Fano 3-folds. II

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.
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