IDEALS GENERATED BY SUBMAXIMAL MINORS.

JAN O. KLEPPE, ROSA M. MIRÓ-ROIG^{∗}

Abstract. The goal of this paper is to study irreducible families W_{t,t}^{t−1}(\underline{b}; \underline{a}) of codimension 4, arithmetically Gorenstein schemes X ⊂ ℙ^{n} defined by the submaximal minors of a t × t homogeneous matrix A whose entries are homogeneous forms of degree a_{j} − b_{i}. Under some numerical assumption on a_{j} and b_{i} we prove that the closure of W_{t,t}^{t−1}(\underline{b}; \underline{a}) is an irreducible component of Hilb^{p(x)}(ℙ^{n}), we show that Hilb^{p(x)}(ℙ^{n}) is generically smooth along W_{t,t}^{t−1}(\underline{b}; \underline{a}) and we compute the dimension of W_{t,t}^{t−1}(\underline{b}; \underline{a}) in terms of a_{j} and b_{i}. To achieve these results we first prove that X is determined by a regular section of I_{Y} /I_{Y}^{2} (s) where s = deg(det(A)) and Y ⊂ ℙ^{n} is a codimension 2, arithmetically Cohen-Macaulay scheme defined by the maximal minors of the matrix obtained deleting a suitable row of A.

Contents

1. Introduction 1 2. Preliminaries 3 2.1. Local cohomology 4 2.2. Basic deformation theory 4 2.3. Useful exact sequences 5 3. Deformations of quotients of regular sections 5 4. Ideals generated by submaximal minors of square matrices 9 5. Examples 16 References 21

1. Introduction

In this paper we deal with determinantal schemes. 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 with c = (p−r +1)(q −r +1). When r = min(p, q) we say that X is standard determinantal. Given integers r ≤ p ≤ q, a_{1} ≤ a_{2} ≤ ... ≤ a_{p} and b_{1} ≤ b_{2} ≤ ... ≤ b_{q} we denote by W_{p,q}^{r}(\underline{b}; \underline{a}) ⊂ Hilb^{p(x)}(ℙ^{n}) the locus of determinantal schemes X ⊂ ℙ^{n} of codimension c = (p − r + 1)(q − r + 1) defined by the r × r minors of a p × q matrix (f_{ji})_{j=1,...,p}^{i=1,...,q} where f_{ji} ∈ k[x_{0}, x_{1}, ..., x_{n}] is a homogeneous polynomial of degree a_{j} − b_{i}.

The study of determinantal schemes has received considerable attention in the literature (See, for instance, [3], [5], [6] and [23]). Some classical schemes that can be constructed in

1991 Mathematics Subject Classification. Primary 14M12, 14C05, 14H10, 14J10; Secondary 14N05.

∗ The second author was partially supported by MTM2007-61104.

2 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

this way are the Segre varieties, rational normal scrolls, and the Veronese varieties. The main goal of this paper is to contribute to the classification of determinantal schemes and we address in the case p = q = t, r = t − 1 the following three fundamental problems:

(1) To determine the dimension of W_{p,q}^{r}(\underline{b}; \underline{a}) in terms of a_{j} and b_{i},

(2) To determine whether the closure of W_{p,q}^{r}(\underline{b}; \underline{a}) is an irreducible component of Hilb^{p(x)}(ℙ^{n}), and

(3) To determine when Hilb^{p(x)}(ℙ^{n}) is generically smooth along W_{p,q}^{r}(\underline{b}; \underline{a}).

