Consider the Grassmanian $G(k,n)$ ($k\le \frac{n}{2}$) and take its Plucker embedding. Consider now the space of all normal rational curves of degree $k$, contained in the Plucker embedding of the Grassmanian. Finally, take the closure of this space. I would like to know if this closure has some nice description.


Example. The Grassmanian $G(2,4)$ is a quadric $Q$ in $\mathbb P^5$. A normal curve of degree two in this quadric is just the intersection of $Q$ with a $\mathbb P^2$. So the corresponding space in this case is $G(3,6)$.



Rational normal curves on Grassmanians
In your example it seems that you are identifying the Hilbert scheme $H$ of conics in $G(2,4)$ with $G(3,6)$. There is a morphism from $H$ to $G(3,6)$ mapping a conic $C$ to the plane spanned by $C$. However, there are planes contained in $G(2,4)$. If $L$ is the locus in $G(3,6)$ parametrizing planes contained in $G(2,4)$, then you have a description of $H$ as the blow-up of $G(3,6)$ along $L$.


In general there is quite a nice scheme parametrizing rational curves in $G(k,n)$: the moduli space of stable maps $\overline{M}\_{0,0}(G(k,n),d)$ parametrizing rational curves of degree $d$ in $G(k,n)$. Since you are interested in curves of genus zero in a homogeneous variety this space is an irreducible scheme with rational quotient singularities. Many things about $\overline{M}\_{0,0}(G(k,n),d)$ are know. For instance the generators of its Picard group. You may look at:


<http://arxiv.org/abs/alg-geom/9608011>


and


<http://homepages.math.uic.edu/~coskun/utah-notes.pdf>


