The rationality problem for conic bundles

Yu. G. Prokhorov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 73 (2018) no. 3, pp. 375-456

Résumé

This expository paper is concerned with the rationality problem for three-dimensional algebraic varieties with a conic bundle structure. The main methods of this theory are discussed, proofs of certain principal results are sketched, and some recent achievements are presented. Many open problems are also stated. Bibliography: 209 titles.

Détail

Keywords: algebraic threefolds, conic bundles, rationality, singularities, intermediate Jacobian, Prym variety.
Mots-clés : invariants, birational transformations, Sarkisov programme

Bibliographie

[1] H. Ahmadinezhad, T. Okada, Stable rationality of higher dimensional conic bundles, 2018 (v1 – 2016), 10 pp., arXiv: 1612.04206

[2] V. A. Alekseev, “Ob usloviyakh ratsionalnosti trekhmernykh mnogoobrazii s puchkom poverkhnostei Del-Petstso stepeni $4$”, Matem. zametki, 41:5 (1987), 724–730 | MR | Zbl

[3] V. Alexeev, “Boundedness and $K^2$ for log surfaces”, Internat. J. Math., 5:6 (1994), 779–810 | DOI | MR | Zbl

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

[5] V. Alexeev, A. Borisov, “On the log discrepancies in toric Mori contractions”, Proc. Amer. Math. Soc., 142:11 (2014), 3687–3694 | DOI | MR | Zbl

[6] A. B. Altman, S. L. Kleiman, “Foundations of the theory of Fano schemes”, Compositio Math., 34:1 (1977), 3–47 | MR | Zbl

[7] A. Alzati, M. Bertolini, “On the rationality of Fano $3$-folds with $B_2\ge2$”, Matematiche (Catania), 47:1 (1992), 63–74 | MR | Zbl

[8] T. Ando, “On extremal rays of the higher dimensional varieties”, Invent. Math., 81:2 (1985), 347–357 | DOI | MR | Zbl

[9] M. Andreatta, J. A. Wiśniewski, “A view on contractions of higher dimensional varieties”, Algebraic geometry (Santa Cruz, CA, 1995), Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997, 153–183 | MR | Zbl

[10] M. Artin, D. Mumford, “Some elementary examples of unirational varieties which are not rational”, Proc. Lond. Math. Soc. (3), 25 (1972), 75–95 | DOI | MR | Zbl

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

[12] A. Beauville, “Prym varieties and Schottky problem”, Invent. Math., 41:2 (1977), 149–196 | DOI | MR | Zbl

[13] A. Beauville, “Variétés de {P}rym et jacobiennes intermédiaires”, Ann. Sci. École Norm. Sup. (4), 10:3 (1977), 309–391 | DOI | MR | Zbl

[14] A. Beauville, “Les singularités du diviseur $\Theta $ de la jacobienne intermédiaire de l'hypersurface cubique dans $\mathbb P^4$”, Algebraic threefolds (Varenna, 1981), Lecture Notes in Math., 947, Springer, Berlin–New York, 1982, 190–208 | DOI | MR | Zbl

[15] A. Beauville, “Prym varieties: a survey”, Theta functions – Bowdoin 1987, Part 1 (Brunswick, ME, 1987), Proc. Sympos. Pure Math., 49, Part 1, Amer. Math. Soc., Providence, RI, 1989, 607–620 | DOI | MR | Zbl

[16] A. Beauville, “Determinantal hypersurfaces”, Michigan Math. J., 48 (2000), 39–64 | DOI | MR | Zbl

[17] A. Beauville, “A very general sextic double solid is not stably rational”, Bull. Lond. Math. Soc., 48:2 (2016), 321–324 | DOI | MR | Zbl

[18] A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, P. Swinnerton-Dyer, “Variétés stablement rationnelles non rationnelles”, Ann. of Math. (2), 121:2 (1985), 283–318 | DOI | MR | Zbl

[19] M. Beltrametti, “On the Chow group and the intermediate Jacobian of a conic bundle”, Ann. Mat. Pura Appl. (4), 141 (1985), 331–351 | DOI | MR | Zbl

[20] C. Birkar, Singularities of linear systems and boundedness of Fano varieties, 2016, 33 pp., arXiv: 1609.05543

[21] C. Birkar, “Singularities on the base of a Fano type fibration”, J. Reine Angew. Math., 2016:715 (2016), 125–142 | DOI | MR | Zbl

[22] 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

[23] J. Blanc, S. Lamy, “Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links”, Proc. Lond. Math. Soc. (3), 105:5 (2012), 1047–1075 | DOI | MR | Zbl

