FAMILIES OF LOW DIMENSIONAL DETERMINANTAL SCHEMES.

JAN O. KLEPPE

Abstract. A scheme X ⊂ ℙ^{n} of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t × t minors of a homogeneous t × (t + c − 1) matrix (f_{ij}). Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we denote by W_{s}(\underline{b}; \underline{a}) ⊂ Hilb(ℙ^{n}) the stratum of standard determinantal schemes where f_{ij} are homogeneous polynomials of degrees a_{j} − b_{i} and Hilb(ℙ^{n}) is the Hilbert scheme (if n − c > 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}(\underline{b}; \underline{a}) in Hilb(ℙ^{n}) and we show that Hilb(ℙ^{n}) is generically smooth along W_{s}(\underline{b}; \underline{a}) under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of dim W_{s}(\underline{b}; \underline{a}) appearing in Kleppe and Miró-Roig [25].

1. Introduction

The goal of this paper is to study maximal families of determinantal schemes. Recall that a scheme X ⊂ ℙ^{n} of codimension c is called determinantal if its homogeneous saturated ideal can be generated by the r × r minors of a homogeneous p × q matrix (f_{ij}) with c = (p − r + 1)(q − r + 1). If r = min(p, q), then X is called standard determinantal. X is called good determinantal if it is standard determinantal and a generic complete intersection.

Let Hilb(ℙ^{n}) be the Hilbert scheme (resp. postulation Hilbert scheme, i.e. the Hilbert scheme of constant Hilbert function) parameterizing closed subschemes of ℙ^{n} of dimension n − c > 0 (resp. n − c = 0). Given integers a_{1} ≤ a_{2} ≤ ... ≤ a_{p} and b_{1} ≤ ... ≤ b_{q}, we denote by W (\underline{b}; \underline{a}) (resp. W_{s}(\underline{b}; \underline{a})) the stratum in Hilb(ℙ^{n}) consisting of good (resp. standard) determinantal schemes where f_{ij} are homogeneous polynomials of degrees a_{j} − b_{i}. Then W_{s}(\underline{b}; \underline{a}) is irreducible and W (\underline{b}; \underline{a}) ≠ ∅ if and only if W_{s}(\underline{b}; \underline{a}) ≠ ∅ (Corollary 2.1).

In this paper we focus, notably for zero dimensional schemes, on the following problems.

(1) Determine when the closure of W (\underline{b}; \underline{a}) is an irreducible component of Hilb(ℙ^{n}).

(2) Find the codimension of W (\underline{b}; \underline{a}) in Hilb(ℙ^{n}) if its closure is not a component.

(3) Determine when Hilb(ℙ^{n}) is generically smooth along W (\underline{b}; \underline{a}).

This paper generalizes and completes several results of [24] and [25] for schemes of dimension 0 or 1. Moreover we announced in [25], Rem. 6.3 that [24], §10 contains inaccurate results in the zero dimensional case, which we fully correct in this paper (Remark 4.26).

By successively deleting columns of the matrix associated to a determinantal scheme X, we get a nest (“flag”) of closed subschemes X = X_{c} ⊂ X_{c−1} ⊂ ... ⊂ X_{2} ⊂ ℙ^{n}. We

Date: November 3, 2010.

1991 Mathematics Subject Classification. Primary 14M12, 14C05, 13D10; Secondary 14H10, 14J10.

2 JAN O. KLEPPE

prove our results inductively by considering the smoothness of the Hilbert flag scheme of pairs and its natural projections into the Hilbert schemes. Note that, for c = 2, one knows that the closure \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb(ℙ^{n}) (i.e. Hilb(ℙ^{n}) is smooth along some non-empty open subset U of Hilb(ℙ^{n}) satisfying U ⊂ W (\underline{b}; \underline{a})), see Theorem 4.10.

In this approach we need to prove that certain (kernels of) Ext^{1}-groups vanish or to compute its dimensions. If dim X = 1 (resp. 0), then one (resp. 2 or 3) of these Ext^{1}- groups may be non-zero and its dimension (resp. the sum of its dimensions) is precisely the codimension of W (\underline{b}; \underline{a}) in Hilb(ℙ^{n}) under certain assumptions, see Theorem 4.19, Proposition 4.24 and Proposition 4.15 of Section 4. These are main results of this paper, together with Theorem 4.6 which through Proposition 4.13 and Lemma 4.4 give the tools we need in the proofs. As a consequence, if the mentioned Ext^{1}-groups vanish and c ≤ 5 or 6, we get that the closure \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb(ℙ^{n}) and that every deformation of a general X of W (\underline{b}; \underline{a}) comes from deforming the defining matrix (f_{ij}) of X. Note that this conclusion holds if dim X ≥ 2 and 3 ≤ c ≤ 4 because the above Ext^{1}-groups vanish by [25] and [24]. If the codimension of W (\underline{b}; \underline{a}) in Hilb(ℙ^{n}) is positive, there are deformations of X which do not come from deforming the matrix (f_{ij}). In the proofs we use results of [25] and [24] (see Section 3 which also contains a counterexample to the Conjectures of [25] in the case dim X = 0), as well as the Eagon-Northcott and Buchsbaum-Rim complexes ([9],[6], [10]). We give many examples, supported by Macaulay 2 computations [13], to illustrate the results.

As an application we use the results for zero dimensional schemes X = Proj(A) of this paper, together with the main result of [22] in which artinian Gorenstein rings are obtained by dividing A with ideals being isomorphic to a fixed twist of the canonical module of A, to contribute to the classification of Gorenstein quotients of a polynomial ring of e.g. codimension 4 from the point of view of determining PGor(H), cf. [19], [21].

Some of the results of this paper were lectured at the ”4th World Conference on 21st Century Mathematics 2009” in Lahore in March 2009. The author thanks the organizers for their hospitality. Moreover I thank prof. R.M. Miró-Roig at Barcelona for interesting comments and our discussion on the Conjectures 3.1 and 3.2 and the counterexample 3.3.

Notation: In this paper ℙ := ℙ^{n} will be the projective n-space over an algebraically closed field k, R = k[x_{0}, x_{1}, …, x_{n}] is a polynomial ring and 𝔪 = (x_{0}, …, x_{n}).

We mainly keep the notations of [25]. If X ⊂ Y are closed subschemes of ℙ^{n}, we denote by I_{X/Y} (resp. N_{X/Y}) the ideal (resp. normal) sheaf of X in Y. Note that by the codimension, codim_{Y} X, of an irreducible X in a not necessarily equidimensional scheme Y we mean dim O_{Y,x}−dim X, where x is a general k-point of X. For any closed subscheme X of ℙ^{n} of codimension c, we denote by I_{X} its ideal sheaf, N_{X} its normal sheaf, I_{X} = H_{∗}^{0}(I_{X}) its saturated homogeneous ideal and we let ω_{X} = Ext_{O_{ℙ^{n}}}^{c}(O_{X}, O_{ℙ^{n}})(−n − 1). When we write X = Proj(A) we take A := R/I_{X} and K_{A} = Ext_{R}^{c}(A, R)(−n − 1) for the canonical module of A or X. We denote the group of morphisms between coherent O_{X}-modules by Hom_{O_{X}}(F, G) while Hom_{O_{X}}(F, G) denotes the sheaf of local morphisms. Moreover we set hom(F, G) = dim_{k} Hom(F, G) and we correspondingly use small letters for the dimension, as a k-vector space, of similar groups.

We denote the Hilbert scheme by Hilb^{p}(ℙ^{n}), p the Hilbert polynomial [14], and (X) ∈ Hilb^{p}(ℙ^{n}) for the point which corresponds to the subscheme X ⊂ ℙ^{n}. We denote by GradAlg(H), or Hilb^{H}(ℙ^{n}), the representing object of the functor which parameterizes flat families of graded quotients A of R of depth_{𝔪} A ≥ min(1, dim A) and with Hilbert function H, H(i) = dim A_{i} ([21], [22]), and we call it “the postulation Hilbert scheme” ([23], §1.1) even though it may be different from the parameter space studied by Gotzmann, Iarrobino and others ([12], [19]) who study the “same” scheme with the reduced scheme structure (ours may be non-reduced and is equivalent to the Hilbert scheme of constant postulation considered in [33] in the curve case. They are both special cases of the multigraded Hilbert scheme of Haiman and Sturmfels [17]). Again we let (A), or (X) where X = Proj(A), denote the point of GradAlg(H) which corresponds to A. Note that if depth_{𝔪} A ≥ 1 and 0Hom_{R}(I_{X}, H_{𝔪}^{1}(A)) = 0, then

(1.1)  GradAlg(H) ≃ Hilb^{p}(ℙ^{n}) at (X) ,

and hence we have an isomorphism _{0}Hom(I_{X}, A) ≃ H^{0}(N_{X}) of their tangent spaces (cf.

[11] for the case depth_{𝔪} A ≥ 2, and [22], (9) for the general case). If (1.1) holds and X is generically a complete intersection, then _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) is an obstruction space of GradAlg(H) and hence of Hilb^{p}(ℙ^{n}) at (X) ([22], §1.1). When we simply write Hilb(ℙ^{n}), we interpret it as the Hilbert scheme (resp. postulation Hilbert scheme) if n−c > 0 (resp. n − c = 0). By definition X (resp. A) is unobstructed if Hilb^{p}(ℙ^{n}) (resp. Hilb^{H}(ℙ^{n})) is smooth at (X). Note that we called X H-unobstructed in [24] if A was unobstructed.

We say that X is general in some irreducible subset W ⊂ Hilb(ℙ^{n}) if (X) belongs to a sufficiently small open subset U of W such that any (X) in U has all the openness properties that we want to require.

2. Background

In this section we recall some basic results on standard (resp. good) determinantal schemes needed in the sequel, see [3], [4], [10] and [34] for more details. Let

(2.1)  φ : F = ⨁_{i=1}^{t} R(b_{i}) ⟶ G := ⨁_{j=0}^{t+c−2} R(a_{j})

be a graded morphism of free R-modules and let A = (f_{ij})_{i=1,...t}^{j=0,...,t+c−2}, deg f_{ij} = a_{j} − b_{i}, be a t × (t + c − 1) homogeneous matrix which represents the dual φ^{∗} := Hom_{R}(φ, R). Let I(A) = I_{t}(A) (or I_{t}(φ)) be the ideal of R generated by the maximal minors of A. In the following we always suppose

(2.2)  c ≥ 2, t ≥ 2, b_{1} ≤ ... ≤ b_{t} and a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2}.

Recall that a codimension c subscheme X ⊂ ℙ^{n} is standard determinantal if I_{X} = I(A) for some homogeneous t × (t + c − 1) matrix A as above. Moreover X ⊂ ℙ^{n} is a good determinantal scheme if additionally, A contains a (t − 1) × (t + c − 1) submatrix (allowing a change of basis if necessary) whose ideal of maximal minors defines a scheme of codimension c + 1. Note that if X is standard determinantal and a generic complete intersection in ℙ^{n}, then X is good determinantal, and conversely [27], Thm. 3.4.

4 JAN O. KLEPPE

Given integers b_{i} and a_{j} satisfying (2.2) we let W (\underline{b}; \underline{a}) (resp. W_{s}(\underline{b}; \underline{a})) be the stratum in Hilb(ℙ^{n}) consisting of good (resp. standard) determinantal schemes as above. Since we will not require A to be minimal (i.e. f_{ij} = 0 when b_{i} = a_{j}) for X = Proj(R/I_{t}(A)) to belong to W (\underline{b}; \underline{a}) or W_{s}(\underline{b}; \underline{a}), we need to reconsider [25], Cor. 2.6. Indeed looking to its proof and to [25], Rem. 3.7 and the end of p. 2877 (see the Preliminaries of [26] for details), we get

Corollary 2.1. The closures of W (\underline{b}; \underline{a}) and W_{s}(\underline{b}; \underline{a}) in Hilb(ℙ^{n}) are equal and irreducible. Moreover

W (\underline{b}; \underline{a}) ≠ ∅ ⇔ W_{s}(\underline{b}; \underline{a}) ≠ ∅ ⇔ a_{i−1} ≥ b_{i} for all i and a_{i−1} > b_{i} for some i .

