Liaison Invariants and the Hilbert Scheme of Codimension 2 Subschemes in ℙ^{n + 2}

Abstract

In this paper we study the Hilbert scheme Hilb^p(v)(ℙ) of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in ℙ⁴ and 3-folds in ℙ⁵, and the Hilbert scheme stratification H_{γ, ρ} of constant cohomology. For every (X) ∈ Hilb^p(v)(ℙ) we define a number δ _x in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + δ _x — dim_(X) H_{γ, ρ} and 1 + δ _x — dimT _{γ, ρ} are CI-biliaison invariants where T _{γ, ρ} is the tangent space of II_{γ, ρ} at (X). As a corollary we get a formula for the dimension of any generically smooth component of Hilb^p(v)(ℙ) in terms of δ _x and the CI-biliaison invariant. Both invariants are equal in this case.

Recall that, for space curves C, Martin-Deschamps and Perrin have proved the smoothness of the “morphism” ϕ: H_{γ, ρ} → E _ρ:= isomorphism classes of graded modules M satisfying dimM _v = ρ(v), given by sending C onto its Rao module. For surfaces X in ℙ⁴ we have two Rao modules M _i ≃⊕ H ⁱ(I _x (v)) of dimension ρ _i (v), ρ:= (ρ1, ρ2) and an induced extension b ∈ ₀Ext²(M ₂, M ₁) and a result of Horrocks and Rao saying that a triple D:= (M ₁, M ₂, b) of modules M _i of finite length and an extension b as above determine a surface X up to biliaison. We prove that the corresponding “morphism” φ: H_{γ, ρ} → V_ρ = isomorphism classes of graded modules M _i satisfying dim(M _i ) _v = ρ _i (v) and commuting with b, is smooth, and we get a smoothness criterion for H_{γ, ρ}, i.e., for the equality of the two biliaison invariants. Moreover we get some smoothness results for Hilb^p(v)(ℙ), valid also for 3-folds, and we give examples of obstructed surfaces and 3-folds. The linkage result we prove in this paper turns out to be useful in determining the structure and dimension of H_{γ, ρ}, and for proving the main biliaison theorem above.