[24] F. A. Bogomolov, “Gruppa Brauera faktorprostranstv lineinykh predstavlenii”, Izv. AN SSSR. Ser. matem., 51:3 (1987), 485–516 ; F. A. Bogomolov, “The Brauer group of quotient spaces by linear group actions”, Math. USSR-Izv., 30:3 (1988), 455–485 | MR | Zbl | DOI

[25] F. Bogomolov, Yu. Prokhorov, “On stable conjugacy of finite subgroups of the plane {C}remona group. I”, Cent. Eur. J. Math., 11:12 (2013), 2099–2105 | DOI | MR | Zbl

[26] C. Böhning, H.-C. Graf von Bothmer, “On stable rationality of some conic bundles and moduli spaces of Prym curves”, Comment. Math. Helv., 93:1 (2018), 133–155 ; (2016), 23 pp., arXiv: 1605.03029 | DOI | MR | Zbl

[27] G. Brown, A. Corti, F. Zucconi, “Birational geometry of $3$-fold Mori fibre spaces”, The Fano conference, Univ. Torino, Torino, 2004, 235–275 | MR | Zbl

[28] G. Brown et al., Graded ring database http://www.grdb.co.uk/

[29] F. Catanese, M. Franciosi, K. Hulek, M. Reid, “Embeddings of curves and surfaces”, Nagoya Math. J., 154 (1999), 185–220 | DOI | MR | Zbl

[30] I. A. Cheltsov, “Rassloeniya na koniki s bolshim diskriminantom”, Izv. RAN. Ser. matem., 68:2 (2004), 215–221 | DOI | MR | Zbl

[31] I. Cheltsov, “Birationally rigid del Pezzo fibrations”, Manuscripta Math., 116:4 (2005), 385–396 | DOI | MR | Zbl

[32] I. Cheltsov, “Points in projective spaces and applications”, J. Differential Geom., 81:3 (2009), 575–599 | DOI | MR | Zbl

[33] I. Cheltsov, V. Przyjalkowski, C. Shramov, Which quartic double solids are rational?, 2015, 30 pp., arXiv: 1508.07277

[34] C. H. Clemens, “Applications of the theory of Prym varieties”, Proceedings of the International congress of mathematicians (Vancouver, BC, 1974), v. 1, Canad. Math. Congress, Montreal, QC, 1975, 415–421 | MR | Zbl

[35] C. H. Clemens, “Double solids”, Adv. in Math., 47:2 (1983), 107–230 | DOI | MR | Zbl

[36] K. G. Klemens, F. A. Griffits, “Promezhutochnyi yakobian trekhmernoi kubicheskoi giperpoverkhnosti”, Matematika, 16:6 (1972), 3–32 ; 17:1 (1973), 3–41 ; C. H. Clemens, P. A. Griffiths, “The intermediate Jacobian of the cubic threefold”, Ann. of Math. (2), 95:2 (1972), 281–356 | Zbl | Zbl | DOI | MR | Zbl

[37] J.-L. Colliot-Thélène, A. Pirutka, “Hypersurfaces quartiques de dimension 3: non-rationalité stable”, Ann. Sci. Éc. Norm. Supér. (4), 49:2 (2016), 371–397 | DOI | MR | Zbl

[38] Zh.-L. Kolë-Telen, E. V. Piryutko, “Tsiklicheskie nakrytiya, kotorye ne yavlyayutsya stabilno ratsionalnymi”, Izv. RAN. Ser. matem., 80:4 (2016), 35–48 | DOI | MR | Zbl

[39] J.-L. Colliot-Thélène, J.-J. Sansuc, “The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)”, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., 19, Tata Inst. Fund. Res., Mumbai, 2007, 113–186 | MR | Zbl

[40] A. Conte, J. P. Murre, “On quartic threefolds with a double line. I”, Nederl. Akad. Wetensch. Proc. Ser. A, 80, =Indag. Math., 39:3 (1977), 145–160 | MR | Zbl

[41] A. Conte, J. P. Murre, “On quartic threefolds with a double line. II”, Nederl. Akad. Wetensch. Proc. Ser. A, 80, =Indag. Math., 39:3 (1977), 161–175 | DOI | MR | Zbl

[42] D. F. Coray, M. A. Tsfasman, “Arithmetic on singular Del Pezzo surfaces”, Proc. Lond. Math. Soc. (3), 57:1 (1988), 25–87 | DOI | MR | Zbl

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

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

[45] A. Corti, “Singularities of linear systems and $3$-fold birational geometry”, Explicit birational geometry of $3$-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 259–312 | MR | Zbl