Let A := R/I_{t}(A) and M := coker(φ^{∗}). Using the generalized Koszul complexes associated to a codimension c standard determinantal scheme X, one knows, for A minimal, that the Eagon-Northcott complex yields the following minimal free resolution

(2.3)  0 ⟶ ∧^{t+c−1}G^{∗} ⊗ S_{c−1}(F) ⊗ ∧^{t}F ⟶ ∧^{t+c−2}G^{∗} ⊗ S_{c−2}(F) ⊗ ∧^{t}F ⟶ …
⟶ ∧^{t}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{t}F ⟶ R ⟶ A ⟶ 0

of A and that the Buchsbaum-Rim complex yields a minimal free resolution of M;

(2.4)  0 ⟶ ∧^{t+c−1}G^{∗} ⊗ S_{c−2}(F) ⊗ ∧^{t}F ⟶ ∧^{t+c−2}G^{∗} ⊗ S_{c−3}(F) ⊗ ∧^{t}F ⟶ …
⟶ ∧^{t+1}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{t}F ⟶ G^{∗} ⟶ F^{∗} ⟶ M ⟶ 0.

See, for instance [4]; Thm. 2.20 and [10]; Cor. A2.12 and Cor. A2.13. Note that (2.3) show that any standard determinantal scheme is arithmetically Cohen-Macaulay (ACM).

Let B be the matrix obtained deleting the last column of A and let B be the k-algebra given by the maximal minors of B. Let Y = Proj(B). The transpose of B induces a map ϕ : F = ⊕_{i=1}^{t}R(b_{i}) → G′ := ⊕_{j=0}^{t+c−3}R(a_{j}). Let M_{B} be the cokernel of ϕ^{∗} = Hom_{R}(ϕ, R), let M_{A} = M and suppose c > 2. In this situation we recall that there is an exact sequence

(2.5)  0 ⟶ B ⟶ M_{B}(a_{t+c−2}) ⟶ M_{A}(a_{t+c−2}) ⟶ 0

in which B ⟶ M_{B}(a_{t+c−2}) is a regular section given by the last column of A. Moreover,

(2.6)  0 ⟶ M_{B}(a_{t+c−2})^{∗} := Hom_{B}(M_{B}(a_{t+c−2}), B) ⟶ B ⟶ A ⟶ 0

is exact by [27] or [24], (3.1), i.e. we may put I_{X/Y} := M_{B}(a_{t+c−2})^{∗}. Due to (2.4), M is a maximal Cohen-Macaulay A-module, and so is I_{X/Y} by (2.6). By [10] we have K_{A}(n + 1) ≃ S_{c−1}M_{A}(ℓ_{c}), and hence K_{B}(n + 1) ≃ S_{c−2}M_{B}(ℓ_{c−1}), where

(2.7)  ℓ_{i} := ∑_{j=0}^{t+i−2} a_{j} − ∑_{k=1}^{t} b_{k} for 2 ≤ i ≤ c.

Hence by successively deleting columns from the right hand side of A, and taking maximal minors, one gets a flag of determinantal subschemes

(2.8)  (X.) : X = X_{c} ⊂ X_{c−1} ⊂ ... ⊂ X_{2} ⊂ ℙ^{n}

where each X_{i+1} ⊂ X_{i} (with ideal sheaf I_{X_{i+1}/X_{i}} = I_{i}) is of codimension 1, X_{i} ⊂ ℙ^{n} is of codimension i and where there exist O_{X_{i}}-modules M_{i} fitting into short exact sequences

(2.9)  0 → O_{X_{i}}(−a_{t+i−1}) → M_{i} → M_{i+1} → 0 for 2 ≤ i ≤ c − 1,

such that I_{i}(a_{t+i−1}) is the O_{X_{i}}-dual of M_{i} for 2 ≤ i ≤ c, and M_{2} is a twist of the canonical module of X_{2}. In this context we let D_{i} := R/I_{X_{i}}, I_{D_{i}} = I_{X_{i}} and I_{i} := I_{D_{i+1}/I_{D_{i}}}.

Remark 2.2. Let α be a positive integer. If X is general in W (\underline{b}; \underline{a}) and a_{i−min(α,t)}−b_{i} ≥ 0 for min(α, t) ≤ i ≤ t, then X_{j}, for all j = 2, ⋯, c, is non-singular except for a subset of codimension at least min{2α − 1, j + 2}, i.e.

(2.10)  codim_{X_{j}} Sing(X_{j}) ≥ min{2α − 1, j + 2}.

As observed in Rem. 2.7 of [25], this follows from the Theorem of [7] by arguing as in [7], Example 2.1. See also [38], Prop. 1, [1], Sect. 2 and [35] for a related cases. In particular, if α ≥ 3, we get that for each i > 0, the closed embeddings X_{i} ⊂ ℙ^{n} and X_{i+1} ⊂ X_{i} are local complete intersections outside some set Z_{i} of codimension at least min(4, i + 1) in X_{i+1} (depth_{Z_{i}} O_{X_{i+1}} ≥ min(4, i + 1)), cf. next paragraph.

In what follows we always let Z ⊂ X (and similarly for Z_{i} ⊂ X_{i}) be some closed subset such that U := X − Z ↪ ℙ^{n} (resp. U_{i} := X_{i} − Z_{i} ↪ ℙ^{n}) is a local complete intersection (l.c.i.). Using that the 1^{st} Fitting ideal of M is equal to I_{t−1}(φ), we get that M̃ is locally free of rank one precisely on X − V (I_{t−1}(φ)) [3], Lem. 1.4.8. Since the set of non locally complete intersection points of X ↪ ℙ^{n} is exactly V (I_{t−1}(φ)) by e.g. [39], Lem. 1.8, we get that U ⊂ X − V (I_{t−1}(φ)) and that M̃ is locally free on U. Indeed M_{i} and I_{X_{i}}/I_{X_{i}}^{2} are locally free on U_{i}, as well as on U_{i−1} ∩ X_{i}. Note that since V (I_{t−1}(B)) ⊂ V (I_{t}(A)), we may suppose Z_{i−1} ⊂ X_{i}!

Finally notice that there is a close relation between M(a_{t+c−2}) and the normal module N_{X/Y} of the quotient B ≃ R/I_{Y} → A ≃ R/I_{X}. If we suppose depth_{I(Z)} B ≥ 2 where now Y − Z ↪ ℙ^{n} is an l.c.i., we get by applying Hom_{B}(I_{X/Y} , −) to (2.6), that

(2.11)  0 ⟶ B ⟶ M_{B}(a_{t+c−2}) ⟶ N_{X/Y}

is exact. Hence we have an injection M_{A}(a_{t+c−2}) ↪ N_{X/Y}, which in the case depth_{I(Z)} B ≥ 3 leads to an isomorphism M_{A}(a_{t+c−2}) ≃ N_{X/Y}. Indeed, this follows from the more general fact (by letting L = N = I_{X/Y}) that if L and N are finitely generated B-modules such that depth_{I(Z)} L ≥ r + 1 and Ñ is locally free on U := Y − Z, then the natural map

(2.12)  Ext_{B}^{i}(N, L) ⟶ H_{∗}^{i}(U, Hom_{O_{Y}} (Ñ, L̃))

is an isomorphism (resp. an injection) for i < r (resp. i = r), and H_{∗}^{i}(U, Hom_{O_{Y}} (Ñ, L̃)) ≃ H_{I(Z)}^{i+1}(Hom_{B}(N, L)) for i > 0, cf. [15], exp. VI. Note that we interpret I(Z) as m if Z = ∅.

3. The dimension of the determinantal locus

In [25] we conjectured the dimension of W (\underline{b}; \underline{a}) in terms of the invariant

(3.1) λ_{c} := ∑_{i,j} \binom{a_{i} − b_{j} + n}{n}+∑_{i,j} \binom{b_{j} − a_{i} + n}{n}−∑_{i,j} \binom{a_{i} − a_{j} + n}{n}−∑_{i,j} \binom{b_{i} − b_{j} + n}{n}+1.

Here the indices belonging to a_{j} (resp. b_{i}) range over 0 ≤ j ≤ t + c − 2 (resp. 1 ≤ i ≤ t), \binom{a}{n} = 0 if a < n and we always suppose W (\underline{b}; \underline{a}) ≠ ∅ in the following, cf. Corollary 2.1.

6 JAN O. KLEPPE

Conjecture 3.1. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we set ℓ_{i} := ∑_{j=0}^{t+i−2} a_{j} − ∑_{k=1}^{t} b_{k} and h_{i−3} := 2a_{t+i−2} − ℓ_{i} + n, for i = 3, 4, ..., c. Assume a_{i−min([c/2]+1,t)} ≥ b_{i} for min([c/2] + 1, t) ≤ i ≤ t. Then we have

dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ... + K_{c} , ( dim W (\underline{b}; \underline{a}) = λ_{2} if c = 2 )

where K_{3} = \binom{h_{0}}{n} and K_{4} = ∑_{j=0}^{t+1} \binom{h_{1}+a_{j}}{n} − ∑_{i=1}^{t} \binom{h_{1}+b_{i}}{n} and in general

K_{i+3} = ∑_{{r+s=i \atop r,s≥0}} ∑_{{0≤i_{1}<...<ir≤t+i \atop 1≤j_{1}≤...≤js≤t}}(−1)^{i−r}\binom{h_{i} + a_{i_{1}} + ⋯ + a_{i_{r}} + b_{j_{1}} + ⋯ + b_{j_{s}}}{n} for 0 ≤ i ≤ c − 3.

For the special case where all the entries of A have the same degree, this means:

Conjecture 3.2. Let W (\underline{0}; \underline{d}) be the locus of good determinantal schemes in ℙ^{n} of codimension c given by the maximal minors of a t×(t+c−1) matrix with entries homogeneous forms of degree d. Then,

dim W (\underline{0}; \underline{d}) = t(t + c − 1)\binom{d + n}{n} − t^{2} − (t + c − 1)^{2} + 1.

In [25] we proved that the right hand side in the formula for dim W (\underline{b}; \underline{a}) in the Conjectures is always an upper bound for dim W (\underline{b}; \underline{a}) ([25], Thm. 3.5), and moreover, that the Conjectures hold in the range

(3.2)  2 ≤ c ≤ 5 and n − c > 0 ( supposing chark = 0 if c = 5 ) ,

as well as for large classes in the range c ≥ 2 (without assuming n > c), cf. [25], §4.

Example 3.3 (Counterexample to the Conjectures in the range n = c ≥ 3).

Let A be a general 2 × (c + 1) matrix of linear entries. The vanishing of all 2 × 2 minors defines a reduced scheme X of c + 1 different points in ℙ^{c}. The conjectured dimension is c(c + 1) + c − 2 while the dimension of the postulation Hilbert scheme, dim_{(X)} Hilb^{H}(ℙ^{c}) is at most c(c + 1). Hence

dim W (0, 0; 1, 1, ..., 1) ≤ c(c + 1) .

This contradicts both conjectures for every c ≥ 3.

We have, however, looked at many examples in the range a_{0} > b_{t} where we have used Macaulay 2 to compute necessary invariants (cf. (3.3) below), without finding more counterexamples. The counterexample we have is only for zero dimensional schemes. Mainly because of this example the Conjecture is slightly changed (for zero-schemes) in [26], Conj. 4.1, excluding Example 3.3 from the new conjecture.

