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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 $G(3,6)$.

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    $\begingroup$ Buch -- Kresch -- Tamvakis $\endgroup$ – Jason Starr Jul 16 '14 at 0:41
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    $\begingroup$ I'm confused: I would have thought that [BKT] would give a birational isomorphism of this space with $G(2k,n)$, which obviously isn't happening here. $\endgroup$ – Allen Knutson Jul 16 '14 at 4:21
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    $\begingroup$ @AllenKnutson: [BKT] (birationally) identify the Kontsevich space of 3-pointed curves, $\overline{M}_{0,3}(G(k,n),k)$, with a fiber bundle over $G(2k,n)$ whose fiber is the parameter space of 3-tuples of points in $G(k,2k)$. $\endgroup$ – Jason Starr Jul 16 '14 at 11:09
  • $\begingroup$ Thank you Jason. As far as I've got, the idea is simple, these normal curves have the shape $(u_1+tv_1)\wedge ... \wedge (u_k+tv_k)$, where $(u_i,v_i)$ are bases in two $k$-planes, and $t$ is the parameter on the normal curve. $\endgroup$ – aglearner Jul 16 '14 at 11:58
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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

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