Jan O. Kleppe
In this paper we study the Hilbert scheme, Hilb(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 Hc of constant cohomology. For every (X) in Hilb(P) we define a number \delta X in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + \delta X - \dim(X) Hc and 1 + \delta X - \dim Tc are CI-biliaison invariants where Tc is the tangent space of Hc at (X). As a corollary we get a formula for the dimension of any generically smooth component of Hilb(P) in terms of \delta 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'', Hc -> E = isomorphism classes of graded artinian modules, given by sending C onto its Rao-module. For surfaces X in P4 we have two Rao-modules Mi and an induced extension b in Ext2(M2,M1) and a result of Horrocks and Rao saying that a triple D := (M1,M2,b) of modules Mi of finite length and an extension b as above determine a surface X up to biliaison. We prove that the corresponding ``morphism'', Hc -> V = isomorphism classes of graded artinian modules Mi commuting with b, is smooth, and we get a smoothness criterion for Hc. Moreover we get some smoothness results for Hilb(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 Hc, and for proving the main biliaison theorem above.