On some numerical characteristics of multidimensional Fano varieties

On the basis of homological properties of the homogeneous coordinate ring $R(V)$ of an $n$-dimensional projective Fano variety $V$ over the field $\mathbf C$ estimates are obtained for the degrees of generators of the algebra $R(V)$ and the ideal $I(V)$. All possible values are found for the dimension and the degree of a variety $V$ of codimension 3 in $\mathbf P^N$ which is not a complete intersection. We give a description of multidimensional Fano varieties of codimension 4 in $\mathbf P^N$ whose linear sections are canonical curves of genus 6.
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