The first important contribution to these problems was made by G. Ellingsrud [8] in 1975, who proved that every arithmetically Cohen-Macaulay, codimension 2 closed subscheme X of ℙ^{n} is unobstructed (i.e. the corresponding point in the Hilbert scheme Hilb^{p(x)}(ℙ^{n}) is smooth) provided n ≥ 3. He also computed the dimension of the Hilbert scheme at (X). Recall also that the homogeneous ideal of an arithmetically Cohen-Macaulay, codimension 2 closed subscheme X of ℙ^{n} is given by the maximal minors of a (t − 1) × t homogeneous matrix, the Hilbert-Burch matrix. That is, such a scheme is standard determinantal. The purpose of this work is to extend Ellingsrud’s Theorem, viewed as a statement on standard determinantal schemes of codimension 2, to arbitrary determinantal schemes. The case of codimension 3 standard determinantal schemes, was mainly solved in [19]; Proposition 1.12; and the case of standard determinantal schemes of arbitrary codimension was studied and partially solved in [20]. In [21], we treated the case of codimension 3 determinantal schemes X ⊂ ℙ^{n} defined by the submaximal minors of a symmetric homogeneous matrix. In our opinion, it is difficult to solve the above three questions in full generality and, in this paper, we will focus our attention to the first unsolved case, namely, we will deal with codimension 4 determinantal schemes X ⊂ ℙ^{n}, n ≥ 5, defined by the submaximal minors of a homogeneous square matrix. As in [19], [20] and [21], we prove our results by considering the smoothness of the Hilbert flag scheme of pairs or, more generally, the Hilbert flag scheme of chains of closed subschemes obtained by deleting suitable rows, and its natural projections into the usual Hilbert schemes. We wonder if a similar strategy could facilitate the study of the general case.

Next we outline the structure of the paper. In section 2, we recall the basic facts on local cohomology and deformation theory needed in the sequel. In section 3, we describe the deformations of the codimension 4 arithmetically Gorenstein schemes X ⊂ ℙ^{n} defined as the degeneracy locus of a regular section of the twisted conormal sheaf I_{Y} /I_{Y}^{2} (s) of a codimension 2, arithmetically Cohen-Macaulay scheme Y ⊂ ℙ^{n} of dimension ≥ 3. Section 4 is the heart of the paper. In section 4, we determine the dimension of W_{t,t}^{t−1}(\underline{b}; \underline{a}) in terms of b_{i} and a_{j} provided a_{i} ≥ b_{i+3} for 1 ≤ i ≤ t − 3 (and a_{1} ≥ b_{t} if t ≤ 3), a_{t} > a_{t−1} +a_{t−2} −b_{1} and dim X ≥ 1. We also prove that, under this numerical restriction, Hilb^{p(x)}(ℙ^{n}) is generically smooth along W_{t,t}^{t−1}(\underline{b}; \underline{a}) and the closure of W_{t,t}^{t−1}(\underline{b}; \underline{a}) is an irreducible component of Hilb^{p(x)}(ℙ^{n}) (cf. Theorem 4.6).

The key point for proving our result is the fact that any codimension 4, determinantal scheme X ⊂ ℙ^{n} defined by the submaximal minors of a homogeneous square matrix A is arithmetically Gorenstein and determined by a regular section of I_{Y} /I_{Y}^{2} (s) where s = deg(det(A)) and Y ⊂ ℙ^{n} is a codimension 2, arithmetically Cohen-Macaulay scheme defined by the maximal minors of the matrix N obtained deleting a suitable row of A (cf. Proposition 4.3). Conversely, any codimension 4, arithmetically Gorenstein scheme X = Proj(A) ⊂ ℙ^{n} defined by a regular section σ of I_{Y} /I_{Y}^{2} (s) where Y = Proj(B) ⊂ ℙ^{n} is a codimension 2, arithmetically Cohen-Macaulay scheme, fits into an exact sequence of the following type

0 ⟶ K_{B}(n + 1 − 2s) ⟶ N_{B}(−s) ⟶^{σ^{∗}} B ⟶ A ⟶ 0

and it is determined by the submaximal minors of a t×t homogeneous matrix A obtained by adding a suitable row to the Hilbert-Burch matrix of Y (cf. Proposition 4.3). In the last section, we include some examples which illustrate that the numerical hypothesis in Theorem 4.6, a_{t} > a_{t−1} + a_{t−2} − b_{1}, cannot be avoided.

Notation. Throughout this paper k will be an algebraically closed field k, R = k[x_{0}, x_{1}, …, x_{n}], 𝔪 = (x_{0}, …, x_{n}) and ℙ^{n} = Proj(R). As usual, the sheafification of a graded R-module M will be denoted by M̃ and the support of M by Supp(M).

