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Is there a nice description of the variety $G(r,2r) \setminus \sqcup_{i+j=r}(G(i,r) \times G(j,r))$ in terms of blow ups or a sub-variety of a secant variety or any other natural construction to see it in a better way ? Where the product of Grassmannians $G(i,r) \times G(j,r)$ embedded in the bigger Grassmannian $G(r,2r)$ naturally.

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  • $\begingroup$ How exactly do you embed? I see a map, but it is not an embedding. $\endgroup$ – მამუკა ჯიბლაძე Apr 10 '17 at 20:51
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    $\begingroup$ $G(i,V) \times G(j,W) \rightarrow G(i+j, V \oplus W)$ by $((v_1 \wedge v_2 \cdots \wedge v_i), (w_1 \wedge w_2 \cdots \wedge w_j)) \mapsto (v_1 \wedge v_2 \cdots \wedge v_i \wedge w_1 \wedge w_2 \cdots \wedge w_j)$ by choosing bases. I am assuming $V \cap W = \emptyset$. $\endgroup$ – icmes Apr 10 '17 at 21:02
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Per my very recent answer on another question (https://mathoverflow.net/a/266282/66), consider an invertible linear transformation with two eigenspaces, both of dimension $n$ (for example, $\mathrm{diag}(2,\dots, 2,1,\dots,1)$). The variety you mention is the non-fixed points of this transformation (by the converse of the logic described in that answer).

EDIT: Is "a construction in terms of an incidence correspondence" covered by saying that it is also $\{ U\subset V\oplus W \mid \dim U=n, \dim (U\cap V)+\dim (U\cap W)<n\}.$ That is, it is the subspaces $U$ which are not spanned by $U\cap V$ and $U\cap W$.

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  • $\begingroup$ Thank you for your answer. I was looking for a construction in terms of incidence correspondence or in terms of scrolls or blow ups at some point. $\endgroup$ – icmes Apr 11 '17 at 21:48
  • $\begingroup$ @icmes Fair enough; I do my best to please. $\endgroup$ – Ben Webster Apr 12 '17 at 1:07

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