For the purposes of my research, I want to know if for any choices of dimensions $d_i,d_I\in \mathbb N\cup\{-1\},$ $$\mathcal V:=\{(L_1,\ldots,L_m):L_i\in \mathrm{Gr}(d_i,\mathbb P^n), \dim L_I\ge d_I\},$$ is an irreducible variety, where $L_I=\cap_{i\in I}L_i$ is the intersection of the linear spaces $L_i$ for $i\in I\subseteq [m]$.
Putting $d_I$ to $-1$ here means that there is no condition on how $L_i$ for $i\in I$ intersect.
I'm sure the answer to this question is stated in the literature, but I don't know where to look. Any help is appreciated.
I found the following example showing that $\mathcal V$ is not always irreducible. In $\mathbb P^3$ consider three lines $L_1,L_2,L_3$ with the condition that they pairwise intersect in points. This variety has two components: one where all $L_i$ are coplanar, and one where $L_i$ all meet in one point.