Linear dependence on the Grassmannian

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?

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