR | Zbl

[238] J. Varouchas, “Kähler spaces and proper open morphisms”, Math. Ann., 283:1 (1989), 13–52 | DOI | MR | Zbl

[239] J. Waldron, “The LMMP for log canonical 3-folds in characteristic $p>5$”, Nagoya Math. J., 230 (2018), 48–71 | DOI | MR | Zbl

[240] A. Weil, “On algebraic groups of transformations”, Amer. J. Math., 77:2 (1955), 355–391 | DOI | MR | Zbl

[241] Chenyang Xu, “Finiteness of algebraic fundamental groups”, Compositio Math., 150:3 (2014), 409–414 | DOI | MR | Zbl

[242] Yu. G. Zarhin, “Theta groups and products of abelian and rational varieties”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 299–304 | DOI | MR | Zbl

[243] Yu. G. Zarkhin, “Kompleksnye tory, teta-gruppy i ikh svoistva Zhordana”, Algebra, teoriya chisel i algebraicheskaya geometriya, Sbornik statei. Posvyaschaetsya pamyati akademika Igorya Rostislavovicha Shafarevicha, Trudy MIAN, 307, MIAN, M., 2019, 32–62 | DOI | MR | Zbl