The purpose of this article is to develop further a method to classify varieties X subset of P-N having canonical curve section, using Gaussian map computations. In a previous article we applied these techniques to classify 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 and Mukai varieties, i.e., varieties of dimension four or more with canonical curve sections. First we determine when the Hilbert scheme Ht of such varieties X is nonempty. Moreover, in the case of Picard number one, we prove that H is irreducible and that the examples of Fano-Iskovskih and Mukai form a dense open subset of smooth points of H. [References: 22]
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