Now we recall a few statements from the proof of (3.2) and a variation which we will need in the next section. In the proof we used induction on c by successively deleting columns of the largest possible degree. Hence we computed the dimension of W (\underline{b}; \underline{a}), \underline{a} = a_{0}, a_{1}, ..., a_{t+c−2} in terms of dimension of W (\underline{b}; \underline{a′}), where \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3}. As in §2, we let X = Proj(A) belong to W (\underline{b}; \underline{a}) and we let Y = Proj(B), (Y) ∈ W (\underline{b}; \underline{a′}), be obtained by deleting the last column of A. We have Proposition 3.4. Let c ≥ 3, let (X) ∈ W (\underline{b}; \underline{a}) and suppose dim W (\underline{b}; \underline{a′}) ≥ λ_{c−1} + K_{3} + K_{4} + ... + K_{c−1} and depth_{I(Z)} B ≥ 2 for a general Y = Proj(B) ∈ W (\underline{b}; \underline{a′}). If

(3.3)  _{0}hom_{R}(I_{Y} , I_{X/Y}) ≤ ∑_{j=0}^{t+c−3} \binom{a_{j} − a_{t+c−2} + n}{n} ,

then dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ... + K_{c}. We also get equality in (3.3), as well as

dim W (\underline{b}; \underline{a}) = dim W (\underline{b}; \underline{a′}) + dim_{k} M_{B}(a_{t+c−2})_{0} − 1 − _{0}hom_{R}(I_{Y} , I_{X/Y}).

Proof. Indeed the proof of Thm. 4.5 of [25] contains the ideas we need, but since the assumptions of Thm. 4.5 are different, we include a proof. First we remark that we have

λ_{c} + K_{3} + K_{4} + ... + K_{c} ≥ dim W (\underline{b}; \underline{a})

by [25], Prop. 3.13 which combined with the assumption on dim W (\underline{b}; \underline{a′}) yields

λ_{c} − λ_{c−1} − K_{c} ≥ dim W (\underline{b}; \underline{a}) − dim W (\underline{b}; \underline{a′}).

Next by [25], Prop. 4.1 we have the inequality

dim W (\underline{b}; \underline{a}) − dim W (\underline{b}; \underline{a′}) ≥ dim_{k} M_{B}(a_{t+c−2})_{0} − 1 − _{0}hom_{R}(I_{Y} , I_{X/Y}).

Since K_{c} = _{0}hom(coker φ, R(a_{t+c−2})) by definition (see [25], Prop. 3.12 and (3.14)) we can use (2.4) and (2.5) to get

(3.4)  dim M_{B}(a_{t+c−2})_{0} − 1 = dim M_{A}(a_{t+c−2})_{0} =
dim F^{∗}(a_{t+c−2})_{0} − dim G^{∗}(a_{t+c−2})_{0} + _{0}hom(coker φ, R(a_{t+c−2})) =

= ∑_{i=1}^{t} \binom{a_{t+c−2} − b_{i} + n}{n} − ∑_{j=0}^{t+c−2} \binom{a_{t+c−2} − a_{j} + n}{n} + K_{c}.

Now looking at (3.1) and noticing that λ_{c−1} is defined by the analogous expression where a_{j} (resp b_{i}) ranges over 0 ≤ j ≤ t+c−3 (resp. 1 ≤ i ≤ t), it follows after a straightforward computation that

λ_{c} − λ_{c−1} = ∑_{i=1}^{t} \binom{a_{t+c−2} − b_{i} + n}{n} − ∑_{j=0}^{t+c−2} \binom{a_{t+c−2} − a_{j} + n}{n} − ∑_{j=0}^{t+c−3} \binom{a_{j} − a_{t+c−2} + n}{n}.

Combining with (3.3), we get

dim M_{B}(a_{t+c−2})_{0} − 1 − _{0}hom_{R}(I_{Y} , I_{X/Y}) ≥ λ_{c} − λ_{c−1} + K_{c} .

Hence all inequalities of displayed formulas in this proof are equalities and we are done. □

Theorem 3.5. The Conjectures (and if c > 2, the final dimension formula of Proposition 3.4) hold provided

2 ≤ c ≤ 5 and n − c > 0 ( supposing chark = 0 if c = 5 ) .

Indeed this is mainly [25], Thm. 4.5, Cor. 4.7, Cor. 4.10, Cor. 4.14 and [11] (c = 2) and [24] (c = 3). Moreover since the proofs of [25] also show (3.3), we get the final dimension formula of Proposition 3.4. Moreover we have (valid also for n = c and chark ≠ 0):

Remark 3.6. Assume a_{0} > b_{t}. Then (3.3) for X general, and Conjecture 3.1 hold provided 3 ≤ c ≤ 5 (resp. c > 5) and a_{t+c−2} > a_{t−2} (resp. a_{t+3} > a_{t−2}) by [26], Thm. 3.2.

8 JAN O. KLEPPE

4. the codimension of the determinantal locus

In this section we consider the problem of when the closure of W (\underline{b}; \underline{a}) is an irreducible component of Hilb(ℙ^{n}). If it is not a component, we determine its codimension in Hilb(ℙ^{n}) under certain assumptions. We also examine when Hilb(ℙ^{n}) is generically smooth along W (\underline{b}; \underline{a}). Moreover we have chosen to introduce the notion “every deformation of X comes from deforming A” because it gives the main reason for why \overline{W (\underline{b}; \underline{a})} is not always an irreducible component of Hilb(ℙ^{n}).

In the case the determinantal schemes are of dimension zero or one, then \overline{W (\underline{b}; \underline{a})} is not necessarily an irreducible component of Hilb(ℙ^{n}), as the following example shows.

Example 4.1 ( \overline{W (\underline{b}; \underline{a})} not an irreducible component in the range 0 ≤ n − c ≤ 1, c ≥ 3).

Let B be a general 2 × c matrix of linear entries and let A = [B, v] where the entries of the column v are general polynomials of the same degree 2. The vanishing all 2 × 2 minors of A defines a determinantal scheme X of codimension c in ℙ^{n}.

(i) Let n = c. Then X = Proj(A) is a reduced scheme of 2c+1 points in ℙ^{c} and with hvector (dim A_{i})_{i=0}^{∞} = (1, c + 1, 2c + 1, 2c + 1, ...). It follows that _{v}H_{m}^{1}(A) ≃ H^{1}(I_{X}(v)) = 0 for v ≥ 2 and we get _{0}Hom_{R}(I_{X}, H_{m}^{1}(A)) = 0. By (1.1) the postulation Hilbert scheme is isomorphic to the usual Hilbert scheme at (X), whose dimension is c(2c + 1). Moreover since the dimension of \overline{W (\underline{b}; \underline{a})} is at most the conjectured value c^{2} + 4c − 2, and since

c^{2} + 4c − 2 < c(2c + 1) for every c ≥ 3,

it follows that \overline{W (0, 0; 1, 1, ..., 1, 2)} is not an irreducible component of Hilb^{H}(ℙ^{c}).

(ii) Let n = c + 1. Then X is a smooth connected curve in ℙ^{c+1} of degree d = 2c + 1 and genus g = c. Since dim \overline{W (\underline{b}; \underline{a})} is at most the conjectured value, which is c^{2} + 7c + 2, and since dim_{(X)} Hilb^{p}(ℙ^{c+1}) is at least (n + 1)d + (n − 3)(1 − g) = c^{2} + 8c, it follows that \overline{W (0, 0; 1, 1, ..., 1, 2)} is not an irreducible component of Hilb^{p}(ℙ^{c+1}) for every c ≥ 3.

In what follows we briefly say “T a local ring” (resp. “T artinian”) for a local k-algebra (T, 𝔪_{T}) essentially of finite type over k = T/𝔪_{T} (resp. such that 𝔪_{T}^{r} = 0 for some integer r). Moreover we say “T → S is a small artinian surjection” provided there is a morphism (T, 𝔪_{T}) → (S, 𝔪_{S}) of local artinian k-algebras whose kernel 𝔞 satisfies 𝔞 · 𝔪_{T} = 0.

Let A = R/I_{t}(A). If T is a local ring, we denote by A_{T} = (f_{ij,T}) a matrix of homogeneous polynomials belonging to the graded polynomial algebra R_{T} := R ⊗_{k} T, satisfying f_{ij,T} ⊗_{T} k = f_{ij} and deg f_{ij,T} = a_{j} − b_{i} for all i, j. Note that all elements from T are considered to be of degree zero.

Once having such a matrix A_{T}, we get an induced morphism

(4.1)  φ_{T} : F_{T} := ⊕_{i=1}^{t}R_{T} (b_{i}) → G_{T} := ⊕_{j=0}^{t+c−2}R_{T} (a_{j})

and we put M_{T} = coker φ_{T}^{∗}.

Lemma 4.2. If X = Proj(A), A = R/I_{t}(A), is a standard determinantal scheme, then A_{T} := R_{T} /I_{t}(A_{T}) and M_{T} are (flat) graded deformations of A and M respectively for every choice of A_{T} as above. In particular X_{T} = Proj(A_{T}) ⊂ ℙ_{T}^{n} := Proj(R_{T}) is a deformation of X ⊂ ℙ^{n} to T with constant Hilbert function.

Proof. Consulting (2.3) and (2.4) we see that the Eagon-Northcott and Buchsbaum-Rim complexes are functorial in the sense that, over R_{T}, all free modules and all morphisms in these complexes are induced by φ_{T}, i.e. they are determined by A_{T}. Since these complexes become free resolutions of A and M respectively when we tensor with k over T, it follows that A_{T} and M_{T} are flat over T and satisfy A_{T} ⊗_{T} k = A and M_{T} ⊗_{T} k = M. □

Definition 4.3. Let X = Proj(A), A = R/I_{t}(A), be a standard determinantal scheme. We say “every deformation of X comes from deforming A” if for every local ring T and every graded deformation R_{T} → A_{T} of R → A to T, then A_{T} is of the form A_{T} = R_{T} /I_{t}(A_{T}) for some A_{T} as above. Note that by (1.1) we can in this definition replace “graded deformations of R → A” by “deformations of X ↪ ℙ^{n}” provided dim X ≥ 1.

Lemma 4.4. Let X = Proj(A), A = R/I_{t}(A), be a standard determinantal scheme, (X) ∈ W (\underline{b}; \underline{a}). If every deformation of X comes from deforming A, then A (and hence X if dim X ≥ 1) is unobstructed. Moreover \overline{W (\underline{b}; \underline{a})} is an irreducible component of Hilb(ℙ^{n}).

Proof. Let T → S be a small artinian surjection and let A_{S} be a deformation of A to S. By assumption, A_{S} = R_{S}/I_{t}(A_{S}) for some matrix A_{S}. Since T → S is surjective, we can lift each f_{ij,S} to a polynomial f_{ij,T} with coefficients in T such that f_{ij,T} ⊗_{T} S = f_{ij,S}. By Lemma 4.2 it follows that A_{T} := R_{T} /I_{t}(A_{T}) is flat over T. Since A_{T} ⊗_{T} S = A_{S} we get the unobstructedness of A, as well as the unobstructedness of X in the case dim X ≥ 1 by (1.1).

Finally let T be the local ring of Hilb(ℙ^{n}) at (X) and let A_{T}, or Proj(A_{T}) if dim X ≥ 1, be the pullback of the universal object of Hilb(ℙ^{n}) to Spec(T). Then there is a matrix A_{T} = (f_{ij,T}) such that A_{T} = R_{T} /I_{t}(A_{T}) by assumption. We can extend f_{ij,T} to polynomials f_{ij,D} with coefficients in D where Spec(D) ⊂ Hilb(ℙ^{n}) is an open neighborhood of (X) for which the Eagon-Northcott complex associated to the matrix A_{D} = (f_{ij,D}) is exact at any (X′) ∈ Spec(D) (cf. [32], Lem. 6.3 or [11], proof of Thm. 1; in our case the existence of Spec(D) is quite easy since the Eagon-Northcott complex of the homogeneous coordinate ring of a standard determinantal scheme is always exact). It follows that Spec(D) ⊂ W (\underline{b}; \underline{a}), and since Spec(D) is open in Hilb(ℙ^{n}) we are done. □

Remark 4.5. The arguments of these lemmas, which rely on the fact that we get T - flat schemes by just parameterizing the polynomials of A over a local ring T, is mostly known, see e.g. Laksov’s papers [29], [28] where he looks to flat families of determinantal schemes and their singular loci for arbitrary determinantal schemes. Indeed we may expect from Laksov’s papers (or prove by other arguments, as we remember Laksov did in a talk at the university of Oslo in the 70’s) that families of arbitrary determinantal schemes obtained by parameterizing polynomials as above are T -flat; thus he mainly shows the unobstructedness part of Lemma 4.4. The unobstructedness may also easily be deduced from of Schaps’ paper [36], Remark to Prop. 1. Since we in this paper only look at determinantal schemes defined through maximal minors, our proof only uses the exactness of the Eagon-Northcott complex which somehow contains the argument about generators and relations indicated in [36] as a main ingredient. Surprisingly enough the corresponding unobstructedness result of the R-module M (of maximal grade [5]) seems less known. Indeed one may prove the unobstructedness of M as we did for A in Lemma 4.4 because we from the Buchsbaum-Rim complex may see that every deformation of M to

10 JAN O. KLEPPE

T comes from deforming A. We have learned, by distributing a preliminary version of this paper to specialists in deformations of modules, that the unobstructedness of M was proved in Runar Ile’s PhD thesis, cf. [20], ch. 6 (Lem. 6.1.2 or Cor. 6.1.4).

