Fano 3-folds.~I

V. A. Iskovskikh

Izvestiya. Mathematics , Tome 11 (1977) no. 3, pp. 485-527.

Résumé

This article contains a classification of special Fano varieties; we give a description of the projective models of Fano 3-folds of index $r\geqslant2$ and of hyperelliptic Fano 3-folds. Bibliography: 31 titles.

Bibliographie

[1] Altman A. B., Kleiman S. L., Foundations of the theory of Fano schemas, preprint

[2] Baum P., Fulton W., Mac Pherson R., Riemann–Roch for signular varieties, preprint

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

[4] Fano G., “Sopra alcune varieta algebriche a tre dimensioni aventi tutti i generi nulli”, Torino Atti, 43 (1908), 973–984 | Zbl

[5] Fano G., “Osservazioni sopra alcune varieta non razionali aventi tutti i generi nulli”, Torino Atti, 50 (1915), 1067–1072 | Zbl

[6] Fano G., “Sulle varieta algebriche a tre dimensioni aventi tutti i generi nulli”, Atti Congresso Bologna, 4 (1931), 115–121 | Zbl

[7] Fano G., “Su alcune varietà algebriche a tre dimensioni aventi curve-sezioni canoniche”, Seritti Matematici offerti a L. Berzolari, Paria, Tip. Rossett, 1936, 329–341

[8] Fano G., “Su alcune varieta algebriche a tre dimensioni razionali, e aventi curvesezioni canoniche”, Comment. Math. Helv., 14 (1941), 202–211 | DOI | MR | Zbl

[9] Fano G., “Sulle varieta algebriche a tre dimensioni a curve-sezioni canoniche”, Mem. Accad. Ital., 8 (1937), 23–64 | Zbl

[10] Fano G., “Nuove ricerche sulle varieta algebriche a tre dimensioni a curve-sezioni canoniche”, Comment. Pontif. Acad. Sci., 11 (1947), 635–720 | MR | Zbl

[11] Grothendieck A., “Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schémas de Hilbert”, Sem. Bourbaki (1960/61), 13, no. 221, 1961, 28

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

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

[14] Manin Yu. I., “Lektsii o $K$-funktore v algebraicheskoi geometrii”, Uspekhi matem. nauk, 24:5 (1969), 3–86 | MR | Zbl

[15] Manin Yu. I., Kubicheskie formy, Nauka, M., 1972 | MR

[16] Mori S., “On a generalisation of complete intersections”, J. Math. Kyoto Uniov., 15:3 (1975), 619–646 | MR | Zbl

[17] Mamford D., Lektsii o krivykh na algebraicheskoi poverkhnosti, Mir, M., 1968

[18] Mumford D., “Varieties defined by quadratic equations”, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, 29–100 | MR

[19] Murre J. P., “Reduction of the proof of the nonrationality of a nonsingular cubic threefold to a result of Mumford”, Comp. Math., 27 (1973), 63–82 | MR | Zbl

[20] Nagata M., “On rational surfaces, I”, Memoirs of the College of science, Kyoto, 1963; Matematika, 8:1 (1964), 55–71

[21] Ramanujam C. P., “Remarks on the Kodaira vanishing theorem”, J. Indian Math. Soc., 36 (1972), 41–51 | MR | Zbl

[22] Reid M., The complete intersection of two on mone quadrics, These D. Ph., Cambridge University, 1972

[23] Reid M., “Hyperelliptic linear systems on a K3 surface”, J. London Math. Soc. Ser. 2, 13:3 (1976), 425–437 | MR

[24] Roth L., Algebraic theefolds with special regard to problems of rationality, Springer, Berlin, Heidelberg, New York, 1955 | MR

[25] Saint-Donat B., “Projective models of K3 surfaces”, Amer J. Math., 96:4 (1974), 602–639 | DOI | MR | Zbl

[26] Saint-Donat B., “Sur les équations définissant une courbe algébrique”, C. R. Acad. Sci., 274:4 (1972), A324–A327 | MR

[27] Tennison B., “On the quartic threefold”, Proc. London Math. Soc., 29 (1974), 714–734 | DOI | MR | Zbl

[28] Tyurin A. N., “Pyat lektsii o trekhmernykh mnogoobraziyakh”, Uspekhi matem. nauk, 27:5(186) (1972), 3–50 | Zbl

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

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

[31] Shokurov V. V., “Teorema Netera–Enrikvesa o kanonicheskikh krivykh”, Matem. sb., 86:3 (1971), 367–408 | Zbl