Let $X\subset\mathbb{P}^N$ be a non-degenerate smooth complex projective variety (manifold for short). Recall that $X$ is called prime Fano if $Pic(X)=\mathbb{Z}\langle \mathcal{O}(1)\rangle$ and if the anticanonical divisor $−K_X$ is ample. The index of $X$ is the positive integer defined by $−K_X= i(X)H$, with $H$ a hyperplane section of $X\subset\mathbb{P}^N$; the coindex of $X$ is $c(X)=\dim(X)-i(X)+1$.

Prime Fano manifolds of coindex 2 and 3 are completely classified; they are respectively the so-called Del-Pezzo manifolds and Mukai manifolds.

What is known when $c(X)=4$ or $c(X)=5$?

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    $\begingroup$ Some families having coindex $4$ were studied by Occhetta, Chierici and Novelli. Look at their papers on the arXiv. $\endgroup$ – Francesco Polizzi Jul 9 '11 at 7:18

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