Geodesic
Numerical invariants of families of lines on some Fano varieties
Sbornik. Mathematics, Tome 44 (1983) no. 2, pp. 239-260 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is known that one-dimensional families of lines lie on irrational Fano 3-folds of index 1. These families, which are the object of study in this paper, can be used to describe the birational geometry of the varieties themselves. The author computes such invariants as the genus of the parametrizing curve, the degree of the ruled surface swept out by the lines, and the number of lines lying on the variety and intersecting a given line. The theory of Chern characteristic classes and the Schubert calculus on Grassmannians are used in the computations. Bibliography: 13 titles. 
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     title = {Numerical invariants of families of lines on some {Fano} varieties},
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D. G. Markushevich. Numerical invariants of families of lines on some Fano varieties. Sbornik. Mathematics, Tome 44 (1983) no. 2, pp. 239-260. http://geodesic.mathdoc.fr/item/SM_1983_44_2_a7/

[1] Fano G., “Sulle varietà algebriche à tre dimensioni aventi curve – sezioni canoniche”, Mem. R. Acad. d'ltalia, 8 (1937), 23–64 | Zbl

[2] Pskovskikh V. A., “Antikanonicheskie modeli trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 12, VINITI, M., 1979, 59–158

[3] Pskovskikh V. A., “Biratsionalnye avtomorfizmy trekhmernykh algebraicheskikh mnogoobrazii”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 12, VINITI, M., 1979, 159–236

[4] Tyurin A. N., “Pyat lektsii o trekhmernykh mnogoobraziyakh”, UMN, XXVII:5 (1972), 3–50

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

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

[7] Griffiths P., Harris J., Principles of algebraic geometry, J. Wiley Sons, New York, Chichester, Brisbane, Toronto, 1978 | MR | Zbl

[8] Kleiman S. L., Laksov D., “Shubert calculus”, Amer. Math. Monthly, 79 (1972), 1061–1082 | DOI | MR | Zbl

[9] Grothendieck A., “La theorie des classes de Chern”, Bull. Soc. Math. France, 86 (1958), 137–154 | MR | Zbl

[10] Hartshorne R., Algebraic geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1977 | MR | Zbl

[11] Shokurov V. V., “Suschestvovanie pryamoi na mnogoobraziyakh Fano”, Izv. AN SSSR, ser. matem., 43:4 (1979), 922–964 | MR | Zbl

[12] Bloch S., Murre J. P., “On the Chow group of certain types of Fano threefolds”, Comp. Math., 39 (1979), 47–105 | MR | Zbl

[13] Tikhomirov A. S., “Srednii yakobian dvoinogo prostranstva $P^3$, razvetvlennogo v kvartike”, Izv. AN SSSR, ser. matem., 44 (1980), 1329–1377 | MR | Zbl