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

[47] O. Debarre, “Sur le problème de {T}orelli pour les variétés de Prym”, Amer. J. Math., 111:1 (1989), 111–134 | DOI | MR | Zbl

[48] O. Debarre, “Sur le théorème de Torelli pour les solides doubles quartiques”, Compositio Math., 73:2 (1990), 161–187 | MR | Zbl

[49] O. Debarre, A. Iliev, L. Manivel, “On nodal prime Fano threefolds of degree 10”, Sci. China Math., 54:8 (2011), 1591–1609 | DOI | MR | Zbl

[50] O. Debarre, A. Iliev, L. Manivel, “On the period map for prime Fano threefolds of degree 10”, J. Algebraic Geom., 21:1 (2012), 21–59 | DOI | MR | Zbl

[51] I. V. Dolgachev, “O ratsionalnykh poverkhnostyakh s puchkom ellipticheskikh krivykh”, Izv. AN SSSR. Ser. matem., 30:5 (1966), 1073–1100 | MR | Zbl

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

[53] S. Yu. Endryushka, “Neratsionalnost obschego mnogoobraziya Enrikvesa”, Matem. sb., 123(165):2 (1984), 269–275 ; S. Yu. Èndryushka, “Nonrationality of the general Enriques variety”, Math. USSR-Sb., 51:1 (1985), 267–273 | MR | Zbl | DOI

[54] D. Festi, R. van Luijk, “Unirationality of del Pezzo surfaces of degree 2 over finite fields”, Bull. Lond. Math. Soc., 48:1 (2016), 135–140 | DOI | MR | Zbl

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

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

[57] M. M. Grinenko, “Biratsionalnye svoistva puchkov poverkhnostei Del Petstso stepenei $1$ i $2$”, Matem. sb., 191:5 (2000), 17–38 | DOI | MR | Zbl

[58] M. M. Grinenko, “O dvoinom konuse nad poverkhnostyu Veroneze”, Izv. RAN. Ser. matem., 67:3 (2003), 5–22 | DOI | MR | Zbl

[59] M. M. Grinenko, “Struktury Mori na trekhmernom mnogoobrazii Fano indeksa $2$ i stepeni $1$”, Algebraicheskaya geometriya: Metody, svyazi i prilozheniya, Sbornik statei. Posvyaschaetsya pamyati chlena-korrespondenta RAN Andreya Nikolaevicha Tyurina, Tr. MIAN, 246, Nauka, MAIK “Nauka/Interperiodika”, M., 2004, 116–141 | MR | Zbl

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

[61] C. D. Hacon, J. McKernan, C. Xu, “ACC for log canonical thresholds”, Ann. of Math. (2), 180:2 (2014), 523–571 | DOI | MR | Zbl

[62] G. Halphén, “Sur les courbes planes du sixième degré à neuf points doubles”, Bull. Soc. Math. France, 10 (1882), 162–172 | DOI | MR | Zbl

[63] 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

[64] B. Hassett, A. Kresch, Yu. Tschinkel, “Stable rationality and conic bundles”, Math. Ann., 365:3-4 (2016), 1201–1217 | DOI | MR | Zbl

[65] T. Hayakawa, “Blowing ups of {$3$}-dimensional terminal singularities”, Publ. Res. Inst. Math. Sci., 35:3 (1999), 515–570 | DOI | MR | Zbl

[66] T. Hayakawa, “Blowing ups of $3$-dimensional terminal singularities. II”, Publ. Res. Inst. Math. Sci., 36:3 (2000), 423–456 | DOI | MR | Zbl

[67] A. Iliev, L. Katzarkov, V. Przyjalkowski, “Double solids, categories and non-rationality”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 145–173 | DOI | MR | Zbl

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

[69] 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

[70] V. A. Iskovskikh, “Biratsionalnye svoistva poverkhnosti stepeni 4 v $\mathbf {P}^4_k$”, Matem. sb., 88(130):1(5) (1972), 31–37 | MR | Zbl

[71] 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

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

[73] V. A. Iskovskikh, “On the rationality problem for conic bundles”, Duke Math. J., 54:2 (1987), 271–294 | DOI | MR | Zbl

[74] V. A. Iskovskikh, “K probleme ratsionalnosti rassloenii na koniki”, Matem. sb., 182:1 (1991), 114–121 ; V. A. Iskovskikh, “On the rationality problem for conic bundles”, Math. USSR-Sb., 72:1 (1992), 105–111 | MR | Zbl | DOI

