The Hilbert scheme of a pair of linear spaces

The Grassmannian is a smooth moduli space with very rich geometry that parameterizes simple varieties, namely, linear spaces. One can study a natural generalization, the component of a Hilbert scheme that parameterizes a pair of linear spaces in P^n. In this talk, I will describe the geometry of this component and show that they are smooth Mori dream spaces. Along the way, we will obtain a complete classification of the degenerations of a pair of linear spaces.