3
$\begingroup$

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 $G(3,6)$.

$\endgroup$
4
  • 2
    $\begingroup$ Buch -- Kresch -- Tamvakis $\endgroup$ Jul 16, 2014 at 0:41
  • 1
    $\begingroup$ 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. $\endgroup$ Jul 16, 2014 at 4:21
  • 2
    $\begingroup$ @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 $G(k,2k)$. $\endgroup$ Jul 16, 2014 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, 2014 at 11:58

1 Answer 1

1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.