general points in Grassmannian

Question

what is the definition of general points or very general points in Grassmannian Gr(k,n)?

because two points in grassmannian may not be connected by a line.

Answer 1

As Mohan said in the comments, when someone says "For a set of $k$ points in $X$ in general position, bla is true", that means there is a proper closed subset $Z$ inside the symmetric product $X^{[k]}$ such that bla holds for all $k$-tuples outside $Z$. (For "very general" the story is similar, but now $Z$ might be a countable union of proper closed subsets.)

As Ted Shifrin said, what exactly the closed subset $Z$ is, depends on the problem in question.

In the case you are asking about, the author is talking in some places about two points in general position in $G(k,n)$, and he is assuming (by duality) that $n \geq 2k$. Looking at the proof of Theorem 3.8, we see that what this means here is that the two $k$-dimensional subspaces corresponding to the two points intersect only in the trivial subspace.

In the special case of $G(2,4)$, that is exactly the same as saying that the two points do not lie on a line, as you suggested in the comments.