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), of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in P^4 and 3-folds in P^5, and the Hilbert scheme stratification H_{c} of constant cohomology. For every (X) in Hilb(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_{c} and 1 + δ_X - \dim T_{c} are CI-biliaison invariants where T_{c} is the tangent space of H_{c} at (X). As a corollary we get a formula for the dimension of any generically smooth component of Hilb(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'', H_{c} -> E = isomorphism classes of graded artinian modules, given by sending C onto its Rao-module. For surfaces X in P^4 we have two Rao-modules M_i and an induced extension b in Ext^2(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'', H_{c} -> V = isomorphism classes of graded artinian modules M_i commuting with b, is smooth, and we get a smoothness criterion for H_{c}. 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 H_{c}, and for proving the main biliaison theorem above.