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 linear forms in ${\mathbb{P}}^{n-1}$, and we study this birational transformation in some detail. Then for $n$ even, we prove that $X$ is unirational and birational to a hypersurface defined by a pfaffian determinant of order $n$ of linear forms in ${\mathbb{P}}^{n-1}$. Moreover we prove that the hypersurface of ${\mathbb{P}}^n$ described by the lines corresponding to the points in $X$ has degree $n-1$ and is rational, and we find a rational parametrization of it.