On Fano 3-folds with \(B_ 2\geq 2\). (English) Zbl 0537.14026

Algebraic varieties and analytic varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 101-129 (1983).
[For the entire collection see Zbl 0504.00008.]
This paper contains a clear exposition of a method for the classification of Fano threefolds X with \(B_ 2(X)\geq 2\) and states, in particular, the following result: \(B_ 2(X)\leq 10\) and there are 87 types of Fano threefolds with \(B_ 2(X)\geq 2\), up to deformations. The complete classification can be found in a paper of the authors [Manuscr. Math. 36, 147-162 (1981; Zbl 0478.14033)]. - Such a method is essentially based on the following tools: (i) extremal rays, according to Mori’s theory; (ii) blowing up of Fano threefolds; (iii) conic bundles; (iv) Sokurov’s results on the family of lines contained in the anticanonical model of a Fano threefold.
For the classification of Fano threefolds with \(B_ 2=1\) see: V. A. Iskovskih, Math. USSR, Izv. 12, 469-506 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 506-549 (1978; Zbl 0407.14016).
Reviewer: L.Picco Botta


14J30 \(3\)-folds