[75] V. A. Iskovskikh, “O probleme ratsionalnosti dlya trekhmernykh algebraicheskikh mnogoobrazii, rassloennykh na poverkhnosti Del Petstso”, Teoriya chisel, algebra i algebraicheskaya geometriya, Sbornik statei. K semidesyatiletiyu so dnya rozhdeniya akademika Igorya Rostislavovicha Shafarevicha, Tr. MIAN, 208, Nauka, M., 1995, 128–138 | MR | Zbl

[76] V. A. Iskovskikh, “O kriterii ratsionalnosti dlya rassloenii na koniki”, Matem. sb., 187:7 (1996), 75–92 | DOI | MR | Zbl

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

[78] V. A. Iskovskikh, “Biratsionalnaya zhestkost giperpoverkhnostei Fano v ramkakh teorii Mori”, UMN, 56:2(338) (2001), 3–86 | DOI | MR | Zbl

[79] V. A. Iskovskikh, “b-Divizory i funktsionalnye algebry po Shokurovu”, Biratsionalnaya geometriya: lineinye sistemy i konechno porozhdennye algebry, Sbornik statei, Tr. MIAN, 240, Nauka, MAIK “Nauka/Interperiodika”, M., 2003, 8–20 | MR | Zbl

[80] V. A. Iskovskikh, “On the Noether–Fano inequalities”, The Fano conference, Univ. Torino, Torino, 2004, 25–35 | MR | Zbl

[81] V. A. Iskovskikh, Yu. I. Manin, “Trekhmernye kvartiki i kontrprimery k probleme Lyurota”, Matem. sb., 86(128):1(9) (1971), 140–166 | MR | Zbl

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

[83] V. A. Iskovskikh, A. V. Pukhlikov, “Mnogoobraziya Fano”, Itogi nauki i tekhn. Ser. Sovrem. matem. i ee pril. Temat. obz., 19, VINITI, M., 2001, 5–139 ; V. A. Iskovskikh, A. V. Pukhlikov, “Birational automorphisms of multidimensional algebraic manifolds”, J. Math. Sci. (N. Y.), 82:4 (1996), 3528–3613 | DOI | MR | Zbl

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

[85] N. Jacobson, Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, 1996, viii+278 pp. | DOI | MR | Zbl

[86] P. Jahnke, T. Peternell, I. Radloff, “Threefolds with big and nef anticanonical bundles. II”, Cent. Eur. J. Math., 9:3 (2011), 449–488 | DOI | MR | Zbl

[87] 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

[88] M.-C. Kang, “The rationality problem of finite group actions”, First international congress of Chinese mathematicians (Beijing, 1998), AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, 2001, 211–218 | DOI | MR | Zbl

[89] S. Kantor, “Die Typen der linearen Complexe rationaler Curven im $R_r$”, Amer. J. Math., 23:1 (1901), 1–28 | DOI | MR | Zbl

[90] 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

[91] 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

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

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

[94] 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 | MR | Zbl

[95] J. Kollár, “Higher direct images of dualizing sheaves. I”, Ann. of Math. (2), 123:1 (1986), 11–42 | DOI | MR | Zbl

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

[97] J. Kollár, “Nonrational hypersurfaces”, J. Amer. Math. Soc., 8:1 (1995), 241–249 | DOI | MR | Zbl

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

[99] J. Kollár, “Unirationality of cubic hypersurfaces”, J. Inst. Math. Jussieu, 1:3 (2002), 467–476 | DOI | MR | Zbl

[100] J. Kollár, “Conic bundles that are not birational to numerical {C}alabi–{Y}au pairs”, Épijournal Geom. Algébrique, 1 (2017), 1, 14 pp. ; (2017 (v1 – 2016)), 14 pp., arXiv: 1605.04763 | MR

[101] J. Kollár, M. Mella, “Quadratic families of elliptic curves and unirationality of degree $1$ conic bundles”, Amer. J. Math., 139:4 (2017), 915–936 | DOI | MR | Zbl

[102] J. Kollár, Y. Miyaoka, S. Mori, “Rationally connected varieties”, J. Algebraic Geom., 1:3 (1992), 429–448 | MR | Zbl

[103] J. Kollár, S. Mori, “Classification of three-dimensional flips”, J. Amer. Math. Soc., 5:3 (1992), 533–703 | DOI | MR | Zbl

[104] J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998, viii+254 pp. | DOI | MR | Zbl

[105] J. Kollár, K. E. Smith, A. Corti, Rational and nearly rational varieties, Cambridge Stud. Adv. Math., 92, Cambridge Univ. Press, Cambridge, 2004, vi+235 pp. | DOI | MR | Zbl

