4
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ How exactly do you embed? I see a map, but it is not an embedding. $\endgroup$ Apr 10, 2017 at 20:51
  • 1
    $\begingroup$ $G(i,V) \times G(j,W) \rightarrow G(i+j, V \oplus W)$ 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, 2017 at 21:02

1 Answer 1

4
$\begingroup$

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 $U\cap W$.

$\endgroup$
2
  • $\begingroup$ 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, 2017 at 21:48
  • $\begingroup$ @icmes Fair enough; I do my best to please. $\endgroup$
    – Ben Webster
    Apr 12, 2017 at 1:07

Your Answer

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