  Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The
purpose of this article is to further develop a method to classify varieties
having $C$ as their curve section, using Gaussian map computations. In a
previous article a careful analysis of the degeneration to the cone over the
hyperplane section was made for _prime_ Fano threefolds, that is Fano
threefolds whose Picard group is generated by the hyperplane bundle. In this
article we extend this method and classify Fano threefolds of higher index
(which still have Picard number one). We are also able to classify Mukai
varieties, i.e. varieties of dimension four or more with canonical curve
sections.
