Hilbert Scheme of a Pair of Codimension Two Linear Subspaces

Abstract

We study the component Hn of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in P n for n ≥ 3. We show that Hn is smooth and isomorphic to the blow-up of the symmetric square of G(n − 2, n) along the diagonal. Further Hn 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 Hn is a Mori dream space.

Mori dream spaces

“…It has been proved recently that the component of the Hilbert scheme which parametrises pairs of codimension two linear subspaces of P n is a Mori dream space, [6].…”

Hypersurfaces in P^5 containing unexpected subvarieties

“…In other words, there is a surjective map G −→ H, where H denotes the Hilbert scheme component of unions of two planes in P 5 . See [3] for the case of pairs of subspaces of codimension 2.…”

Detaching embedded points

“…The first named author applied the Mori program to the 12-dimensional component of twisted cubic curves, working out the the effective cone decomposition and the corresponding models, exhibiting it as a flip of the Kontsevich moduli space of stable maps over the Chow variety [1]. Similarly the Hilbert scheme component of unions of a pair of codimension two linear subspaces of P N is a smooth Mori dream space [2].…”

“…Proof. For d = 2 or 3, this was proved in [2] and [21], even though H d is not the full Hilbert scheme in these cases. For d = 1 and d ≥ 4, H d is the full Hilbert scheme, and it suffices to compute the global sections H 0 (N D ) of the normal sheaf associated to a point [D] ∈ H d , so let D be the union of plane curve C and the point p = (0, 0, 0, 1).…”