The Hilbert scheme of a pair of linear spaces

Let $$\mathcal {H}_{a,b}^n$$ 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 $${\mathbf {P}}^n$$ P n . We show that $$\mathcal {H}_{a,b}^n$$ H a , b n is smooth and isomorphic to successive blow ups of $$\mathbf {Gr}(a,n) \times \mathbf {Gr}(b,n)$$ Gr ( a , n ) × Gr ( b , n ) or $$\text {Sym}^2 \mathbf {Gr}(a,n)$$ Sym 2 Gr ( a , n ) along certain incidence correspondences. We classify the subschemes parameterized by $$\mathcal {H}_{a,b}^n$$ 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 $$\mathcal {H}_{a,b}^n$$ H a , b n and determine when the component is Fano. Moreover, we show that $$\mathcal {H}_{a,b}^n$$ H a , b n is a Mori dream space for all values of a, b, n.