  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 \delta_X in
terms of the graded Betti numbers of the homogeneous ideal of X and we prove
that 1 + \delta_X - \dim_{(X)} H_{c} and 1 + \delta_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 \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'', 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.
