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The family of lines on the Fano threefold V5

Published online by Cambridge University Press:  22 January 2016

Mikio Furushima  and
Noboru Nakayama
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A smooth projective algebraic 3-fold V over the field C is called a Fano 3-fold if the anticanonical divisor — Kv is ample. The integer g = g(V) = ½(- Kv)3 is called the genus of the Fano 3-fold V. The maximal integer r ≧ 1 such that ϑ(— Kv)≃ ℋ r for some (ample) invertible sheaf ℋ ε Pic V is called the index of the Fano 3-fold V. Let V be a Fano 3-fold of the index r = 2 and the genus g = 21 which has the second Betti number b2(V) = 1. Then V can be embedded in P6 with degree 5, by the linear system |ℋ|, where ϑ(— Kv)≃ ℋ2 (see Iskovskih [5]). We denote this Fano 3-fold V by V5.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 116 , December 1989 , pp. 111 - 122
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Fujita, T., On the structure of polarized manifolds with total deficiency one II, J. Math. Soc. Japan, 33 (1981), 415–434.Google Scholar
[2] Furushima, M., Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C3 , Nagoya Math. J., 104 (1986), 1–28.CrossRefGoogle Scholar
[3] Furushima, N. - Nakayama, N., A new construction of a compactification of C3 , Tôhoku Math. J., 41 (1989), 543–560.Google Scholar
[4] Hirzebruch, F., Some problems on differentiate and complex manifolds, Ann. Math., 60 (1954), 213–236.CrossRefGoogle Scholar
[5] Iskovskih, V. A., Fano 3-fold I, Math. U.S.S.R. Izvestija, 11 (1977), 485–527.Google Scholar
[6] Peternell, Th.–Schneider, M., Compactifications of C3(I), Math. Ann., 280 (1988), 129–146.Google Scholar
[7] Miyanishi, M., Algebraic methods in the theory of algebraic threefolds, Advanced study in Pure Math. 1, 1983 Algebraic varieties and Analytic varieties, 66–99.Google Scholar
[8] Mori, S., Threefolds whose canonical bundle are not numerical effective, Ann. Math., 116 (1982), 133–176.Google Scholar
[9] Mukai, S. - Umemura, H., Minimal rational threefolds, Lecture Notes in Math., 1016, Springer-Verlag (1983), 490–518.Google Scholar