Geodesic
Three-dimensional quartics and counterexamples to the Lüroth problem
Sbornik. Mathematics, Tome 15 (1971) no. 1, pp. 141-166 
V. A. Iskovskikh; Yu. I. Manin. Three-dimensional quartics and counterexamples to the Lüroth problem. Sbornik. Mathematics, Tome 15 (1971) no. 1, pp. 141-166. http://geodesic.mathdoc.fr/item/SM_1971_15_1_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we prove that any birational mapping between smooth hypersurfaces of degree four is an isomorphism. Since B. Segre constructed examples of smooth unirational quartics, this leads to a negative resolution of the three-dimensioal Lüroth problem. Bibliography: 13 titles.

[1] S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, Academic Press, New York–London, 1966 | MR | Zbl

[2] A. Borel, J. P. Serre, “Le theoreme de Riemann–Roch”, Bull. Soc. Math. France, 86 (1958), 97–136 | MR | Zbl

[3] G. Fano, “Sopra aleune varieta algebriche a tre dimensioni aventi tutti i generi nulli”, Atti Ace. Torino, 43 (1907–1908), 973–977

[4] G. Fano, “Osservazioni sopra aleune varieta non razionali aventi tutti i generi nulli”, Atti Ace. Torino, 50 (1915), 1067–1071

[5] G. Fano, “Nuove ricerche sulle varieta algebriche a tre dimensione a curve-sezioni canoniche”, Comm. Pont. Ac. Sci., 11 (1947), 635–720 | MR | Zbl

[6] A. Grothendiesk, Sur quelques proprietes fondamentales en theorie des intersection, Seminaire Chevalley “Anneaux de Chow”, Paris, 1958

[7] H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero”, Ann. Math., 79 (1964), 109–326 | DOI | MR | Zbl

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

[9] Yu. I. Manin, “Sootvetstviya, motivy i monoidalnye preobrazovaniya”, Matem. sb., 77(119) (1968), 475–507 | MR | Zbl

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

[11] L. Roth, Algebraic threefolds with special regard to problems of rationality, Springer, Berlin, 1955 | MR

[12] B. Segre, “Variazione continua ad omotopia in geometria algebrica”, Ann. Mat. Pura ed Appl, Ser. IV, L (1960), 149–186 | DOI | MR

[13] O. Zariski, “Reduction of singularities of algebraic threedimensional varieties”, Ann. Math., 45 (1944), 472–542 | DOI | MR | Zbl