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

Abstract

AbstractIn this paper we study the Hilbert scheme Hilbp(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 Hγ, ρ of constant cohomology. For every (X) ∈ Hilbp(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) Hγ, ρ and 1 + δ x — dimT γ, ρ are CI-biliaison invariants where T γ, ρ is the tangent space of IIγ, ρ at (X). As a corollary we get a formula for the dimension of any generically smooth component of Hilbp(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: Hγ, 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: Hγ, ρ → Vρ = 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 Hγ, ρ, i.e., for the equality of the two biliaison invariants. Moreover we get some smoothness results for Hilbp(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 Hγ, ρ, and for proving the main biliaison theorem above.Mathematics Subject Classification (2000)14C0514D1514M0614M0714B1513D02KeywordsHilbert schemesurfaces in 4-space3-folds in 5-spaceunobstructednessgraded Betti numbersliaisonnormal sheaf