[106] I. Krylov, Birational geometry of del Pezzo fibrations with terminal quotient singularities, 2016, 27 pp., arXiv: 1606.03252

[107] I. Krylov, “Rationally connected non-Fano type varieties”, Eur. J. Math., 4:1 (2018), 335–355 | DOI | MR | Zbl

[108] B. È. Kunyavskij, A. N. Skorobogatov, M. A. Tsfasman, “Del Pezzo surfaces of degree four”, Mém. Soc. Math. France (N. S.), 37 (1989), 1–113 | DOI | MR | Zbl

[109] A. G. Kuznetsov, “O mnogoobraziyakh Kyukhle s chislom Pikara, bolshim $1$”, Izv. RAN. Ser. matem., 79:4 (2015), 57–70 | DOI | MR | Zbl

[110] 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

[111] K. Loginov, Standard models of degree $1$ del Pezzo fibrations, 2017, 22 pp., arXiv: 1710.02482

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

[113] 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

[114] L. Masiewicki, “Universal properties on Prym varieties with an application to algebraic curves of genus five”, Trans. Amer. Math. Soc., 222 (1976), 221–240 | DOI | MR | Zbl

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

[116] M. Mella, On the unirationality of $3$-fold conic bundles, 2014, 9 pp., arXiv: 1403.7055

[117] A. S. Merkurev, “O simvole normennogo vycheta stepeni 2”, Dokl. AN SSSR, 261:3 (1981), 542–547 | MR | Zbl

[118] T. Millevoi, “Sulla classificazione cremoniana delle congruenze di curve razionali dello spazio”, Rend. Sem. Mat. Univ. Padova, 30 (1960), 194–214 | MR | Zbl

[119] M. Miyanishi, “Algebraic methods in the theory of algebraic threefolds – surrounding the works of Iskovskikh, Mori and Sarkisov”, Algebraic varieties and analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983, 69–99 | MR | Zbl

[120] M. Miyanishi, D.-Q. Zhang, “Gorenstein log del {P}ezzo surfaces of rank one”, J. Algebra, 118:1 (1988), 63–84 | DOI | MR | Zbl

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

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

[123] 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

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

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

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

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

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

[129] 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

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

[131] S. Mori, Yu. G. Prokhorov, “Threefold extremal contractions of type (IIA). II”, Geometry and physics, A festschrift in honour of Nigel Hitchin, Oxford Univ. Press (to appear); 2017, 27 pp., arXiv: 1707.05047

[132] D. R. Morrison, “The birational geometry of surfaces with rational double points”, Math. Ann., 271:3 (1985), 415–438 | DOI | MR | Zbl

[133] D. Mumford, “Theta characteristics of an algebraic curve”, Ann. Sci. École Norm. Sup. (4), 4:2 (1971), 181–192 | DOI | MR | Zbl

[134] D. Mumford, “Prym varieties. I”, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, 325–350 | MR | Zbl

[135] I. A. Panin, “Ratsionalnost svyazok konik s krivoi vyrozhdeniya stepeni pyat i chetnoi teta-kharakteristikoi”, Moduli i lineinye gruppy, Zap. nauch. sem. LOMI, 103, Nauka, Leningr. otd., L., 1980, 100–105 ; I. A. Panin, “Rationality of bundles of conics with degenerate curve of degree five and even theta-characteristic”, J. Soviet Math., 24:4 (1984), 449–452 | MR | Zbl | DOI

[136] J. Park, “Birational maps of del Pezzo fibrations”, J. Reine Angew. Math., 2001:538 (2001), 213–221 | DOI | MR | Zbl

[137] J. Park, “A note on del {P}ezzo fibrations of degree $1$”, Comm. Algebra, 31:12 (2003), 5755–5768 | DOI | MR | Zbl

[138] L. Picco Botta, A. Verra, “The non rationality of the generic Enriques' threefold”, Compositio Math., 48:2 (1983), 167–184 | MR | Zbl

[139] J. Piontkowski, “Theta-characteristics on singular curves”, J. Lond. Math. Soc. (2), 75:2 (2007), 479–494 | DOI | MR | Zbl

[140] V. L. Popov, “Konechnye podgruppy grupp diffeomorfizmov”, Izbrannye voprosy matematiki i mekhaniki, Sbornik statei. K 150-letiyu so dnya rozhdeniya akademika Vladimira Andreevicha Steklova, Tr. MIAN, 289, MAIK “Nauka/Interperiodika”, M., 2015, 235–241 | DOI | Zbl

