Well, actually, what are the dimensions of the following two subvarieties of the Grassmannian. Let $N$ be a positive integer.
Let $V \subseteq \mathbb{C}^N$ be a linear subspace of dimension $N-k$ for some positive inter $k \leq N$. What is the dimension of the subvariety of $Gr(k,\mathbb{C}^N)$ consisting of those subspaces $W$ for which $V \cap W \neq \{0\}$?
Let $d$ and $l$ be positive integers with $l \leq N$. What is the dimension of the subvariety of $$ Gr\Bigg(\binom{l+d-1}{d},S^d(\mathbb{C}^N)\Bigg) $$
consisting of those subspaces that have a basis
$$\tag{1} \{v_{a_1} \cdot \dots \cdot v_{a_d} : a_1\leq a_2\leq \dots\leq a_d \in \{1,\dots, l\}\} $$ for some $v_1,\dots, v_l \in \mathbb{C}^N$? Here, $S^d(\mathbb{C}^N)$ denotes the symmetric subspace of $(\mathbb{C}^N)^{\otimes d}$, and the product $\cdot$ appearing in the above set is the symmetric product, $$ v_{a_1} \cdot \dots \cdot v_{a_d}=\frac{1}{d!} \sum_{\sigma \in S_d} v_{a_{\sigma(1)}} \otimes \dots \otimes v_{a_{\sigma(d)}}. $$
