Geodesic
Fano 3-folds.~II
Izvestiya. Mathematics , Tome 12 (1978) no. 3, pp. 469-506

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In this paper Fano 3-folds of the principal series $V_{2g-2}$ in $\mathbf P^{g+1}$ are studied. A classification is given of trivial (i.e. containing a trigonal canonical curve) 3-folds of this kind. Among all Fano 3-folds of the principal series these are distinguished by the property that they are not the intersection of the quadrics containing them. It turns out that the genus $g$ of such 3-folds does not exceed 10. Fano 3-folds of genus one (i.e. with $\operatorname{Pic}V\simeq\mathbf Z$) containing a line are described. It is proved that they exist for $g\leqslant10$ and $g=12$. Their rationality for $g=7$ and $g\geqslant9$ is established by direct construction. Bibliography: 18 titles. 
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     author = {V. A. Iskovskikh},
     title = {Fano {3-folds.~II}},
     journal = {Izvestiya. Mathematics },
     pages = {469--506},
     publisher = {mathdoc},
     volume = {12},
     number = {3},
     year = {1978},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1978_12_3_a3/}
}
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V. A. Iskovskikh. Fano 3-folds.~II. Izvestiya. Mathematics , Tome 12 (1978) no. 3, pp. 469-506. http://geodesic.mathdoc.fr/item/IM2_1978_12_3_a3/