[141] V. L. Popov, “Borelevskie podgruppy grupp Kremony”, Matem. zametki, 102:1 (2017), 72–80 | DOI | MR | Zbl

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

[143] Yu. G. Prokhorov, “On extremal contractions from threefolds to surfaces: the case of one non-Gorenstein point”, Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math., 207, Amer. Math. Soc., Providence, RI, 1997, 119–141 | DOI | MR | Zbl

[144] Yu. G. Prokhorov, “Mori conic bundles with a reduced log-terminal boundary”, J. Math. Sci. (N. Y.), 94:1 (1999), 1051–1059 | DOI | MR | Zbl

[145] Yu. G. Prokhorov, “On extremal contractions from threefolds to surfaces: the case of one non-Gorenstein point and a nonsingular base surface”, J. Math. Sci. (N. Y.), 95:1 (1999), 1986–1995 | DOI | MR | Zbl

[146] Yu. G. Prokhorov, “Stepen trekhmernykh mnogoobrazii Fano s kanonicheskimi gorenshteinovymi osobennostyami”, Matem. sb., 196:1 (2005), 81–122 | DOI | MR | Zbl

[147] Yu. Prokhorov, “Gap conjecture for $3$-dimensional canonical thresholds”, J. Math. Sci. Univ. Tokyo, 15:4 (2008), 449–459 | MR | Zbl

[148] Yu. G. Prokhorov, “Fields of invariants of finite linear groups”, Cohomological and geometric approaches to rationality problems, Progr. Math., 282, Birkhäuser Boston, Inc., Boston, MA, 2010, 245–273 | DOI | MR | Zbl

[149] Yu. Prokhorov, “On stable conjugacy of finite subgroups of the plane Cremona group. II”, Michigan Math. J., 64:2 (2015), 293–318 | DOI | MR | Zbl

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

[151] Yu. G. Prokhorov, “Ratsionalnye poverkhnosti”, Lekts. kursy NOTs, 24, MIAN, M., 2015, 3–76 | DOI | MR

[152] Yu. G. Prokhorov, “Trekhmernye mnogoobraziya $\mathbb Q$-Fano indeksa $7$”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki. II, Sbornik statei, Tr. MIAN, 294, MAIK “Nauka/Interperiodika”, M., 2016, 152–166 | DOI | MR | Zbl

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

[154] Yu. Prokhorov, M. Reid, “On $\mathbb Q$-Fano $3$-folds of Fano index $2$”, Minimal models and extremal rays (Kyoto, 2011), Adv. Stud. Pure Math., 70, Math. Soc. Japan, Tokyo, 2016, 397–420 | MR | Zbl

[155] Yu. G. Prokhorov, V. V. Shokurov, “Towards the second main theorem on complements”, J. Algebraic Geom., 18:1 (2009), 151–199 | DOI | MR | Zbl

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

[157] Yu. Prokhorov, C. Shramov, “Finite groups of birational selfmaps of threefolds”, Math. Res. Lett. (to appear); 2017 (v1 – 2016), 9 pp., arXiv: 1611.00789

[158] Yu. Prokhorov and C. Shramov, “$p$-subgroups in the space Cremona group”, Math. Nachr., 291:8-9 (2018), 1374–1389 ; (2017 (v1 – 2016)), 27 pp., arXiv: 1610.02990 | DOI

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

[160] V. V. Przhiyalkovskii, I. A. Cheltsov, K. A. Shramov, “Giperellipticheskie i trigonalnye trekhmernye mnogoobraziya Fano”, Izv. RAN. Ser. matem., 69:2 (2005), 145–204 | DOI | MR | Zbl

[161] V. V. Przhiyalkovskii, K. A. Shramov, “Dvoinye kvadriki s bolshimi gruppami avtomorfizmov”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki. II, Sbornik statei, Tr. MIAN, 294, MAIK “Nauka/Interperiodika”, M., 2016, 167–190 | DOI | MR | Zbl

[162] A. V. Pukhlikov, “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii s puchkom poverkhnostei del Petstso”, Izv. RAN. Ser. matem., 62:1 (1998), 123–164 | DOI | MR | Zbl

[163] A. V. Pukhlikov, “Posloinye biratsionalnye sootvetstviya”, Matem. zametki, 68:1 (2000), 120–130 | DOI | MR | Zbl

[164] A. V. Pukhlikov, “Essentials of the method of maximal singularities”, Explicit birational geometry of $3$-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 73–100 | MR | Zbl

[165] A. Pukhlikov, Birationally rigid varieties, Math. Surveys Monogr., 190, Amer. Math. Soc., Providence, RI, 2013, vi+365 pp. | DOI | MR | Zbl

