Singularities of orbit closures in module varieties and cones over rational normal curves

Let $N$ be a point of an orbit closure $\overline{O_M}$ in a module variety such that its orbit $O_N$ has codimension two in $\overline{O_M}$. We show that under some additional conditions the pointed variety $(\overline{O_M},N)$ is smoothly equivalent to a cone over a rational normal curve.