It is easy to see that in $3$-dimensional Euclidean space, given $4$ lines in general position, there exists precisely one line who intersects with each of the $4$ lines. We call the $4$ lines determined this line.
I want to know the analogue of this statement in higher dimension, i.e. for given positive integers $N=n+m+1$, how many $n$-planes in general position will determine one $m$-plane in $N$-dimensional total space? If we denote the number by $d(n,m)$, for example the first paragraph just says $d(1,1)=4$. Then is there a formula for $d(n,m)$?
I just realized that we may not get one $m$-plane in general. It should be finite instead.
Now I found the finite intersection may also not happen. The question now we should ask is, for which $n,m$ we will have the finite interstion case?
My approach is to compute the dimension of $X(V^n)\subset \mathbb G(m,N)$ where $X(V^n)$ consists of all $m$-plane in $\mathbb P^N$ which intersects with the $n$-plane $V^n$. But I don't know how to do it.