Geodesic
Three-dimensional varieties with rational sections
Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 569-579 
B. V. Martynov. Three-dimensional varieties with rational sections. Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 569-579. http://geodesic.mathdoc.fr/item/SM_1970_10_4_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper we prove the following Theorem. A three-dimensional nonsingular variety over the field of complex numbers whose generic hyperplane sections are nonsingular rational surfaces is birationally equivalent either to projective three-space or to a cubic variety in projective four-space. Bibliography: 11 titles.

[1] G. Fano, “Sulle varieta algebriche a tre dimensioni a superficie-sezioni razionali”, Ann. Math. Pura ed Appl., 24 (1915), 49–88

[2] U. Morin, “Sulla classificazione proiettiva della varieta a superficie sezioni e razionali”, Ann. Mat. Pura ed Appl., 18 (1939), 147–161 | DOI | MR

[3] L. Roth, “Algebraic threefolds”, Ergeb. Math., 1955, 65–69

[4] Chzhen Schen Shen, Kompleksnye mnogoobraziya, IL, Moskva, 1961

[5] R. Uoker, Algebraicheskie krivye, IL, Moskva, 1952

[6] B. V. Martynov, “O vyrozhdennykh sloyakh trekhmernykh mnogoobrazii, rassloennykh na ratsionalnye poverkhnosti”, Matem. zametki, 7:2 (1970), 191–202 | MR | Zbl

[7] T. Matsusaka, “On the theorem of Castelnuovo–Enriques”, Science reports of Ochanomizu University, 4 (1953), 164–171 | MR

[8] O. Zarisski, “Pencils on an algebraic variety and a new proof of a theorem of Bertini”, Trans. Amer. Math. Soc., 50 (1951), 48–70 | DOI | MR

[9] M. Baldassarri, Algebraicheskie mnogoobraziya, IL, Moskva, 1961 | MR

[10] Algebraicheskie poverkhnosti, Trudy matem. in-ta im. V. A. Steklova AN SSSR, 75, 1965 | MR | Zbl

[11] Yu. I. Manin, “Ratsionalnye poverkhnosti nad sovershennymi polyami”, Publ. Math. IHES, 1966, no. 30 | MR