The following result is a key result to our work in this section. Here the morphisms of the Ext^{1}-groups are induced by the inclusion I_{X/Y} ↪ B, e.g.

τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) → _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B).

Theorem 4.6. Let X = Proj(A) ⊂ Y = Proj(B) be good determinantal schemes defined by the vanishing of the maximal minors of A and B respectively where B is obtained by deleting the last column of A. Let Z ⊂ Y be a closed subset such that U := Y − Z ↪ ℙ^{n} is a local complete intersection and suppose

(1) depth_{I(Z)} B ≥ 3, \underline{o}r depth_{I(Z)} B ≥ 2 and ρ^{1} : _{0}Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) → _{0}Ext_{B}^{1}(I_{X/Y} , B) is injective,

(2) τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) → _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) is injective, and

(3) every deformation of Y comes from deforming B.

Then every deformation of X comes from deforming A. Moreover

dim_{(X)} Hilb(ℙ^{n}) = dim_{(Y)} Hilb(ℙ^{n}) + dim M_{B}(a_{t+c−2})_{0} − 1 − _{0}hom_{R}(I_{Y} , I_{X/Y}).

Remark 4.7. If depth_{I(Z)} B ≥ 3, then depth_{I(Z)} I_{X/Y} ≥ 3 and it follows from (2.12) that

0Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) = _{0}Ext_{B}^{1}(I_{X/Y} , B) = 0.

Remark 4.8. Let GradAlg(H_{2}, H_{1}) be the “postulation Hilbert-flag scheme”, i.e. the representing object of the functor deforming surjections (B → A) of graded quotients of R of positive depth at m, or equivalently flags (X ⊂ Y) of closed subschemes of ℙ^{n} with Hilbert functions H_{Y} = H_{B} = H_{2} and H_{X} = H_{A} = H_{1}. In [22], Prop. 4 (iii), we use theorems of Laudal on deformations of a category ([30]) to show that the forgetful morphism

GradAlg(H_{2}, H_{1}) ⟶ GradAlg(H_{1})

induced by (X ⊂ Y) ⟶ (X), is smooth and has fiber dimension_{0}hom_{R}(I_{Y} , I_{X/Y}) at (X ⊂ Y) provided B is unobstructed and (2) of Theorem 4.6 holds. By (1.1) this conclusion holds for the corresponding forgetful map from the Hilbert-flag scheme into the usual Hilbert scheme provided X and Y are ACM and dim X ≥ 1.

Proof. Let R_{T} → A_{T} be any graded deformation of R → A to a local ring T. By the smoothness of the forgetful map of Remark 4.8, there is a graded deformation R_{T} → B_{T} of R → B and a morphism B_{T} → A_{T}. By the assumption (3) there exists a matrix B_{T} such that B_{T} = R_{T} /I_{t}(B_{T}). By Lemma 4.2 B_{T} also defines a deformation M_{B_{T}} of M_{B}.

We will prove that there is a matrix A_{T} such that A_{T} = R_{T} /I_{t}(A_{T}) and such that we get B_{T} by deleting the last column of A_{T}. Looking at (2.5) and the text before and after (2.5), we see that if we can find a section B_{T} → M_{B_{T}} (a_{t+c−2}) which reduces to B → M_{B}(a_{t+c−2}) via (−) ⊗_{T} k, we can use this section to define a column v_{T} which allows us to put A_{T} := [B_{T} , v_{T}]. Now since we have a deformation B_{T} → A_{T} of B → A, it follows that I_{X_{T} /Y_{T}} := ker(B_{T} → A_{T}) is a deformation of I_{X/Y} ≃ M_{B}(a_{t+c−2})^{∗}. If we sheafify and restrict to U we get an isomorphism Ĩ_{X/Y} |_{U} ≃ M̃_{B}(a_{t+c−2})^{∗}|_{U} of invertible sheaves. Hence the flat sheaves (Ĩ_{X_{T} /Y_{T}})^{∗} and M̃_{B_{T}} (a_{t+c−2}) are also isomorphic on the set U_{T} which corresponds to U. Taking global sections, H_{∗}^{0}(U_{T} , −), of B̃_{T} → (Ĩ_{X_{T} /Y_{T}})^{∗}, we get a map which fits into a commutative diagram

[figure: B_{T}; H_{∗}^{0}(U_{T} , M̃_{B_{T}} (a_{t+c−2})); ≃ M_{B_{T}} (a_{t+c−2}); B; H_{∗}^{0}(U, M̃_{B}(a_{t+c−2})); ≃ M_{B}(a_{t+c−2})]

where the lower isomorphism follows from the fact that M_{B} is maximally CM, i.e. from

depth_{I(Z)} M_{B} = depth_{I(Z)} B ≥ 2 .

Note that since an M_{B}-regular sequence lifts to an M_{B_{T}} -regular sequence, we also have sufficient depth to get the upper isomorphism. Hence we get a section B_{T} → M_{B_{T}} (a_{t+c−2}) and an induced matrix A_{T}, as required.

Let A′ = R_{T} /I_{t}(A_{T}). We claim that A′ = A_{T}, i.e. that I′_{T} = I_{X_{T} /Y_{T}} where I′_{T} = ker(B_{T} → A′). Let T_{r} := T/𝔪_{T}^{r}, B_{T_{r}} := B_{T} ⊗_{T} T_{r},

I_{X_{Tr} /Y_{Tr}} := ker(B_{T_{r}} → A_{T} ⊗_{T} T_{r}) , I′_{T_{r}} = ker(B_{T_{r}} → A′ ⊗_{T} T_{r})

and S := T_{r−1}. To prove the claim we first show that I′_{T_{r}} = I_{X_{Tr} /Y_{Tr}} for every integer r > 0. To see that this follows from the assumption (1), we suppose by induction that I′_{S} = I_{X_{S}/Y_{S}} as ideals of B_{S}. Then I′_{T_{r}} and I_{X_{Tr} /Y_{Tr}} are two deformations of the same ideal I_{X_{S}/Y_{S}} ↪ B_{S} to T_{r} and their difference, as graded B_{T_{r}} modules, corresponds to an element,

diff ∈ _{0}Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) ⊗_{k} (𝔪_{T}^{r−1} · T_{r})

which via ρ^{1} maps to a difference, o_{1} − o_{2} ∈ _{0}Ext_{B}^{1}(I_{X/Y} , B) ⊗_{k} (𝔪_{T}^{r−1} · T_{r}) where o_{i} are the following obstructions. One of them, say o_{1} (resp. the other o_{2}) is the obstruction of deforming the graded morphism I′_{S} ↪ B_{S} (i.e. the ideal) to a graded morphism between I′_{T} and B_{T} (resp. between I_{X_{Tr} /Y_{Tr}} and B_{T_{r}}), cf. [22], Rem. 3 for a similar situation. Since I′_{T} and I_{X_{Tr} /Y_{Tr}} are ideals in B_{T_{r}}, such graded morphisms exist. Hence o_{i} = 0 for i = 1, 2, whence diff = 0 by the injectivity of ρ^{1}, and we conclude that I′_{T_{r}} = I_{X_{Tr} /Y_{Tr}}.

To get the claim let A′′ := B_{T} /(I′_{T} + I_{X_{T} /Y_{T}}). It suffices to show that the natural maps A′ → A′′ and A_{T} → A′′ are isomorphisms. Note that A′′ ⊗_{T} k ≃ A′′ ⊗_{R_{T}} R ≃ A and that we have similar isomorphisms for A′ and A_{T}. Notice also that every maximal ideal of R_{T} lies over 𝔪_{T}. Hence we get both isomorphisms by the lemma of Nakayama, Azumaya and Krull ([31], Lem. 1.M), i.e. by “Nakayama’s lemma” if we can show that A′′ is T -flat. But by the proof in the paragraph above the induced maps A′ ⊗_{T} T_{r} → A′′ ⊗_{T} T_{r} are isomorphism for every r > 0. It follows that A′′ ⊗_{T} T_{r} is T_{r}-flat since A′ ⊗_{T} T_{r} is! Since A′′ is idealwise separated for 𝔪_{T}, we get that A′′ is T -flat by Bourbaki’s local criterion of flatness (see Thm. 20.C of [31]).

12 JAN O. KLEPPE

It remains to prove the dimension formula. Recall that there is a “standard” diagram (whose square is cartesian) (4.2)

0 ↓ 0Hom_{R}(I_{Y} , I_{X/Y}) ↓

A^{1}  ⟶^{T_{pr_{1}}}  _{0}Hom_{R}(I_{Y} , B)
↓  □  ↓
0Hom(I_{X/Y} , A) ↪ _{0}Hom(I_{X}, A) ⟶ _{0}Hom_{R}(I_{Y} , A) ⟶^{δ} _{0}Ext_{B}^{1}(I_{X/Y} , A)

↓ 0

which defines the tangent space A^{1} of the Hilbert flag scheme GradAlg(H_{2}, H_{1}) at (B → A) and where the morphisms T_{pr_{1}} and δ are natural maps (cf. [22], (10) and note that the algebra cohomology group _{0}H^{2}(B, A, A) ≃ _{0}Ext_{B}^{1}(I_{X/Y} , A) (cf. [22], §1.1)). Under the assumption (2) the vertical sequence is exact. We claim that δ = 0. To see it, it suffices to prove that T_{pr_{1}} is surjective. The cartesian diagram is, however, well understood in terms of the deformation theory of the Hilbert flag scheme. Indeed if we take an arbitrary deformation B_{S} of B to the dual numbers S := k[t]/(t^{2}), then T_{pr_{1}} is surjective provided we can prove that there is a deformation (B_{S} → A_{S}) of (B → A) to S. The latter follows from the first part of the proof, or simply, from the assumption (3) because we by (3) get B_{S} = R_{S}/I_{t}(B_{S}) for some matrix B_{S} and we can take A_{S} = [B_{S}, v_{S}] where v_{S} is any lifting of the last column of A to S. Letting A_{S} := R_{S}/I_{t}(A_{S}) we get the claim.

Since we have dim_{(X)} Hilb(ℙ^{n}) = _{0}hom(I_{X}, A) and dim_{(Y)} Hilb(ℙ^{n}) = _{0}hom(I_{Y} , B) by Lemma 4.4, we get the dimension formula from the big diagram in which δ = 0 provided we can prove that _{0}hom(I_{X/Y} , A) = dim M_{B}(a_{t+c−2})_{0}−1. To see it we apply Hom(I_{X/Y} , −) onto 0 → I_{X/Y} → B → A → 0. If we use that Hom(I_{X/Y} , B) ≃ M_{B}(a_{t+c−2}), see (2.11), we get the exact sequence

(4.3) 0 → B → M_{B}(a_{t+c−2}) → Hom(I_{X/Y} , A) → Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) → Ext_{B}^{1}(I_{X/Y} , B),

and we conclude by the assumption (1). □

Remark 4.9. Suppose τ_{X/Y} is not injective. Then the vertical sequence in the diagram (4.2) is not exact, and δ may be non-zero. It is, however, easy to enlarge the diagram (4.2) to a diagram of exact horizontal and vertical sequences by including ker τ_{X/Y} and im δ in the diagram. From this enlarged diagram it follows that

0hom(I_{X}, A) = _{0}hom(I_{X/Y} , A) + h^{0}(N_{Y}) − _{0}hom(I_{Y} , I_{X/Y}) + dim ker τ_{X/Y} − dim im δ

since we have Hom(I_{Y} , B) ≃ H_{∗}^{0}(Y, N_{Y}) by (2.12) and depth_{m} B ≥ 2. The displayed formula also holds if τ_{X/Y} is injective.

