Let $V\in G(k,n)$. A family of $k$-dimensional subspaces (of $\Bbb{C}^n$) $\rho:V\to B$ parameterized by a variety $B$ induces a map of sets $B\to G(k,n)$ (taking a point $b\in B$ to the class of the fiber $\rho^{-1}(b)$). This map of sets is infact an algebraic morphism.
- Will every $k$-subspace $V$ have a morphism to any arbitrary variety $B$?
- What is "class of the fiber $\rho^{-1}(b)$? Does it mean that every $b\in B$ is mapped to all its pre-images with respect to $\rho$, across all elements of $G(k,n)$?