Del Pezzo surfaces of degree four (1989)


Kunyavskij, B. Eh.; Skorobogatov, A. N.; Tsfasman, M. A.
Mémoires de la Société Mathématique de France, Tome 37 (1989) 120 p doi : 10.24033/msmf.338
URL stable : http://www.numdam.org/item?id=MSMF_1989_2_37__1_0

Bibliographie

[1] A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, Sir Peter Swinnerton-Dyer. - Variétés stablement rationnelles non rationnelles, Ann. Math., 1985, v. 121, 283-318. MR 86m:14009 | Zbl 0589.14042

[2] S. Bloch. - On the Chow groups of certain rational surfaces, Ann. Sci. Ec. Norm. Sup., 4ème série, 1981, t. 14, N° 1, 41-59. Numdam | MR 83e:14006 | Zbl 0524.14006

[3] N. Bourbaki. - Groupes et algèbres de Lie, Ch. IV-VI, Hermann, Paris, 1968 ; Ch. VII-VIII, Hermann, Paris, 1975.

[4] J.-L. Colliot-Thélène. - Hilbert's theorem 90 for K2, with application the the Chow group of rational surfaces, Invent. Math., 1983, v. 71, 1-20. MR 85d:14016 | Zbl 0527.14011

[5] J.-L. Colliot-Thélène, D. Coray. - L'équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques, Compos. Math., 1979, v. 39, N° 3, 301-332. Numdam | MR 81d:14008 | Zbl 0386.14003

[6] J.-L. Colliot-Thélène, J.-J. Sansuc. - La R-équivalence sur les tores, Ann. Sci. Ec. Norm. Sup., 4ème série, 1977, t. 10, 175-229. Numdam | MR 56 #8576 | Zbl 0356.14007

[7] J.-L. Colliot-Thélène, J.-J. Sansuc. - La descente sur les variétés rationnelles, In : Journées de géométrie algébrique d'Angers, 1979, Sijthoff-Nordhoff, 1980, 223-237. MR 82d:14016 | Zbl 0451.14018

[8] J.-L. Colliot-Thélène, J.-J. Sansuc. - Principal homogeneous spaces under flasque tori, J. Algebra, 1987, v. 106, N° 1, 148-205. MR 88j:14059 | Zbl 0597.14014

[9] J.-L. Colliot-Thélène, J.-J. Sansuc. - On the Chow groups of certain rational surfaces : a sequel to a paper of S. Bloch, Duke Math. J., 1981, v. 48, N° 2, 421-447. MR 83e:14007 | Zbl 0479.14006

[10] J.-L. Colliot-Thélène, J.-J. Sansuc. - La descente sur les variétés rationnelles, II, Duke Math. J., 1987, v. 54, N° 2, 375-492. MR 89f:11082 | Zbl 0659.14028

[11] J.-L. Colliot-Thélène, J.-J. Sansuc. - La descente sur les surfaces rationnelles fibrées en coniques, C.R. Acad. Sci. Paris, 1986, t. 303, Série I, N° 7, 303-306. MR 88h:14043 | Zbl 0596.14016

[12] J.-L. Colliot-Thélène, J.-J. Sansuc, Sir Peter Swinnerton-Dyer. - Intersections of two quadrics and Châtelet surfaces, J. reine angew. Math., 1987, Bd. 373, S. 37-107 and Bd. 374, S. 72-168. MR 88m:11045a | Zbl 0622.14029

[13] J.-L. Colliot-Thélène, A.N. Skorobogatov. - R-equivalence on conic bundles of degree 4, Duke Math. J., 1987, v. 54, N° 2, 671-677. MR 88i:14032 | Zbl 0634.14024

[14] D.F. Coray, M.A. Tsfasman. - Arithmetic on singular Del Pezzo surfaces, Proc. London Math. Soc, 1988, v. 57, N° 1, 25-87. MR 89f:11083 | Zbl 0653.14018

[15] C.W. Curtis, I. Reiner. - Representation theory of finite groups and associative algebras, Wiley, New York, 1962. MR 26 #2519 | Zbl 0131.25601

[16] P. Damey, J.-J. Payan. - Existence et construction des extensions galoisiennes et non-abéliennes de degré 8 d'un corps de caractéristique différente de 2, J. reine angew. Math., 1970, Bd. 244, S. 37-54. MR 43 #6186 | Zbl 0206.34401

[17] S. Endo, T. Miyata. - Integral representations with trivial first cohomology groups, Nagoya Math. J., 1982, v. 85, 231-240. MR 83e:20012 | Zbl 0539.20006

[18] A. Grothendieck. - Groupe de Brauer, III : Exemples et compléments, In : Dix exposés sur la cohomologie des schémas, Amsterdam, North Holland, 1968, 88-188. MR 39 #5586c | Zbl 0198.25901

[19] V.A. Iskovskih. - Rational surfaces with a pencil of rational curves, Mat. Sb., 1967, v. 74, 608-638 (= Math. USSR-Sb., 1967, v. 3, 563-587). Zbl 0181.24003

[20] V.A. Iskovskih. - Birational properties of surfaces of degree 4 in P4k, Mat. Sb., 1972, v. 88, 31-37 (= Math. USSR-Sb, 1972, v. 17, 30-36). Zbl 0253.14002

[21] V.A. Iskovskih. - Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR, Ser. Mat., 1979, v. 43, 19-43. (= Math. USSR-Izv., 1980, v. 14, 17-39). Zbl 0427.14011