Theorem 4.10. Suppose either c = 2 and n ≥ 2, \underline{o}r 3 ≤ c ≤ 4, n − c ≥ 2 and a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t. If W (\underline{b}; \underline{a}) ≠ ∅, then \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb(ℙ^{n}) of dimension

λ_{c} + K_{3} + ... + K_{c} .

Moreover every deformation of a general (X) ∈ W (\underline{b}; \underline{a}) comes from deforming A.

For c > 2 this result is really [25], Thm. 5.1 and Cor. 5.3 except for the final statement. Since, however, the proof of Theorem 4.10 is to apply Theorem 4.6 inductively to the flag (2.8), starting with the codimension 2 case where we by Lemma 4.11 know that the final statement holds, the proof of [25] extends to get Theorem 4.10 for c > 2. If c = 2 we get the other conclusions (and even more) from [11] for n > 2 and from works of Gotzmann and others for n = 2 as explained in [23], Rem. 22 (i), or see [25], Rem. 4.6 for a direct approach to dim W (\underline{b}; \underline{a}) = λ_{2}. (We also get all conclusions for c = 2 by combining Lemma 4.11 and Lemma 4.4.)

Lemma 4.11. If X = Proj(A), A = R/I_{t}(A), is a standard determinantal scheme of codimension 2 in ℙ^{n} and n ≥ 2, then every deformation of X comes from deforming A.

Proof. Let A = (f_{ij}) be a homogeneous t × (t + 1) matrix which represents the morphism φ^{∗} of (2.1) and let R_{T} → A_{T} be a graded deformation of R → A to a local ring T. To see that A_{T} is of the form A_{T} = R_{T} /I_{t}(A_{T}) for some matrix A_{T} reducing to A via (−) ⊗_{T} k, we consider the canonical module K_{A} = Ext_{R}^{2}(A, R)(−n − 1). Note that since c = 2 we have K_{A}(n + 1 − ℓ_{1}) = M, where G^{∗} ⟶^{φ^{∗}} F^{∗} → M → 0 is a part of the Buchsbaum-Rim complex, cf. (2.4) and (2.7). Now we observe that K_{A_{T}} := Ext_{R_{T}}^{2} (A_{T} , R_{T})(−n − 1) is a (flat) graded deformation of K_{A} to T because Ext_{R}^{i}(A, R) = 0 for i ≠ 2 ([8], Proposition (A1)). It follows that K_{A_{T}} (n + 1 − ℓ_{1}) = coker(φ_{T}^{∗}) where φ_{T}^{∗} corresponds to some matrix A_{T} := (f_{ij,T}), as in (4.1).

Let A′ := R_{T} /I_{t}(A_{T}). It suffices to show that A′ and A_{T} are isomorphic as R_{T} quotients. Looking to the Eagon-Northcott complex associated to A′ over R_{T} and dualizing, i.e. applying Hom_{R_{T}} (−, R_{T}) to it, we get back the part of the Buchsbaum-Rim complex where φ_{T}^{∗} appeared (up to twist). It follows that K_{A_{T}} ≃ K_{A′} where K_{A′} := Ext_{R_{T}}^{2} (A′, R_{T})(−n − 1). Note that the Buchsbaum-Rim complex above is a free resolution of K_{A′}(n + 1 − ℓ_{1}) over R_{T} since it is T -flat and reduces to a R-free resolution via a (−) ⊗_{T} k. Applying Hom_{R_{T}} (−, R_{T}), we get A′ ≃ Ext_{R_{T}}^{2} (K_{A′}, R_{T})(−n−1). Similarly by dualizing twice an R_{T} - free resolution of A_{T} we show that A_{T} ≃ Ext_{R_{T}}^{2} (K_{A_{T}} , R_{T})(−n − 1) and we are done. □

Remark 4.12. If we apply Theorem 4.6 successively to the flag (2.8) it is straightforward to generalize Thm. 5.1 of [25] to the zero dimensional case, i.e. we may replace the condition n ≥ 1 of [25], Thm. 5.1 by n ≥ 0 provided we in (i) of Thm. 5.1 (and correspondingly in (ii) of Thm. 5.1) replace the depth ≥ 3 condition by the condition (1) of Theorem 4.6.

There is a variation to Theorem 4.6 that we will use in the case n = c (dim X = 0) in which we mainly replace the injectivity assumption in (1) for the Ext^{1}-groups with the injectivity assumption for the corresponding Ext^{2}-groups. More precisely let

(4.4)  ρ^{i} : _{0}Ext_{B}^{i}(I_{X/Y} , I_{X/Y}) → _{0}Ext_{B}^{i}(I_{X/Y} , B).

be the map induced by I_{X/Y} ↪ B. Then we have

Proposition 4.13. Let X = Proj(A) ⊂ Y = Proj(B) be good determinantal schemes defined by A and B where B is obtained by deleting the last column of A. Let Z ⊂ Y be a closed subset such that U := Y − Z ↪ ℙ^{n} is a local complete intersection and suppose

14 JAN O. KLEPPE

(1) _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 (i.e., ρ^{1} surjective and ρ^{2} injective) and depth_{I(Z)} B = 2,

(2) τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) ↪ _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) is injective, and

(3) Y is unobstructed (this is weaker than (3) in Theorem 4.6).

Then A is unobstructed and the postulation Hilbert scheme Hilb^{H_{A}}(ℙ^{n}) satisfies

dim_{(X)} Hilb^{H_{A}}(ℙ^{n}) = dim_{(Y)} Hilb^{p_{Y}} (ℙ^{n})+dim M_{B}(a_{t+c−2})_{0}−1−_{0}hom_{R}(I_{Y} , I_{X/Y})+dim ker ρ^{1}.

Proof. Let T → S be a small artinian surjection with kernel 𝔞, and let R_{S} → A_{S} be any graded deformation of R → A to S. By Remark 4.8, there is a graded deformation R_{S} → B_{S} of R → B and a morphism B_{S} → A_{S}. By assumption (3) and (1.1) there exists a deformation R_{T} → B_{T} of R_{S} → B_{S} to T. It is well known that the algebra cohomology group _{0}H^{2}(B, A, A) ⊗_{k} 𝔞 contains the obstruction of deforming B_{S} → A_{S} further to B_{T} and that there is an injection _{0}H^{2}(B, A, A) ↪ _{0}Ext_{B}^{1}(I_{X/Y} , A) ([16], exp. VI, [22], §1.1). The rightmost group vanishes by (1), and it follows that A is unobstructed.

Finally we get the dimension formula from the diagram (4.2). Indeed the arguments are almost exactly the same as in the proof of Theorem 4.6 with the variation that (4.3) now implies

0hom(I_{X/Y} , A) = dim M_{B}(a_{t+c−2})_{0} − 1 + dim ker ρ^{1}.
□

Remark 4.14. We say that “A is unobstructed along any graded deformation of B” (call this phrase (*)) if for every small artinian surjection T → S and for every graded deformation B_{S} → A_{S} of B → A to S, there exists, for every graded deformation B_{T} of B_{S} to T, a graded deformation B_{T} → A_{T} reducing to B_{S} → A_{S} via (−) ⊗_{T} S. It is clear from the proof above that _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 implies (*) and moreover that we can generalize Proposition 4.13 by replacing the assumption _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 by (*).

Now we come to the main results of this paper which are direct consequences of Theorem 4.6 and Proposition 4.13. We start with determinantal curves whose result we will need in the zero dimensional case. Note that the result below is known ([24], Cor. 10.15 and Rem. 10.9 for c = 3, [25], Rem. 5.4 and Cor. 5.7 for 4 ≤ c ≤ 5) except for the final statement of (i) and most statements on the codimension in (ii) and (iii). With notations as in Theorem 4.6 we have

Proposition 4.15. Let X = Proj(A), A = R/I_{t}(A), be general in W (\underline{b}; \underline{a}) and suppose a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t, dim X = n − c = 1 and 3 ≤ c ≤ 5 (and chark = 0 if c = 5). If Y = Proj(B) is defined by the vanishing of the maximal minors of B where B is obtained by deleting the last column of A, then the following statements are true:

(i) If τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) →_{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) is injective, then X is unobstructed and \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{n}) of dimension λ_{c} + K_{3} + ... + K_{c}. Moreover every deformation of X comes from deforming A.

(ii) If _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) = 0, then X is unobstructed, dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} and

codim_{Hilb^{p}(ℙ^{n})} \overline{W (\underline{b}; \underline{a})} = dim ker τ_{X/Y} − _{0}ext_{B}^{1}(I_{X/Y} , A) .

(iii) We always have dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} and

codim_{Hilb^{p}(ℙ^{n})} \overline{W (\underline{b}; \underline{a})} ≤ dim ker τ_{X/Y} .

Moreover if _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0, then we have equality in the codimension formula if and only if X is unobstructed.

Proof. In all cases we use Theorem 3.5 to get dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c}.

(i) Since Theorem 4.10 applies to W (\underline{b}; \underline{a′}) ∋ (Y) where \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3}, we get (i) from Theorem 4.6, Remark 2.2 and Lemma 4.4.

(ii) The vanishing of the Ext^{1}-group of (ii) implies that X is unobstructed (X is l.c.i. by Remark 2.2), and moreover that im δ ≃ _{0}Ext_{B}^{1}(I_{X/Y} , A), cf. the diagram (4.2) and continue the horizontal exact sequence as a long exact sequence of algebra cohomology. Since we have h^{0}(N_{Y}) − dim W (\underline{b}; \underline{a′}) = 0 by Theorem 4.10 and _{0}hom(I_{X/Y} , A) = dim M_{B}(a_{t+c−2})_{0} − 1 by (4.3) and Remark 4.7, we conclude by Remark 4.9 and the final dimension formula of Theorem 3.5.

(iii) As in (ii) we get _{0}hom(I_{X}, A) − dim W (\underline{b}; \underline{a}) = dim ker τ_{X/Y} − im δ and hence the inequality of (iii). If the _{0}Ext_{B}^{1}(I_{X/Y} , A) vanishes, then im δ = 0, and since one knows that X is unobstructed if and only if we have equality in h^{0}(N_{X}) ≥ dim_{(X)} Hilb^{p}(ℙ^{n}), we conclude easily. □

Remark 4.16 (for the case where the codimension of X in ℙ^{n} is 3, i.e. c = 3).

(i) Since Y is licci ([24], Def. 2.10) for c = 3 , we always have _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) = 0 by [2] (or see [18] or [24], Prop. 6.17) and hence we get ker τ_{X/Y} ≃ _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}).

(ii) It is shown in [24], Cor. 10.11 (for n − c = 1) and Cor. 10.17 (for n − c = 0) that _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) = 0 provided a_{t+1} > a_{t} + a_{t−1} − b_{1}. Indeed the proofs of [24] (or [25], Cor. 5.10 (i)) show _{0}Ext_{R}^{1}(I_{Y} , I_{X/Y}) = 0 by mainly using the R-free minimal resolution of I_{Y} and the degree of the minimal generators of I_{X/Y} which we get from (2.3). Hence we can conclude by the injection _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) ↪ _{0}Ext_{R}^{1}(I_{Y} , I_{X/Y}). The vanishing of _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) is, however, much more common than given by the above argument. Indeed examining many examples by Macaulay 2 ([13]) in the range a_{0} > b_{t} = b_{1} we almost always got _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) = 0 provided a_{t+1} > 3 + b_{t} − n + c and 0 ≤ n − c ≤ 1.

