Liaison invariants and the Hilbert scheme of codimension 2 subschemes in P^{n+2}

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

Unobstructedness and dimension of families of codimension 3 ACM algebras

The goal of this paper is to study irreducible families of codimension 3, Cohen-Macaulay quotients A of a polynomial ring R=k[x_0,x_1,...,x_n]; mainly, we study families of graded Cohen-Macaulay quotients A of codimension 1 on a codimension 2 Cohen-Macaulay algebra B defined by a regular section of (S^2K_B*)_t, the 2. symmetric power of the dual of canonical modul of B in degree t. We give lower bounds for the dimension of the irreducible components of the Hilbert scheme which contains Proj(A). The components are generically smooth and the bounds are sharp if t >> 0 and n=4 and 5. We also deal with a particular type of codimension 3, Cohen-Macaulay quotients A of R; concretely we restrict our attention to codimension 3 arithmetically Cohen-Macaulay subschemes X of P^n defined by the submaximal minors of a symmetric homogeneous matrix. We prove that such schemes are glicci and we give lower bounds for the dimension of the corresponding component of the Hilbert scheme. In the last part of the paper, we collect some questions/problems which naturally arise in our context

On the normal sheaf of determinantal varieties

Let X be a standard determinantal scheme X of P^n of codimension c, i.e. a scheme defined by the maximal minors of a tx(t+c-1)homogeneous polynomial matrix A. In this paper, we study the main features of its normal sheaf \shN_X. We prove that under some mild restrictions: (1) there exists a line bundle \shL on X-Sing(X) such that \shN_X \otimes \shL is arithmetically Cohen–Macaulay and, even more, it is Ulrich whenever the entries of A are linear forms, (2) \shN_X is simple (hence, indecomposable) and, finally, (3) \shN_X is \mu-(semi)stable provided the entries of A are linear form

The Hilbert scheme of space curves sitting on a smooth surface containing a line

We continue the study of maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For s > 3, we extend the two ranges where W is a unique irreducible (resp. generically smooth) component of H(d,g)_{sc}. In another range, close to the boarder of the nef cone, we describe for s=4 and 5 components W that are non-reduced, leaving open the non-reducedness of only 3 (resp. 2) families for s > 5 (resp. s=5), thus making progress to recent results of Kleppe and Ottem in [28]. For s=3 we slightly extend previous results on a conjecture of non-reduced components, and in addition we show its existence in a subrange of the conjectured range

Families of Artinian and one-dimensional algebras

The purpose of this paper is to study families of Artinian or one dimensional quotients of a polynomial ring R with a special look to level algebras. Let GradAlg^H(R) be the scheme parametrizing graded quotients of R with Hilbert function H. Let B -> A be any graded surjection of quotients of R with Hilbert function H_B=(1,h_1,...,h_j,...) and H_A respectively. If dim A = 0 (resp. dim A = depth A = 1) and A is a "truncation" of B in the sense that H_A=(1,h_1,...,h_{j-1},s,0,0,...) (resp. H_A=(1,h_1,...,h_{j-1},s,s,s,...)) for some s < 1+h_j, then we show there is a close relationship between GradAlg^{H_A}(R) and GradAlg^{H_B}(R) concerning e.g. smoothness and dimension at the points (A) and (B) respectively, provided B is a complete intersection or provided the Castelnuovo-Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to "non-truncated" Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of deformations in a manageable form which we make rather explicit for level algebras of Cohen-Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg^H(R), H=(1,3,6,10,14,10,6,2), whose general elements are Artinian level algebras of type 2

Liaison invariants and the Hilbert scheme of codimension 2 subschemes in P^{n+2}

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

Families of low dimensional determinantal schemes

A scheme X in P^n of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t x t minors of a homogeneous t x (t+c-1) matrix (f_{ij}). Given integers a_0 0, resp. the postulation Hilbert scheme if n-c = 0). Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of W_s(b;a) in Hilb(P^n) and we show that Hilb(P^n) is generically smooth along W_s(b;a) under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of W_s(b;a) appearing in Kleppe and Miro-Roig [25]