  Let $\mathcal{H}_{a,b}^n$ denote the component of the Hilbert scheme whose
general point parameterizes an $a$-plane union a $b$-plane meeting transversely
in $\mathbf{P}^n$. We show that $\mathcal{H}_{a,b}^n$ is smooth and isomorphic
to successive blow ups of $\mathbf{Gr}(a,n) \times \mathbf{Gr}(b,n)$ or
$\text{Sym}^2 \mathbf{Gr}(a,n)$ along certain incidence correspondences. We
classify the subschemes parameterized by $\mathcal{H}_{a,b}^n$ and show that
this component has a unique Borel fixed point. We also study the birational
geometry of this component. In particular, we describe the effective and nef
cones of $\mathcal{H}_{a,b}^n$ and determine when the component is Fano.
Moreover, we show that $\mathcal{H}_{a,b}^n$ is a Mori dream space for all
values of $a,b,n$.