Example 4.17 (determinantal curves in ℙ^{4}, i.e. with c = 3). (i) Let B be a general 2 × 3 matrix of linear entries and let A = [B, v] where the coordinates of the column v are general polynomials of the same degree m, m > 0. The vanishing of all 2 × 2 minors defines a smooth curve X = X_{m} of degree 3m + 1 and genus 3m(m − 1)/2 in ℙ^{4}. By Macaulay 2 (mainly),

0Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 if and only if m ≠ 2.

Its dimension is 1 if m = 2 in which case _{0}Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A) = 0 and _{0}Ext_{B}^{1}(I_{X_{m}/Y} , A) = 0. Note that we above only need to use Macaulay 2 for m ≤ 2 because the condition a_{t+1} > a_{t} + a_{t−1} − b_{1} of Remark 4.16 (ii) is equivalent to m > 2. It follows from Proposition 4.15 (i) that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{4}) of dimension λ_{3} + K_{3} for m ≠ 2, and from either (ii) or (iii) that X_{m} is unobstructed and codim_{Hilb(ℙ^{4})} \overline{W (\underline{b}; \underline{a})} = _{0}ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 1 for m = 2. Hence if m = 2, then dim_{(X_{m})} Hilb(ℙ^{4}) = λ_{3} + K_{3} + 1. This coincides with the c = 3 case of Example 4.1!

Finally computing λ_{3} and K_{3} by their definitions, we get λ_{3}+K_{3} = 17+(m+1)(3m+4)/2 for m > 1 and 21 for m = 1.

16 JAN O. KLEPPE

(ii) Let A be a general 2 × 4 matrix whose columns consist of general polynomials of the same degree, 1, 2, 3 and m, m ≥ 3 respectively. Put A = [B, v] where the coordinates of the column v are all of degree m. The vanishing of all 2 × 2 minors of A defines a smooth curve X =: X_{m} of degree 11m + 6 and genus (11m^{2} + 29m + 8)/2 in ℙ^{4}. By Remark 4.16 we get _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 for m > 5, but a Macaulay 2 computation shows this vanishing also for 3 ≤ m ≤ 5. It follows from Proposition 4.15 (i) that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{4}) of dimension λ_{3} + K_{3} =

85 + m(11m − 5)/2 for m > 3 , and 126 for m = 3.

We can also analyze the cases m = 1 and 2 by using Proposition 4.15 (i). Note that we now delete the column of degree 3 polynomials to define B, i.e. we let Y be defined by the maximal minors of the 2 × 3 matrix B consisting of linear (resp. degree 2 and m) entries in the first (resp. second and third) column. If m = 1 (resp. m = 2) one verifies that _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 by Macaulay 2 and we get that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{H}(ℙ^{3}) of dimension 66 (resp. 96).

Remark 4.18. We have checked the vanishing of _{0}Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A) for several m ≥ 1 in Example 4.17 (ii). It seems that this group is always non-zero for every m ≥ 1. This, we think, shows that the results presented here are quite strong because it is hard to show unobstructedness and to find dim_{(X_{m})} Hilb^{H}(ℙ^{n}) when even the “smallest known obstruction group, _{0}Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A),” does not vanish.

Now we consider zero dimensional determinantal schemes (n − c = 0). Indeed Theorem 4.6 with depth_{I(Z)} B = 2 and Proposition 4.13 are designed to take care of this case. We restrict our attention to a general X which through Remark 2.2 imply that all depth conditions of Theorem 4.6 and Proposition 4.13 are satisfied. Then our result leads e.g. to the unobstructedness of A where X = Proj(A). In fact for special choices of X, A may be obstructed [37]. First we consider codimension c = 3 determinantal subschemes and schemes with c ≥ 4 which we may treat similarly.

Theorem 4.19. Let X = Proj(A), A = R/I_{t}(A), be general in W (\underline{b}; \underline{a}) and let Y = Proj(B) and V = Proj(C) be defined by the vanishing of the maximal minors of B and C respectively where B (resp. C) is obtained by deleting the last column of A (resp. B). Suppose dim X = n − c = 0, a_{i−3} ≥ b_{i} for min(3, t) ≤ i ≤ t and suppose that (3.3) holds. Moreover suppose

either c = 3 or [ 4 ≤ c ≤ 6 and ker τ_{Y/V} = 0 ],

and suppose chark = 0 if c = 6. Then the following statements are true:

(i) If both ρ^{1} : _{0}Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) → _{0}Ext_{B}^{1}(I_{X/Y} , B) and τ_{X/Y} : _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) → 0Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) are injective, then A is unobstructed and \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of the postulation Hilbert scheme Hilb^{H}(ℙ^{c}) of dimension λ_{c} +K_{3} + ... + K_{c}. Moreover every deformation of X comes from deforming A.

(ii) If _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 and ker τ_{X/Y} = 0, then \overline{W (\underline{b}; \underline{a})} belongs to a unique generically smooth irreducible component Q of Hilb^{H}(ℙ^{c}) and the codimension of \overline{W (\underline{b}; \underline{a})} in Hilb^{H}(ℙ^{c}) is dim ker ρ^{1}. Indeed A is unobstructed and

dim Q = λ_{c} + K_{3} + ... + K_{c} + dim ker ρ^{1}.

(iii) If _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) = 0, then A is unobstructed, dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} and

codim_{Hilb^{H}(ℙ^{c})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{X/Y} − _{0}ext_{B}^{1}(I_{X/Y} , A).

(iv) We always have codim_{Hilb^{H}(ℙ^{c})} \overline{W (\underline{b}; \underline{a})} ≤ dim ker ρ^{1} + dim ker τ_{X/Y}. Suppose _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0. Then we have

codim_{Hilb^{H}(ℙ^{c})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{X/Y}

if and only if A is unobstructed.

Proof. In all cases we use Proposition 3.4 to get dim W (\underline{b}; \underline{a}) = λ_{c} +K_{3} +...+K_{c} since the Conjectures hold for W (\underline{b}; \underline{a′}) ∋ (Y) where \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3} by Theorem 3.5. Moreover if we apply Proposition 4.15 (i) to Y ⊂ V ⊂ ℙ^{n}, (Y) ∈ W (\underline{b}; \underline{a′}) (provided c > 3, if c = 3 we apply Theorem 4.10 to W (\underline{b}; \underline{a′}) ∋ (Y) ), it follows that every deformation of Y comes from deforming B.

(i) Using the above statements we easily conclude by Theorem 4.6 and Lemma 4.4. (ii) Now we use Proposition 4.13 instead of Theorem 4.6. Comparing the dimen-

sion formula of Proposition 4.13 with the final one of Proposition 3.4 and using that dim_{(Y)} Hilb^{p}(ℙ^{c}) = dim W (\underline{b}; \underline{a′}) by Lemma 4.4, we get all conclusions of (ii).

(iii) The vanishing of the Ext^{1}-group implies that A is unobstructed and that im δ ≃ 0Ext_{B}^{1}(I_{X/Y} , A), cf. (4.2). We have _{0}hom(I_{X/Y} , A) = dim M_{B}(a_{t+c−2})_{0} − 1 + dim ker ρ^{1} by (4.3) and h^{0}(N_{Y}) − dim W (\underline{b}; \underline{a′}) = 0 by Lemma 4.4. We conclude by Remark 4.9 and the final dimension formula of Proposition 3.4.

(iv) The proof is similar to the last part of (iii), cf. the proof of Proposition 4.15 (iii). □

Remark 4.20. (i) Looking to the proofs we see that we don’t need to suppose (3.3) to get the conclusions of (i) and (ii) which don’t involve dimension and codimension formulas.

(ii) Note the overlap in (ii) and (iv) of the theorem.

In [24], Ex. 10.18 we considered the example A = [B, v] where B was a general 2 × 3 matrix of linear entries and where the coordinates of the column v are general polynomials of the same degree m, m > 2. Using Macaulay 2 one may easily check that 0Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) = 0 for m = 3. Since I_{X/Y} = K_{B}^{∗}(−m − 1) by (2.6) and (2.7), it follows that _{0}Ext_{B}^{1}(I_{X/Y} , I_{X/Y}) is independent of m, and hence vanishes for every m ≥ 3. The families of zero dimensional schemes given in [24], Ex. 10.18 are therefore generically smooth of known dimension by Theorem 4.19 (i). More advanced examples are given in the examples below where several aspects of Theorem 4.19 are used. Note that the condition (3.3) in Theorem 4.19 is taken care of by Remark 3.6.

Example 4.21 (determinantal zero-schemes in ℙ^{3}, using mainly Theorem 4.19 (i)).

Let A = [B, v] be a general 2 × 4 matrix with linear (resp. cubic) entries in the first and second (resp. third) column and where the entries of v are polynomials of the same degree m, m ≥ 3. The vanishing of all 2 × 2 minors defines a reduced scheme X =: X_{m}

18 JAN O. KLEPPE

of 7m + 3 points in ℙ^{3}. For m = 3 one verifies that

0Ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 0 and _{v}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 and v ≤ 0

by Macaulay 2 mainly and since I_{X_{m}/Y} = K_{B}^{∗}(−m+1) by (2.6) and (2.7), we get the same conclusion for every m ≥ 3. It follows from Theorem 4.19 (i) that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{H}(ℙ^{3}) of dimension λ_{3} + K_{3} = 7m + 25 (resp. 45) for m > 3 (resp. m = 3).

If m = 1 or 2 we delete the column of degree 3 polynomials to define B, i.e. we let Y be defined by the maximal minors of the 2 × 3 matrix B consisting of linear (resp. degree m) entries in the first and second (resp. third) column. If m = 1 one verifies (by Macaulay 2) that

0Ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0

and we get by Theorem 4.19 (i) that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{H}(ℙ^{3}) of dimension λ_{3}+K_{3} = 22. If m = 2 one verifies that_{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 and that _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 2. Hence we can not use Theorem 4.19 (i), but we can use Theorem 4.19 (ii)! Such cases are more thoroughly explained in the next example. We verify that _{0}Ext_{B}^{1}(I_{X_{m}/Y} , B) = 0, to get dim ker ρ^{1} = _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}), and that 0Ext_{B}^{1}(I_{X_{m}/Y} , A) = 0. We conclude that \overline{W (\underline{b}; \underline{a})} is contained in a generically smooth irreducible component of Hilb^{H}(ℙ^{3}) of dimension λ_{3} + K_{3} + _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 37.

Example 4.22 (determinantal zero-schemes in ℙ^{3}, using mainly Theorem 4.19 (ii)).

Similar to Example 4.17 (ii) we let A = [B, v] be a general 2 × 4 matrix whose columns consist of general polynomials of the same degree, 1, 2, 3 and m, m ≥ 3 respectively. The vanishing of all 2 × 2 minors of A defines a reduced scheme X =: X_{m} of 11m + 6 points in ℙ^{3}. This time Macaulay 2 computations show _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 2 and vExt_{B}^{1}(I_{X_{m}/Y} , B) = 0 for every m ≥ 3 and every v ≥ 0 (we only need to check it for m = 3 because I_{X_{m}/Y} = K_{B}^{∗}(−m + 2) by (2.6) and (2.7)). It follows that

dim ker ρ^{1} = _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 2

for every m ≥ 3, i.e. we can not use Theorem 4.19 (i) at all. We have, however, 0Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 for m > 5 by Remark 4.16 (ii) and _{0}ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 2 (resp. 0) for m = 3 (resp. 3 < m ≤ 5) by Macaulay 2. Since _{0}Ext_{B}^{2}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 0 for m = 3 and hence for every m ≥ 3, we get _{0}Ext_{B}^{1}(I_{X_{m}/Y} , A) = 0 for m ≥ 3. We can therefore apply Theorem 4.19 (ii) in this situation except when m = 3. In the latter case 0Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A) = 0 and Theorem 4.19 (iii) applies. Hence Theorem 4.19 applies for every m ≥ 3, and we get that \overline{W (\underline{b}; \underline{a})} belongs to a unique generically smooth irreducible component of Hilb^{H}(ℙ^{3}) of codimension 2 (resp. 4) for m > 3 (resp. m = 3). Indeed A is unobstructed and

dim \overline{W (\underline{b}; \underline{a})} = λ_{3} + K_{3} = 11m + 35 for m > 3 and 67 for m = 3.

