# The Hilbert scheme of a pair of linear spaces

## TL;DR

This paper studies the Hilbert scheme component parameterizing pairs of linear spaces in projective space, revealing its smoothness, geometric structure, and birational properties, including its Mori dream space status.

## Contribution

It provides a detailed geometric and birational analysis of the Hilbert scheme of pairs of linear spaces, including classifications, blow-up descriptions, and Mori dream space characterization.

## Key findings

- The component is smooth and isomorphic to blow-ups of Grassmannians.
- Classifies subschemes parameterized by the component.
- Determines the effective and nef cones, and when it is Fano.

## 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 $a,b,n$.