[166] A. V. Pukhlikov, “Biratsionalnaya geometriya algebraicheskikh mnogoobrazii, rassloennykh na dvoinye prostranstva Fano”, Izv. RAN. Ser. matem., 81:3 (2017), 160–188 | DOI | MR | Zbl

[167] S. Recillas, “Jacobians of curves with $g^1_4$'s are the Prym's of trigonal curves”, Bol. Soc. Mat. Mexicana (2), 19:1 (1974), 9–13 | MR | Zbl

[168] M. Reid, “Surfaces of small degree”, Math. Ann., 275:1 (1986), 71–80 | DOI | MR | Zbl

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

[170] M. Reid, Birational geometry of {$3$}-folds according to Sarkisov, Preprint, Univ. of Warwick, Warwick, 1991

[171] C. Salgado, D. Testa, A. Várilly-Alvarado, “On the unirationality of del Pezzo surfaces of degree $2$”, J. Lond. Math. Soc. (2), 90:1 (2014), 121–139 | DOI | MR | Zbl

[172] D. J. Saltman, “Noether's problem over an algebraically closed field”, Invent. Math., 77:1 (1984), 71–84 | DOI | MR | Zbl

[173] V. G. Sarkisov, “Biratsionalnye avtomorfizmy rassloenii konik”, Izv. AN SSSR. Ser. matem., 44:4 (1980), 918–945 ; V. G. Sarkisov, “Birational automorphisms of conic bundles”, Math. USSR-Izv., 17:1 (1981), 177–202 | MR | Zbl | DOI

[174] 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

[175] V. G. Sarkisov, Klassifikatsiya biratsionalnykh avtomorfizmov rassloenii na koniki. I. Prisoedinennye tsikly, Preprint IAE-4446/15, In-t atomnoi energii im. I. V. Kurchatova, M., 1987

[176] V. G. Sarkisov, Birational maps of standard {Q}-{F}ano fiberings, Preprint IAE-4901/1, I. V. Kurchatov Inst., Moscow, 1989

[177] I. R. Shafarevich, “O probleme Lyurota”, Teoriya Galua, koltsa, algebraicheskie gruppy i ikh prilozheniya, Sbornik statei, Tr. MIAN SSSR, 183, Nauka, Leningr. otd., L., 1990, 199–204 ; I. R. Shafarevich, “Lüroth's problem”, Proc. Steklov Inst. Math., 183 (1991), 241–246 | MR | Zbl

[178] N. I. Shepherd-Barron, “Stably rational irrational varieties”, The Fano conference, Univ. Torino, Torino, 2004, 693–700 | MR | Zbl

[179] V. V. Shokurov, “Distinguishing Prymians from Jacobians”, Invent. Math., 65:2 (1981/82), 209–219 | DOI | MR | Zbl

[180] V. V. Shokurov, “Mnogoobraziya Prima: teoriya i prilozheniya”, Izv. AN SSSR. Ser. matem., 47:4 (1983), 785–855 ; V. V. Shokurov, “Prym varieties: theory and applications”, Math. USSR-Izv., 23:1 (1984), 83–147 | MR | Zbl | DOI

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

[182] 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

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

[184] V. V. Shokurov, Log adjunction: effectiveness and positivity, 2013, 51 pp., arXiv: 1308.5160

[185] V. V. Shokurov, “Kriterii poluobilnosti”, Izv. RAN. Ser. matem., 81:4 (2017), 167–230 | DOI | MR | Zbl

[186] V. V. Shokurov, S. R. Choi, “Geography of log models: theory and applications”, Cent. Eur. J. Math., 9:3 (2011), 489–534 | DOI | MR | Zbl

[187] K. A. Shramov, “K voprosu ratsionalnosti neosobykh trekhmernykh mnogoobrazii s puchkom poverkhnostei del Petstso stepeni $4$”, Matem. sb., 197:1 (2006), 133–144 | DOI | MR | Zbl

[188] K. A. Shramov, “Obryv vozrastayuschikh tsepochek v mnozhestve kanonicheskikh porogov toricheskikh mnogoobrazii”, Matem. sb., 197:4 (2006), 151–160 | DOI | MR | Zbl

[189] I. V. Sobolev, “Biratsionalnye avtomorfizmy odnogo klassa mnogoobrazii, rassloennykh na kubicheskie poverkhnosti”, Izv. RAN. Ser. matem., 66:1 (2002), 203–224 | DOI | MR | Zbl

