The Hilbert scheme of a pair of linear spaces is a Mori dream space

TL;DR

This paper investigates the geometric properties of the Hilbert scheme component parametrizing pairs of linear spaces, establishing its Mori dream space status and describing its effective and nef cones.

Contribution

It provides a comprehensive description of the effective and nef cones of the Hilbert scheme of pairs of linear spaces and proves it is a Mori dream space for all parameters.

Findings

The effective and nef cones of the Hilbert scheme component are explicitly described.

The Hilbert scheme component is shown to be a Mori dream space for all parameter values.

Conditions under which the component is Fano are determined.

Abstract

Let $H_{a,b}^n$ 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 describe the effective and nef cones of $H_{a,b}^n$ and determine when the component is Fano. Moreover, we show that $H_{a,b}^n$ is a Mori dream space for all values of $a,b,n$.