We remark that we have checked a possible vanishing of _{0}Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A) for several m ≥ 3, and in the range 3 < m ≤ 6 this group is non-zero.

Finally to be complete we consider the cases m = 1 and m = 2 in which case we will delete the column of degree 3 polynomials to define B and hence Y. If m = 1 we get by Macaulay 2 dim ker ρ^{1} = _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 2, _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 and 0Ext_{B}^{1}(I_{X_{m}/Y} , A) = 0. We have dim \overline{W (\underline{b}; \underline{a})} = λ_{3} + K_{3} = 35 and

codim_{Hilb^{H}(ℙ^{3})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} = 2

by Theorem 4.19 (ii). Moreover if m = 2 we get dim ker ρ^{1} = _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 4, 0ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 1,_{0}Ext_{B}^{1}(I_{X_{m}/Y} , A) = 0 and _{0}Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A) = 0 by Macaulay 2. By Theorem 4.19 (iii) we find dim W (\underline{b}; \underline{a}) = λ_{3} + K_{3} = 53 and

codim_{Hilb^{H}(ℙ^{3})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{X/Y} = 4 + 1 = 5.

Example 4.23 (Using Theorem 4.19 (i) with non-vanishing obstruction groups).

We let A = [B, v] be a general 2×4 matrix whose columns consist of general polynomials of the same degree, 2, 2, 4 and m, m ≥ 4 respectively. The vanishing of all 2 × 2 minors of A defines a reduced scheme X =: X_{m} of 20m + 16 points in ℙ^{3}. This time Macaulay 2 computations show _{0}ext_{B}^{1}(I_{X_{m}/Y} , I_{X_{m}/Y}) = 0 and _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X_{m}/Y}) = 0 (resp. = 1) for every m > 4 (resp. m = 4). It follows from Theorem 4.19 (i) that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{H}(ℙ^{3}) of dimension λ_{3} + K_{3} = 20m + 49 for m > 4. For m = 4 we have verified that _{0}ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A) = 3 and in this particular case we have not been able to verify whether A is unobstructed or not. But for every m > 4, A is unobstructed by Theorem 4.19 (i)! Moreover we have checked a possible vanishing of _{0}Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A) for many m, and combined with some theoretical arguments (which we don’t take here) we can conclude that this group is always non-zero for every m ≥ 4. Again, we think, this shows that the results presented here are quite strong because it is really hard to show unobstructedness when even the “smallest known obstruction group, 0Ext_{A}^{1}(I_{X_{m}}/I_{X_{m}}^{2}, A),” does not vanish.

In the final case 4 ≤ c ≤ 6 and ker τ_{Y/V} ≠ 0 where a general X = Proj(A) ⊂ Y = Proj(B) ⊂ V = Proj(C) is given by deleting columns as above we can not apply Theorem 4.6 to X ⊂ Y because there is no reason to expect condition (3) of Theorem 4.6 to be true (that condition is closely related to ker τ_{Y/V} = 0). But we can still use Proposition 4.13 since condition (3) of Proposition 4.13 is weakened to ”Y unobstructed”. The natural condition for ”Y unobstructed” which also give a formula for h^{0}(N_{Y})−dim W (\underline{b}; \underline{a′}) is _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) = 0, cf. the proof of Proposition 4.15 (ii). We get

Proposition 4.24. With notations as above, suppose 4 ≤ c ≤ 6 (let chark = 0 if c = 6), dim X = n − c = 0, a_{i−3} ≥ b_{i} for min(3, t) ≤ i ≤ t and suppose that (3.3) holds. Then dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} and the following statements are true:

(i) If _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0, _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) = 0 and _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) = 0 then A is unobstructed. Moreover W (\underline{b}; \underline{a}) is contained in a unique generically smooth irreducible component of Hilb^{H}(ℙ^{c}) of codimension dim ker ρ^{1} + dim ker τ_{Y/V} − _{0}ext_{C}^{1}(I_{Y/V} , B).

(ii) We always have codim_{Hilb^{H}(ℙ^{c})} \overline{W (\underline{b}; \underline{a})} ≤ dim ker ρ^{1} + dim ker τ_{X/Y} + dim ker τ_{Y/V}. Suppose _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 and _{0}Ext_{C}^{1}(I_{Y/V} , B) = 0. Then we have

codim_{Hilb^{H}(ℙ^{c})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{X/Y} + dim ker τ_{Y/V}

if and only if A is unobstructed (e.g. _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) = 0).

20 JAN O. KLEPPE

Proof. We have dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + ... + K_{c} by Proposition 3.4 since Theorem 3.5 applies to W (\underline{b}; \underline{a′}) where \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3}.

(i) This follows from Proposition 4.13 and Proposition 4.15 (ii) and by comparing the dimension formula of Proposition 4.13 with the final one of Proposition 3.4.

(ii) Combining Remark 4.9 and the final formula of Proposition 3.4 with (4.3), we get

0hom(I_{X}, A) − dim W (\underline{b}; \underline{a}) = h^{0}(N_{Y}) − dim W (\underline{b}; \underline{a′}) + dim ker ρ^{1} + dim ker τ_{X/Y} − im δ.

By the same argument we have h^{0}(N_{Y}) − dim W (\underline{b}; \underline{a′}) ≤ dim ker τ_{Y/V} and moreover, if _{0}Ext_{C}^{1}(I_{Y/V} , B) = 0, then equality holds. Hence we get the inequality of (ii), and furthermore, if the two Ext^{1}-groups of (ii) vanish then the inequality of (ii) is an equality if and only if dim_{(X)} Hilb^{H}(ℙ^{c}) = _{0}hom(I_{X}, A) and we are done. □

Example 4.25 (determinantal zero dimensional schemes in ℙ^{4}, i.e. with c = 4).

Let A = [B, v] be a general 2 × 5 matrix with linear (resp. quadratic) entries in the first, second and third (resp. fourth) column and let both entries of the column v be of degree m ≥ 2. Keeping the notations of Proposition 4.24, we get that the vanishing of all 2 × 2 minors defines a reduced scheme X of 7m + 2 points in ℙ^{4}. One verifies that dim ker ρ^{1} = _{0}ext_{B}^{1}(I_{X/Y} , I_{X/Y}) = 3, _{0}Ext_{B}^{1}(I_{X/Y} , A) = 0 and that (3.3) holds by Remark 3.6. Note that we have dim ker τ_{Y/V} = 1 and _{0}Ext_{C}^{1}(I_{Y/V} , B) = 0 from Example 4.17 (i).

Suppose m > 2. Then _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) = _{0}Ext_{B}^{1}(I_{Y} /I_{Y}^{2} , B) = 0 for every m > 2 and it follows from Proposition 4.24 (i) that A is unobstructed and dim W (\underline{b}; \underline{a}) = λ_{4} + K_{3} +K_{4} = 7m+31. Hence W (\underline{b}; \underline{a}) is contained in a unique generically smooth irreducible component of the postulation Hilbert scheme Hilb^{H}(ℙ^{4}) and,

codim_{Hilb^{H}(ℙ^{4})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{Y/V} = 3 + 1 = 4.

Suppose m = 2. Since dim ker τ_{X/Y} = _{0}ext_{B}^{1}(I_{Y} /I_{Y}^{2} , I_{X/Y}) = 4 and _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) = 0, it follows from Proposition 4.24 (ii) that A is unobstructed and that dim W (\underline{b}; \underline{a}) = λ_{4} + K_{3} + K_{4} = 44. Hence W (\underline{b}; \underline{a}) is contained in a unique generically smooth irreducible component of Hilb^{H}(ℙ^{4}) and,

codim_{Hilb^{H}(ℙ^{4})} \overline{W (\underline{b}; \underline{a})} = dim ker ρ^{1} + dim ker τ_{X/Y} + dim ker τ_{Y/V} = 3 + 4 + 1 = 8.

In this case we see that all three kernels of Proposition 4.24 (ii) contribute to the codimension of W (\underline{b}; \underline{a}) in Hilb^{H}(ℙ^{4})!

Remark 4.26. If we apply Theorem 4.6 successively to the flag (2.8) we get Prop. 10.12 and Thm. 10.13 of [24] in a correct version (the injectivity of ρ^{1}, i.e. the assumption (1) of Theorem 4.6 in the case depth_{I(Z)} B = 2 lacked in [24]). Indeed in [25], Rem. 6.3 we announced that some results in §10 of [24] were inaccurate, and in the new hypothesis (*) of Rem. 6.3 we increased the depth assumption of the corresponding hypothesis in [24] by 1 to get valid results. The new hypothesis (*) applies to determinantal schemes of positive dimension, i.e. the results of [24], §10 hold in this case. In the zero dimensional case we introduced, in addition to (*) of Rem. 6.3, an assumption (Rem. 6.3 (ii)), which is equivalent to the injectivity of ρ^{1}. This assumption makes the results of [24], §10 correct in the zero dimensional case. In [25], Rem. 6.3 (ii) we indicate a proof for this claim, and now Theorem 4.6 provides us with another proof. In [25], Rem. 6.3 (i) and (iii), we claimed that e.g. the unobstructedness of A also implied all results of [24], §10, but this is a little inaccurate because the very final result of [24] (Cor. 10.17) uses the injectivity of ρ^{1} to get the dimension formula. E.g. in Example 4.22 for m > 3 (resp. Example 4.21 with m = 2) the formula of Cor. 10.17 gives dim_{(X)} Hilb^{H}(ℙ^{3}) = dim W (\underline{b}; \underline{a}), which should be correct according to Rem. 6.3 (i) (resp. Rem. 6.3 (iii)). The correct dimension is, however, dim_{(X)} Hilb^{H}(ℙ^{3}) = dim W (\underline{b}; \underline{a}) + dim ker ρ^{1}, dim ker ρ^{1} = 2 in both cases. This observation is a reason for writing this paper, namely to provide detailed proofs in the zero dimensional case for the correction “Rem. 6.3 (ii)” and to present several results related to [25], Rem. 6.3 (i) and (iii) (see Proposition 4.13, Theorem 4.19, Proposition 4.24 where we see that we have to add dim ker ρ^{1} to get valid (co)dimension formulas. Note also the obvious misprint in Rem. 6.3, that I_{c} should have been I_{c−1}). Thus, letting D_{i} = R/I_{X_{i}} and I_{i} = I_{X_{i+1}/X_{i}}, the following hypothesis makes all results of [24], §10, true for good determinantal schemes X with dim X ≥ 0;

Given X ⊂ ℙ^{n} a good determinantal scheme of dimension n − c, we will assume that there exists a flag X = X_{c} ⊂ X_{c−1} ⊂ ... ⊂ X_{2} ⊂ ℙ^{n} such that for each i < c, the closed embedding X_{i+1} ↪ X_{i} is l.c.i. outside some set Z_{i} of codimension 2 in X_{i+1} (depth_{Z_{i}} O_{X_{i+1}} ≥ 2). Moreover, we suppose X_{2} ↪ ℙ^{n} is an l.c.i. in codimension ≤ 1 and if c = n we suppose that _{0}Ext_{D_{c−1}}^{1}(I_{c−1}, I_{c−1}) ↪ _{0}Ext_{D_{c−1}}^{1}(I_{c−1}, D_{c−1}) is injective.

5. applications to families of Gorenstein quotients

The results of the preceding section lead to many well described generically smooth components of Hilb(ℙ^{n}) of known dimension. Once we have such a component of Hilb^{H_{A}}(ℙ^{n}) consisting, say, of codimension 3 zero dimensional schemes X = Proj(A), then regular sections of a twist of the anticanonical module of A lead to artinian Gorenstein codimension 4 quotients through the exact sequence

(5.1)  0 → K_{A}(−s) ⟶^{σ} A → D → 0 .