Given a closed subscheme X of ℙ^{n} of codimension c, we denote by I_{X} its ideal sheaf, N_{X} its normal sheaf and I(X) = H_{∗}^{0}(ℙ^{n}, I_{X}) its saturated homogeneous ideal unless X = ∅, in which case we let I(X) = 𝔪 . If X is equidimensional and Cohen-Macaulay of codimension c, we set ω_{X} = Ext_{O_{ℙ^{n}}}^{c}(O_{X}, O_{ℙ^{n}})(−n − 1) to be its canonical sheaf.

In the sequel, for any graded quotient A of R of codimension c, we let I_{A} = ker(R ↠ A), N_{A} = Hom_{R}(I_{A}, A) be the normal module. If A is Cohen-Macaulay of codimension c, we let K_{A} = Ext_{R}^{c}(A, R)(−n − 1) be its canonical module. When we write X = Proj(A), we let A = R/I(X) and K_{X} = K_{A}. If M is a finitely generated graded A-module, let depth_{J} M denote the length of a maximal M-sequence in a homogeneous ideal J and let depth M = depth_{𝔪} M. If Γ_{J}(−) is the functor of sections with support in Spec(A/J), we denote by H_{J}^{i}(−) the right derived functor of Γ_{J}(−).

Let Hilb^{p(x)}(ℙ^{n}) be the Hilbert scheme parameterizing closed subschemes X of ℙ^{n} with Hilbert polynomial p(x) ∈ ℚ[x] (cf. [11]). By abuse of notation we will write (X) ∈ Hilb^{p(x)}(ℙ^{n}) for the k-point which corresponds to a closed subscheme X ⊂ ℙ^{n}. The Hilbert polynomial of X is sometimes denoted by p_{X}. By definition X is called unobstructed if Hilb^{p(x)}(ℙ^{n}) is smooth at (X).

The pullback of the universal family on Hilb^{p}(ℙ^{n}) via a morphism ψ : W ⟶ Hilb^{p}(ℙ^{n}) yields a flat family over W, and we will write (X) ∈ W for a member of that family as well. Suppose that W is irreducible. Then, by definition a general (X) ∈ W has a certain property if there is a non-empty open subset U of W such that all members of U have this property. Moreover, we say that (X) is general in W if it belongs to a sufficiently small open subset U of W (So, any (X) of U has all the openness properties that we want to require).

Finally we let D = D(p_{X}, p_{Y}) be the Hilbert flag scheme parameterizing pairs of closed subschemes (X′ ⊂ Y′) of ℙ^{n} with Hilbert polynomials p_{X′} = p_{X} and p_{Y′} = p_{Y}, respectively.

2. Preliminaries

For convenience of the reader we include in this section the background and basic results on local cohomology and deformation theory needed in the sequel.

4 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

2.1. Local cohomology. Let B = R/I_{B} be a graded quotient of the polynomial ring R, let M and N be finitely generated graded B-modules and let J ⊂ B be an ideal. We say that 0 ≠ M is a Cohen-Macaulay (resp. maximal Cohen-Macaulay) B-module if depth M = dim M (resp. depth M = dim B), or equivalently, H_{𝔪}^{i}(M) = 0 for all i ≠ dim M (resp. i < dim B) since depth_{J} M ≥ r is equivalent to H_{J}^{i}(M) = 0 for i < r. If B is Cohen-Macaulay (CM), we know by Gorenstein duality that the v-graded piece of H_{𝔪}^{i}(M) satisfies

vH_{𝔪}^{i}(M) ≃ _{−v}Ext_{B}^{dim B−i}(M, K_{B})^{∨}.

Let Z be closed in Y := Proj(B) and let U = Y − Z. Then we have an exact sequence

0 → H_{I(Z)}^{0}(M) → M → H_{∗}^{0}(U, M̃) → H_{I(Z)}^{1}(M) → 0

and isomorphisms H_{I(Z)}^{i}(M) ≃ H_{∗}^{i−1}(U, M̃) for i ≥ 2 where as usual we write H_{∗}^{i}(U, M̃) = ⊕_{t}H^{i}(U, M̃(t)). More generally, if depth_{I(Z)} N ≥ i + 1 there is an exact sequence

(2.1) _{0}Ext_{B}^{i}(M, N) ↪ Ext_{O_{U}}^{i} (M̃|_{U}, Ñ|_{U}) → _{0}Hom_{B}(M, H_{I(Z)}^{i+1}(N)) → _{0}Ext_{B}^{i+1}(M, N) →

by [14]; exp. VI, where the middle form comes from a spectral sequence also treated in [14].

