The Hilbert scheme of a pair of linear spaces

Abstract

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 $a,b,n$.

42 pages, 1 figure; To appear in Mathematische Zeitschrift