Hilbert scheme of a pair of codimension two linear subspaces

TL;DR

This paper investigates the geometric structure of the Hilbert scheme component parametrizing pairs of codimension two linear subspaces in projective space, revealing its smoothness, blow-up structure, and Mori dream space properties.

Contribution

It establishes that H_n is smooth, isomorphic to a blow-up of the symmetric square of a Grassmannian, and characterizes its effective cone and modular interpretations.

Findings

H_n is smooth and isomorphic to a blow-up of the symmetric square of G(n-2,n)

H_n intersects only one other component transversely

H_n is a Mori dream space

Abstract

We study the component H_n of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in P^n for n > 2. We show that H_n is smooth and isomorphic to the blow-up of the symmetric square of G(n-2,n) along the diagonal. Further H_n intersects only one other component in the full Hilbert scheme, transversely. We determine the stable base locus decomposition of its effective cone and give modular interpretations of the corresponding models, hence conclude that H_n is a Mori dream space.