 Let $k=n\ge0$. If ${\widetilde}{G}_L$ is non-empty then $d=0$ or $d\geq n$. For $d>n$, ${\widetilde}{G}_L$ is irreducible and $G_L$ is smooth of the expected dimension $dn-n^0+0$. For $d=n$, $G_L$ is empty and ${\widetilde}{G}_L$ is irreducible of dimension $n$ (not of the expected dimension). For $d=0$, $G_L$ is empty and ${\widetilde}{G}_L$ consists of a single point.  