[22] V.A. Iskovskih. - Generators and relations in the group of birational automorphisms of two classes of rational surfaces, Trudy Mat. Inst. Steklov, 1984, v. 165, 67-78. (= Proc. Steklov Inst. Math., 1985, v. 165, 73-84). Zbl 0589.14012

[23] N. Jacobson. - Lectures in abstract algebra, III. Theory of fields and Galois theory, Springer-Verlag, 1975. Zbl 0322.12001

[24] B.È. Kunyavskiĭ, A.N. Skorobogatov. - Maximal tori in semi-simple groups and rational surfaces, Uspekhi Mat. Nauk., 1986, v. 41, N° 4, 217-218 (= Russian Math. Surveys, 1986, v. 41, N° 4, 177-178). Zbl 0618.14028

[25] B.È. Kunyavskiǐ, A.N. Skorobogatov, M.A. Tsfasman. - Combinatorics and geometry of Del Pezzo surfaces of degree 4, Uspekhi Mat. Nauk, 1985, v. 40, N° 6, 145-146 (= Russian Math. Surveys, 1985, v. 40, N° 6, 131-132). Zbl 0612.14039

[26] B.È. Kunyavskiǐ, M.A. Tsfasman. . - Zero-cycles on rational surfaces and Néron-Severi tori, Izv. Akad. Nauk. SSSR, Ser. Mat., 1984, v. 48, N° 3, 631-654 (= Math. USSR-Izv., 1985, v. 24, 583-603). Zbl 0556.14027

[27] Yu.I.Manin. - Rational surfaces over perfect fields (Russian), Inst. des Hautes Etudes Sci. Publ. Math., 1966, t. 30, 55-113 (= Translations AMS (2), 1969, v. 84, 137-186). Numdam | Numdam | Zbl 0171.41701

[28] Yu.I.Manin. - Cubic forms : algebra, geometry, arithmetic, Second edit., North-Holland, Amsterdam, 1986. Zbl 0582.14010

[29] Yu.I.Manin, M.A. Tsfasman. - Rational varieties : algebra, geometry and arithmetic, Uspekhi Math. Nauk 1986, v. 41, N° 2, 43-94 (= Russian Math. Surveys, 1986, v. 41, N° 2, 51-116). Zbl 0621.14029

[30] I. Naruki. - Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc., 1982, v. 45, N° 1, 1-30. MR 84d:14020 | Zbl 0508.14005

[31] M. Reid. - The complete intersection of two or more quadrics, Ph. D. thesis, Cambridge, 1972.

[32] P. Salberger. - Sur l'arithmétique de certaines surfaces de Del Pezzo, C.R. Acad. Sci. Paris, 1986, t. 303, sér. I, N° 7, 273-276. MR 88h:14042 | Zbl 0612.14038

[33] P. Salberger. - On the arithmetic of conic bundle surfaces, In : Séminaire de théorie des nombres de Paris, 1985-1986, 175-197. MR 90m:11093a | Zbl 0657.14020

[34] P. Salberger. - Letter to A. Skorobogatov of March 27, 1987.

[35] P. Salberger. - Zero-cycles on rational surfaces over number fields, Invent. Math., 1988, v. 91, 505-524. MR 89c:14010 | Zbl 0688.14008

[36] J.-J. Sansuc. - Principe de Hasse, surfaces cubiques et intersections de deux quadriques, In : Journées arithmétiques de Besançon, 1985 (= "Astérisque", 1987, N° 147-148, 183-207). Zbl 0661.14021

[37] J.-P. Serre. - Cours d'arithmétique, Presses Univ. de France, Paris, 1970. MR 41 #138 | Zbl 0225.12002

[38] J.-P. Serre. - Représentations linéaires des groupes finis, Hermann, Paris, 1967. MR 38 #1190 | Zbl 0407.20003

[39] I.R. Shafarevich. - Construction of algebraic number fields with given solvable Galois groups (Russian), Izv. Akad. Nauk. SSSR, Ser. Mat., 1954, v. 18, N° 3, 525-578. Zbl 0057.27401

[40] A.N. Skorobogatov. - Birational invariants of rational surfaces, Mat. Zametki, 1986, v. 39, N° 5, 736-746 (= Math. Notes, 1986, v. 39, N° 5, 404-408). MR 87j:14059 | Zbl 0615.14005

[41] A.N. Skorobogatov. - Splitting groups of rational surfaces and zero-cycles (Russian), In : Algebra, logic and number theory. Moscow Univ. Press., Moscow, 1986, 80-84. MR 1100956

[42] B.L. Van Der Waerden. - Modern algebra, 3rd ed. Ungar, New York, 1950. MR 12,236f | Zbl 0033.10102

[43] V.E. Voskresenskiǐ. - Algebraic tori (Russian), Nauka, Moscow, 1977. Zbl 0499.14013

[44] V.E. Voskresenskiǐ. - Projective invariant Demazure models, Izv. Akad. Nauk. SSSR, Ser. Mat., 1982, v. 46, N° 2, 195-210 (= Math. USSR-Izv., 1983, v. 20, 189-202). Zbl 0523.14030

[45] A. Weil. - Adèles and algebraic groups, Lect. Notes, Inst. Adv. Study, Princeton, N.J., 1961, Progress in Math., v. 23, Birkhäuser (1982). Zbl 0493.14028