Indeed by [22], Thm. 16, we have the following result (true for arbitrary dimension of X).

Theorem 5.1. Let A = R/I_{A} be a graded, CM quotient with canonical module K_{A}, let X := Proj(A) be locally Gorenstein and let D be the Gorenstein algebra given by a regular section of σ ∈ (K_{A}^{∗})_{s} for some integer s. If s >> 0, then D is H_{A}-generic. Moreover A is unobstructed if and only if D is unobstructed, and

dim_{(D)} Hilb^{H_{D}}(ℙ^{n}) = dim_{(X)} Hilb^{H_{A}}(ℙ^{n}) + dim(K_{A}^{∗})_{s} − 1 .

Here D is said to be H_{A}-generic if there is an open subset of Hilb^{H_{D}}(ℙ^{n}) containing (D) whose members D′ are quotients of some quotient A′ of R with Hilbert function H_{A}. In fact if X is general in some irreducible component of Hilb^{H_{A}}(ℙ^{n}) we may suppose D is general in some component of Hilb^{H_{D}}(ℙ^{n}), see [22], Thm. 24 for the entire correspondence.

To make s >> 0 precise, we notice that Theorem 5.1 holds for every integer s satisfying

(5.2)  _{s}Ext_{A}^{1}(S_{2}(K_{A}), K_{A}) = 0 and _{−s}Ext_{R}^{i}(I_{A}, K_{A}) = 0 for i = 0, 1 ,

see Lem. 9, Prop. 10 (ii) and proof of Thm. 27 of [23] to understand that we may replace “_{−s}Hom_{B}(H_{2}(R, A, A), K_{A}) = 0 (e.g. I_{A} is generically syzygetic) and_{−s}Ext_{A}^{1}(I_{A}/I_{A}^{2}, K_{A}) = 0 ” of [22], Prop. 13 and Thm. 16 with “_{−s}Ext_{R}^{1}(I_{A}, K_{A}) = 0 ”. Hence we can still use the arguments of [22] which consider minimal resolutions of I_{A} and K_{A} to find s. We get

Remark 5.2. (i) If X is a zero-scheme of degree d, we may replace dim_{(D)} Hilb^{H_{D}}(ℙ^{n}) by dim_{(D)} PGor(H_{D}), s >> 0 by s ≥ 2 reg(I_{A}) and dim(K_{A}^{∗})_{s} by d (cf. [22], Rem. 22).

(ii) If X is an integral curve of degree d and arithmetic genus p, we may replace s >> 0 by s > max{2 reg(I_{A}) − 2, (4p − 4)/d} and dim(K_{A}^{∗})_{s} by ds + 3 − 3p (cf. [22], Rem. 18).

We finish this paper by taking two far from straightforward examples.

Example 5.3 (Artinian Gorenstein quotients of R of codimension 4).

Take the determinantal zero dimensional scheme X of Example 4.22 in which A = [B, v] was a general 2 × 4 matrix whose columns consisted of polynomials of the same degree, 1, 2, 3 and m respectively. In that example X = Proj(A) was a reduced scheme of 11m + 6 points in ℙ^{3} and \overline{W (\underline{b}; \underline{a})} was a proper closed subset of a generically smooth component V of Hilb^{H_{A}}(ℙ^{3}) of codimension 2 (resp. 4 or 5) for m > 3 and m = 1 (resp. m = 3 or 2). Moreover

dim \overline{W (\underline{b}; \underline{a})} = λ_{3} + K_{3} = 11m + 35 for m > 3

and dim \overline{W (\underline{b}; \underline{a})} = 67, 53 or 35 for m = 3, 2 or 1 respectively. Using (2.3) we find the leftmost part of a minimal resolution of A and hence of the general element R/J of V to be 0 → R(−m − 6)^{3} → ... , from which we deduce reg(J) = m + 4. Moreover writing the Hilbert function H as (H(0), H(1), H(2), ...) we have

H_{R/J} = H_{A} = (1, 4, 10, 19, ..., 11m − 4, 11m + 3, 11m + 6, 11m + 6, ...).

By Theorem 5.1 and Remark 5.2 (i) we get, for every m ≥ 1 and s ≥ 2m+8 a generically smooth component of PGor(H_{D}) of dimension 22m + 42 (resp. 109, 85 or 53) provided m ≥ 4 (resp. m = 3, 2 or 1) where the Hilbert function H_{D} is given by the “(s+1)-tuple”

(1, 4, 10, ..., 11m + 3, 11m + 6, ..., 11m + 6, 11m + 3..., 10, 4, 1, 0, 0, ...),

e.g. the case m = 2 and s = 12 yields

(1, 4, 10, 18, 25, 28, 28, 28, 25, 18, 10, 4, 1, 0, 0, ...).

Example 5.4 (One dimensional Gorenstein quotients of R of codimension 4).

Now we consider the determinantal one dimensional schemes of Example 4.17 in which A is exactly as in Example 5.3 with the exception that the polynomials are taken from R := k[x_{0}, ..., x_{4}]. In this case X = Proj(R/I_{A}) is a smooth curve of degree d = 11m + 6 and genus p = (11m^{2} + 29m + 8)/2 in ℙ^{4}, and reg(I_{A}) = m + 4. By Example 4.17 we see that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{4}) of dimension

λ_{3} + K_{3} = 85 + m(11m − 5)/2 (resp. 126, 96 or 66) for m > 3 (resp. m = 3, 2 or 1).

Since (4p − 4)/d < 2m + 5 we apply Theorem 5.1 and Remark 5.2 (ii). Hence for every m ≥ 1 and s ≥ 2m+7 we get a generically smooth component V of the postulation Hilbert scheme Hilb^{H_{D}}(ℙ^{4}) whose general element is a codimension 4 Gorenstein quotient of R and whose h-vector is the (s + 1)-tuple of Example 5.3. Moreover

dim V = (11m + 6)s − 11m^{2} − 46m + 75 for m > 3

and dim V = (11m + 6)s − q where q = 163, 67 or 4 for m = 3, 2 or 1 respectively.

References

[1] G. Bolondi, J. Migliore, The Lazarsfeld-Rao property on an arithmetically Gorenstein variety, Manuscripta Math. 78 (1993), 347–368.

[2] R.O. Buchweitz, Contributions a la theorie des singularites, Thesis l’Université Paris VII (1981).

[3] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993.

[4] W. Bruns and U. Vetter, Determinantal rings, Lect. Notes in Math., Vol. 1327, Springer-Verlag, New York/Berlin, 1988.

[5] D.A. Buchsbaum and D. Eisenbud, What Annihilates a Module? J. Algebra 47, (1977), 231-243.

[6] D.A. Buchsbaum and D.S. Rim, A generalized Koszul complex, II. Depth and multiplicity, Trans. Amer. Math. Soc., 111 (1964), 457–473.

[7] M.C. Chang, A filtered Bertini-type theorem, Crelle J. 397 (1989), 214-219.

[8] T. de Jong and D. van Straten, Deformations of normalization of hypersurfaces, Math. Ann. 288 (1990), 527–547.

[9] J.A. Eagon and D.G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London 269 (1962), 188-204.

[10] D. Eisenbud, Commutative Algebra. With a view toward algebraic geometry, Springer-Verlag, Graduate Texts in Mathematics 150 (1995).

[11] G. Ellingsrud, Sur le schéma de Hilbert des variétés de codimension 2 dans ℙ^{e} a cône de Cohen-Macaulay, Ann. Scient. Éc. Norm. Sup. 8 (1975), 423-432.

[12] G. Gotzmann. A Stratification of the Hilbert Scheme of Points in the Projective Plane, Math. Z. 199 no. 4 (1988) 539-547.

[13] D. Grayson and M. Stillman, Macaulay 2-a software system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2/ .

[14] A. Grothendieck, Les schémas de Hilbert, Séminaire Bourbaki, exp. 221 (1960).

[15] A. Grothendieck, Cohomologie locale des faisceaux cohérents et Théorèmes de Lefschetz locaux et globaux, North Holland, Amsterdam (1968).

[16] A. Grothendieck, M. Raynaud and D.S. Rim, Groupes de Monodromie en Géométrie Algébrique (SGA 7). Lect. Notes in Math., Vol. 288 (1972) and Vol. 340 (1973) Springer–Verlag.

[17] M. Haiman and B. Sturmfels, Multigraded Hilbert Schemes. J. Algebraic Geom. 13 (2004), 725-769.

[18] J. Herzog, Deformationen von Cohen-Macaulay Algebren, Crelle J. 318 (1980), 83-105.

[19] A. Iarrobino and V. Kanev. Power sums, Gorenstein Algebras and Determinantal Loci. Lect. Notes in Math., Vol. 1721 Springer–Verlag, New York, 1999.

[20] R. Ile, Obstructions to deforming modules. PhD thesis, University of Oslo, 2001.

[21] J.O. Kleppe, The smoothness and the dimension of PGor(H) and of other strata of the punctual Hilbert scheme. J. Algebra, 200 no. 2 (1998), 606–628.

[22] J.O. Kleppe, Maximal families of Gorenstein algebras. Trans. Amer. Math. Soc. 358 no. 7 (2006), 3133–3167.

[23] J.O. Kleppe, Families of Artinian and one-dimensional algebras. J. Algebra 311 (2007), 665-701 [24] J.O. Kleppe, J. Migliore, R.M. Miró-Roig, U. Nagel and C. Peterson, Gorenstein liaison, complete

intersection liaison invariants and unobstructedness, Memoirs A.M.S 732, (2001).

[25] J.O. Kleppe and R.M. Miró-Roig, Dimension of families of determinantal schemes, Trans. Amer. Math. Soc. 357, (2005), 2871-2907.

[26] J.O. Kleppe and R.M. Miró-Roig, Families of determinantal schemes, Preprint 2009. To appear in Proc. Amer. Math. Soc.

[27] J. Kreuzer, J. Migliore, U. Nagel and C. Peterson, Determinantal schemes and Buchsbaum-Rim sheaves, JPAA 150 (2000), 155-174.

[28] D. Laksov, Deformation and transversality, Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen 1978, ed. K. Lønsted), Lect. Notes in Math., Springer-Verlag 732 (1979), 300-316.

[29] D. Laksov, Deformation of determinantal schemes, Compos. Math. 30 (1975), no. 3, 273-292.

[30] O.A. Laudal, Formal Moduli of Algebraic Structures. Lect. Notes in Math., Vol. 754, Springer– Verlag, New York, 1979.

[31] H. Matsumura, Commutative Algebra, Nagoya Univ., W. A. Benjamin, Inc., New York (1970).

[32] Ch. Peskine, L. Szpiro, Liaison des varietes algebrique. Invent. Math. 26 (1974), 271-302.

[33] M. Martin-Deschamps and D. Perrin, Sur la classification des courbes gauches, I. Astrisque, 184– 185 (1990).

[34] R.M. Miró-Roig, Determinantal ideals, Birkhaüser, Progress in Math. 264, (2007).

[35] T. Sauer, Smoothing Projectively Cohen-Macaulay Space Curves. Math. Ann. 272 (1985), 83-90.

[36] M. Schaps, Versal determinantal deformations. Pacific J. Math. Vol. 107, No 1, 1983, 213-221.

[37] A. Siqveland, Generalized Matrix Massey Products for Graded Modules. ArXiv:math/0603425. To appear in Journal og generalized Lie Theory and applications.

[38] F. Steffen, Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2, Manuscripta Math. 82 (1994), 417-431.

[39] B. Ulrich, Ring of invariants and linkage of determinantal ideals, Math. Ann. 274 (1986), 1-17.

Faculty of Engineering, Oslo University College, Pb. 4 St. Olavs plass, N-0130 Oslo, Norway

 E-mail address: JanOddvar.Kleppe@iu.hio.no