On Calabi-Yau varieties that are linear sections of Grassmannians of lines

In this paper we consider the $n-3$ dimensional linear section $X_{\mathrm{\Pi }}$ of the Grassmannian $\mathbb{G}(1,n)$ of lines in ${\mathbb{P}}^n$ with a general linear space $\mathrm{\Pi }$ of codimension $n+1$, with $n=2k+1$ odd and $n\ge 5$, that is a Calabi-Yau variety. Related to $X_{\mathrm{\Pi }}$ there is a degree $k+1$ Pfaffian hypersurface ${\mathcal{C}}_{\mathrm{\Pi }}$ that is the intersection of the dual of $\mathbb{G}(1,n)$ with ${\mathrm{\Pi }}^{\perp }$. We prove that ${\mathcal{C}}_{\mathrm{\Pi }}$ is rational and we provide a birational parametrization of it. Moreover we study in some detail the congruence of lines of ${\mathbb{P}}^n$ given by the centers of the linear complexes parametrised by the points of ${\mathcal{C}}_{\mathrm{\Pi }}$.