2.2. Basic deformation theory. To use deformation theory, we will need to consider the (co)homology groups of algebras H_{2}(R, B, B) and H^{2}(R, B, B). Let us recall their definition. We consider

(2.2)  ... → F_{2} := ⊕_{j=1}^{μ_{2}}R(−n_{2,j}) → F_{1} := ⊕_{i=1}^{μ}R(−n_{1,i}) → R → B → 0

a minimal graded free R-resolution of B and let H_{1} = H_{1}(I_{B}) be the first Koszul homology built on a set of minimal generators of I_{B}. Then we may take the exact sequence

(2.3)  0 → H_{2}(R, B, B) → H_{1} → F_{1} ⊗_{R} B → I_{B}/I_{B}^{2} → 0

as a definition of the second algebra homology H_{2}(R, B, B) (cf. [24]), and the dual sequence,

→ _{v}Hom_{B}(F_{1} ⊗ B, B) → _{v}Hom_{B}(H_{1}, B) → _{v}H^{2}(R, B, B) → 0,

as a definition of graded second algebra cohomology H^{2}(R, B, B). If B is generically a complete intersection, then it is well known that Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ≃ H^{2}(R, B, B) ([1]; Proposition 16.1). We also know that H^{0}(Y, N_{Y}) is the tangent space of Hilb^{p(x)}(ℙ^{n}) in general, while H^{1}(Y, N_{Y}) contains the obstructions of deforming Y ⊂ ℙ^{n} in the case Y is locally a complete intersection (l.c.i.) (cf. [11]). If _{0}Hom_{R}(I_{B}, H_{𝔪}^{1}(B)) = 0 (e.g. depth_{𝔪} B ≥ 2), we have by (2.1) that _{0}Hom_{B}(I_{B}/I_{B}^{2}, B) ≃ H^{0}(Y, N_{Y}) and 0H^{2}(R, B, B) ↪ H^{1}(Y, N_{Y}) is injective in the l.c.i. case, and that _{0}H^{2}(R, B, B) contains the obstructions of deforming Y ⊂ ℙ^{n} ([16]; Remark 3.7). Thus _{0}H^{2}(R, B, B) = 0 suffices for the unobstructedness of a l.c.i. arithmetically Cohen-Macaulay subscheme Y of ℙ^{n} of dim Y ≥ 1 (for this conclusion we may even entirely skip “l.c.i.” by slightly extending the argument, as done in [16]).

2.3. Useful exact sequences. In the last part of this section, we collect some exact sequences frequently used in this paper, in the case that B = R/I_{B} is a generically complete intersection codimension two CM quotient of R. First, applying Hom_{R}(−, R) to the minimal graded free R-resolution of B

(2.4)  0 → F_{2} := ⊕_{j=1}^{μ−1}R(−n_{2,j}) → F_{1} := ⊕_{i=1}^{μ}R(−n_{1,i}) → R → B → 0

we get a minimal graded free R-resolution of K_{B}:

(2.5)  0 → R → ⊕R(n_{1,i}) → ⊕R(n_{2,j}) → K_{B}(n + 1) → 0.

If we apply Hom(−, B) to (2.5) we get the exactness to the left in the exact sequence

(2.6)  0 → K_{B}(n + 1)^{∗} → ⊕B(−n_{2,j}) → ⊕B(−n_{1,i}) → I_{B}/I_{B}^{2} → 0

which splits into two short exact sequences “via ⊕B(−n_{2,j}) ↠ H_{1} ↪ ⊕B(−n_{1,i})”, one of which is (2.3) with H_{2}(R, B, B) = 0. Indeed since H_{1} is Cohen-Macaulay by [2], we get H_{2}(R, B, B) = 0 by (2.3). Moreover since Ext_{R}^{1}(I_{B}, I_{B}) ≃ N_{B} we showed in [22]; pg. 788 that there is an exact sequence of the form

