On Fano varieties of index one that are linear sections of Grassmannians of lines

In this paper we consider the general $n-2$ dimensional linear section $X$ of the Grassmannian $\mathbb G(1,n)$ of lines in $\mathbb{P}^n$, that is a Fano variety of index 1. We first prove that for $n$ odd $X$ is birational to a hypersurface defined by a pfaffian determinant of order $n+1$ of linea