[190] D. A. Stepanov, “Gladkie trekhmernye kanonicheskie porogi”, Matem. zametki, 90:2 (2011), 285–299 | DOI | MR | Zbl

[191] B. Totaro, “Hypersurfaces that are not stably rational”, J. Amer. Math. Soc., 29:3 (2016), 883–891 | DOI | MR | Zbl

[192] S. L. Tregub, “Konstruktsiya biratsionalnogo izomorfizma trekhmernoi kubiki i mnogoobraziya Fano pervogo roda s $g=8$, svyazannaya s normalnoi ratsionalnoi krivoi stepeni 4”, Vestn. Mosk. un-ta. Ser. 1 Matem. Mekh., 1985, no. 6, 99–101 | MR | Zbl

[193] S. L. Tregub, “O rassloeniyakh na koniki, biratsionlno ekvivalentnykh trekhmernoi kubike”, UMN, 45:5(275) (1990), 195–196 | MR | Zbl

[194] A. S. Trepalin, “Rationality of the quotient of $\mathbb{P}^2$ by finite group of automorphisms over arbitrary field of characteristic zero”, Cent. Eur. J. Math., 12:2 (2014), 229–239 | DOI | MR | Zbl

[195] A. Trepalin, “Quotients of conic bundles”, Transform. Groups, 21:1 (2016), 275–295 | DOI | MR | Zbl

[196] A. N. Tyurin, “Geometriya poverkhnosti Fano neosoboi kubiki $F\subset P^4$ i teoremy Torelli dlya poverkhnostei Fano i kubik”, Izv. AN SSSR. Ser. matem., 35:3 (1971), 498–529 | MR | Zbl

[197] A. N. Tyurin, “Pyat lektsii o trekhmernykh mnogoobraziyakh”, UMN, 27:5(167) (1972), 3–50 | MR | Zbl

[198] A. N. Tyurin, “O peresechenii kvadrik”, UMN, 30:6(186) (1975), 51–99 | MR | Zbl

[199] A. N. Tyurin, “Ob invariante svyazki kvadrik”, Izv. AN SSSR. Ser. matem., 39:1 (1975), 23–27 | MR | Zbl

[200] A. N. Tyurin, “Srednii yakobian trekhmernykh mnogoobrazii”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem., 12, VINITI, M., 1979, 5–57 ; A. N. Tyurin, “The middle Jacobian of three-dimensional varieties”, J. Soviet Math., 13:6 (1980), 707–745 | MR | Zbl | DOI

[201] R. Varley, “Weddle's surfaces, Humbert's curves, and a certain $4$-dimensional abelian variety”, Amer. J. Math., 108:4 (1986), 931–951 | DOI | MR | Zbl

[202] C. Voisin, “Sur la jacobienne intermédiaire du double solide d'indice deux”, Duke Math. J., 57:2 (1988), 629–646 | DOI | MR | Zbl

[203] C. Voisin, “Unirational threefolds with no universal codimension $2$ cycle”, Invent. Math., 201:1 (2015), 207–237 | DOI | MR | Zbl

[204] I. I. Voronovich, V. I. Yanchevskii, “Ratsionalnye polya razlozheniya prostykh algebr i uniratsionalnost rassloenii na koniki”, Dokl. AN BSSR, 30:4 (1986), 293–296 ; I. I. Voronovich, V. I. Yanchevskiĭ, “Rational splitting fields of simple algebras and unirationality of conic bundles”, Selected papers in $K$-theory, Amer. Math. Soc. Transl. Ser. 2, 154, Amer. Math. Soc., Providence, RI, 1992, 69–74 | MR | Zbl

[205] V. I. Yanchevskii, “$K$-uniratsionalnost rassloenii na koniki i polya razlozhenii prostykh tsentralnykh algebr”, Dokl. AN BSSR, 29:12 (1985), 1061–1064 | MR | Zbl

[206] V. I. Yanchevskiĭ, “$K$-unirationality of conic bundles over large arithmetic fields”, Journées Arithmétiques, 1991 (Geneva), Astérisque, 209, Soc. Math. France, Paris, 1992, 311–320 | MR | Zbl

[207] E. Yasinsky, “Subgroups of odd order in the real plane Cremona group”, J. Algebra, 461:1 (2016), 87–120 | DOI | MR | Zbl

[208] E. Yasinsky, “The Jordan constant for Cremona group of rank $2$”, Bull. Korean Math. Soc., 54:5 (2017), 1859–1871 | DOI | MR

[209] A. A. Zagorskii, “O trekhmernykh konicheskikh rassloeniyakh”, Matem. zametki, 21:6 (1977), 745–758 | MR | Zbl