(2.7)  0 → F_{1}^{∗} ⊗_{R} F_{2} → ((F_{1}^{∗} ⊗_{R} F_{1}) ⊕ (F_{2}^{∗} ⊗_{R} F_{2}))/R → F_{2}^{∗} ⊗_{R} F_{1} → N_{B} → 0

where F_{i}^{∗} = Hom_{R}(F_{i}, R). Indeed this sequence is deduced from the exact sequence

0 → R → ⊕I_{B}(n_{1,i}) → ⊕I_{B}(n_{2,j}) → N_{B} → 0

which we get by applying Hom_{R}(−, I_{B}) to (2.4), (cf. [22]; (26)). Similarly applying Hom_{R}(−, I_{B}/I_{B}^{2}) to (2.4) and noting that Hom_{R}(I_{B}, I_{B}/I_{B}^{2}) ≃ Hom_{B}(I_{B}/I_{B}^{2}, I_{B}/I_{B}^{2}) we get the exact sequence

(2.8)  0 → Hom_{B}(I_{B}/I_{B}^{2}, I_{B}/I_{B}^{2}) → ⊕I_{B}/I_{B}^{2}(n_{1,i}) → ⊕I_{B}/I_{B}^{2}(n_{2,j}) → N_{B} → 0.

Finally we recall the following frequently used exact sequence (cf. [24])

(2.9) 0 → ∧^{2}(⊕R(−n_{2,j})) → (⊕R(−n_{1,i})) ⊗ (⊕R(−n_{2,j})) → S^{2}(⊕R(−n_{1,i})) → I_{B}^{2} → 0 .

3. Deformations of quotients of regular sections

In [18] the first author studied deformations of a scheme X := Proj(A) defined as the degeneracy locus of a regular section of a “nice” sheaf M̃ on an arithmetically Cohen-Macaulay (ACM) scheme Y = Proj(B). Recall that if we take a regular section of the anticanonical sheaf K̃_{B}^{∗}(s) and Y is a l.c.i. of positive dimension, then we get an exact sequence

0 → K_{B}(−s) → B → A → 0,

in which A is Gorenstein. Indeed the mapping cone construction leads to a resolution of A from which we easily see that A is Gorenstein. In [22], we generalized this way of constructing Gorenstein algebras to sheaves of higher rank and, in [18], we studied the deformations of this “construction”, notably in the rank two case which we now recall.

Let M be a maximal Cohen-Macaulay B-module of rank r = 2 such that M̃|_{U} is locally free and ∧^{2}M̃|_{U} ≃ K̃_{B}(t)|_{U} in an open set U := Y − Z of Y satisfying depth_{I(Z)} B ≥ 2.

Then a regular section σ of M̃^{∗}(s)|_{U} defines an arithmetically Gorenstein scheme X = Proj(A) given by the exact sequence

(3.1)  0 → K_{B}(t − 2s) → M(−s) →^{σ^{∗}} B → A → 0

and M ≃ Hom_{B}(M, K_{B}(t)) ([22]; Theorem 8). In this paper we consider and further develop the case where M = N_{B} and dim B = n − 1 (n + 1 = dim R, n ≥ 5). By [22]; Proposition 13, N_{B} is a maximal Cohen-Macaulay B-module and we have the exact sequence

(3.2) 0 → K_{B}(n + 1 − 2s) → N_{B}(−s) → I_{A/B} → 0 , where I_{A/B} := ker(B → A).

Example 3.1. Set R = k[x_{0}, ⋯, x_{5}] and let B = R/I_{B} be a codimension two quotient with minimal resolution

0 → R(−3)^{2} → R(−2)^{3} → R → B → 0

and suppose Y = Proj(B) is a l.c.i in ℙ^{5}. Let A be given by a regular section of { \atop I_{B}/I_{B}^{2}}(s), s ≥ 3. Thanks to the exact sequences (2.5) and (2.7) and the mapping cone construction applied to both (3.2) and 0 → I_{A/B} → B → A → 0, we get the following resolution of the Gorenstein algebra A,

\begin{matrix}0 → R(−2s) → R(2 − 2s)^{3} ⊕ R(−1 − s)^{6} \\ → R(3 − 2s)^{2} ⊕ R(−s)^{12} ⊕ R(−3)^{2} → R(1 − s)^{6} ⊕ R(−2)^{3} → R → A → 0 . \end{matrix}

Indeed X = Proj(A) is an arithmetically Gorenstein curve of degree d = 3s^{2} − 10s + 9 and arithmetic genus g = 1 + d(s − 3) in ℙ^{5} (see [18]; Example 43).

