On Fano varieties of genus 6

In this paper it is proved that any nonsingular Fano variety $V_{10}$ of genus $6$ in $\mathbf P^7$ with $\operatorname{Pic}V_{10}\simeq\mathbf ZK_V$ is either a section $V_{10}^3$ of the Grassmannian $G(1,4)$ of lines in $\mathbf P^4$ by two hyperplanes and a quadric under the Plücker embedding of $G(1,4)$ in $\mathbf P^9$ or is the intersection ${V_{10}^3}'$ of a quadric and a cone over a section of $G(1,4)$ by a subspace of codimension $3$.
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