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The family of lines on the Fano threefold V 5

  • 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 .

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References

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[1] Fujita, T., On the structure of polarized manifolds with total deficiency one II, J. Math. Soc. Japan, 33 (1981), 415434.
[2] Furushima, M., Singular del Pezzo surfaces and analytic compactifications of 3-dimensional complex affine space C3 , Nagoya Math. J., 104 (1986), 128.
[3] Furushima, N. - Nakayama, N., A new construction of a compactification of C3 , Tôhoku Math. J., 41 (1989), 543560.
[4] Hirzebruch, F., Some problems on differentiate and complex manifolds, Ann. Math., 60 (1954), 213236.
[5] Iskovskih, V. A., Fano 3-fold I, Math. U.S.S.R. Izvestija, 11 (1977), 485527.
[6] Peternell, Th.Schneider, M., Compactifications of C3(I), Math. Ann., 280 (1988), 129146.
[7] Miyanishi, M., Algebraic methods in the theory of algebraic threefolds, Advanced study in Pure Math. 1, 1983 Algebraic varieties and Analytic varieties, 6699.
[8] Mori, S., Threefolds whose canonical bundle are not numerical effective, Ann. Math., 116 (1982), 133176.
[9] Mukai, S. - Umemura, H., Minimal rational threefolds, Lecture Notes in Math., 1016, Springer-Verlag (1983), 490518.
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The family of lines on the Fano threefold V 5

  • Mikio Furushima and Noboru Nakayama

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