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Suppose $V$ is a vector space over $\mathbb{C}$ and $G\subset \textrm{GL}(V)$ is a connected linear algebraic group.

Consider the orbit closure $Y = \overline{G.\mathbb{P} L}$, for a subspace $L\subset V$.

Is there a nice description for all the linear spaces contained in $Y$?

I would be really happy with an answer in terms of the representation theory of $G$ and the stabilizer $G_L$.

I know the answer in the case of the Segre variety (I am pretty sure this is classical), and I guess that the answer for other homogeneous varieties might be known. So I'm really interested when the linear space $L$ is at least 2-dimensional.

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  • $\begingroup$ I think in general the answer is not known. $\endgroup$ – Sasha Dec 7 '13 at 15:22
  • $\begingroup$ What was wrong with the answer given by Peter Crooks? $\endgroup$ – Vít Tuček Dec 8 '13 at 2:31
  • $\begingroup$ The linear subspace suggested in the answer is not contained in the orbit closure. $\endgroup$ – Sasha Dec 8 '13 at 20:21
  • $\begingroup$ I guess this is a difficult question in general, so I guess I should keep working with small examples. $\endgroup$ – Luke Oeding Dec 9 '13 at 19:05

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