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If I have $k+1$ linearly dependent points on the Grassmannian $G_{k-1,n-1}$ (I am using projective dimension), what can I say on the corresponding $(k-1)$-subspaces of the projective space $\mathbb{P}^ {n-1}$?

E.g.: three collinear points of $G_{1,n-1}$ are the plucker embedding of three concurrent lines of $\mathbb{P}^ {n-1}$ contained in a plane.

What can I say in general?

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  • $\begingroup$ By "linearly dependent" you mean "linearly dependent after you embed the Grassmannian into a projective space via the Plücker embedding"? $\endgroup$ – darij grinberg Aug 30 '17 at 12:43
  • $\begingroup$ yes! thank you for pointing that out. $\endgroup$ – user46071 Aug 30 '17 at 12:58

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