The Hilbert scheme of a pair of linear spaces

Ritvik Ramkumar

Published 2019-03-15Version 1

Let $H(c,d,n)$ be the component of the Hilbert scheme whose general point parameterizes a $c$-plane union a $d$-plane in $\mathbf{P}^n$. We show that $H(c,d,n)$ is non-singular and isomorphic to successive blow ups of $\mathbf{G}(c,n) \times \mathbf{G}(d,n)$ or $\text{Sym}^{2}\mathbf{G}(d,n)$. We also show that $H(c,d,n)$ has exactly one borel fixed point, describe the finitely many orbits under the $\text{PGL}_n$ action and describe the effective and ample cones. Using this, we prove that $H(n-3,n-3,n)$ and $H(1,n-2,n)$ are Mori dream spaces. In the case of $H(1,n-2,n)$, we show that the full Hilbert scheme contains only one more component and we give a complete description of its singularities. We end by describing the singularities of some Hilbert schemes parameterizing more than two linear spaces.