With M and A as above, it turns out that [18]; Theorem 1 and Theorem 25 describe the deformations space, GradAlg(R), of the graded quotient A and computes the dimension of GradAlg(R) in terms of a number δ := δ(K_{B})_{t−2s} − δ(M)_{−s} where

(3.3)  δ(N)_{v} := _{v}hom_{B}(I_{B}/I_{B}^{2}, N) − _{v}ext_{B}^{1}(I_{B}/I_{B}^{2}, N).

Here we have used small letters for the k-dimension of _{v}Ext_{B}^{i}(−, −) and of similar groups. If we suppose M = N_{B}, depth_{I(Z)} B ≥ 4 and char(k) ≠ 2, then the conditions of the A (resp. B)-part of [18]; Theorem 25 are satisfied provided _{0}Ext_{B}^{2}(N_{B}, N_{B}) = 0 (resp. −sExt_{B}^{1}(I_{B}/I_{B}^{2}, N_{B}) = 0). In both cases X is unobstructed and (3.4) dim_{(X)} Hilb^{p(x)}(ℙ^{n}) = dim(N_{B})_{0}+ dim(I_{B}/I_{B}^{2})_{s}−_{0}hom_{B}(I_{B}/I_{B}^{2}, I_{B}/I_{B}^{2})+ dim(K_{B})_{t−2s}+δ

where t = n + 1 (see [18]; Corollary 41 and its proof, and Remark 42). Using the exact sequence (2.7) we get _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, N_{B}) = 0 for s > 2 max n_{2,j} − min n_{1,i} which led to Corollary 41 of [18] which we slightly generalize in Corollary 3.2 (i) below. The A)-part was considered in Remark 42 of [18]. By the proof of Theorem 25 of [18] we may replace the vanishing of _{0}Ext_{B}^{2}(N_{B}, N_{B}) by the vanishing of the subgroup _{t}Ext_{B}^{2}(S^{2}(I_{A/B}(s)), K_{B}) and we still get all conclusions of the A)-part. Therefore, we can also prove (ii) of the following corollary to Theorem 25 of [18].

Corollary 3.2. Let B = R/I_{B} be a codimension two CM quotient of R, let U = Proj(B)− Z ↪ ℙ^{n} be a l.c.i. and suppose depth_{I(Z)} B ≥ 4. Let A be given by a regular section of Ñ_{B}^{∗}(s) on U, let η(v) := dim(I_{B}/I_{B}^{2})_{v}, and put

ϵ := η(s) + ∑_{j=1}^{μ−1} η(n_{2,j}) − ∑_{i=1}^{μ} η(n_{1,i}).

i) Let j_{0} satisfy n_{2,j_{0}} = max n_{2,j}. If s > n_{2,j_{0}} + max_{j≠j_{0}} n_{2,j} − min n_{1,i} and char(k) ≠ 2, then X is a p_{Y} -generic unobstructed arithmetically Gorenstein subscheme of ℙ^{n} of codimension 4 and dim_{(X)} Hilb^{p(x)}(ℙ^{n}) = ϵ.

ii) If _{s}Ext_{B}^{1}(N_{B}, A) = 0, char(k) = 0, s > max n_{2,j}/2 and (X ⊂ Y) is general, then X is unobstructed, dim_{(X)} Hilb^{p(x)}(ℙ^{n}) = ϵ + δ and the codimension of the stratum in Hilb^{p(x)}(ℙ^{n}) of subschemes given by (3.1) is _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}). Moreover if s > max n_{2,j} + max n_{1,i} − min n_{1,i} (resp. s > max n_{2,j}), then

0ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, N_{B}) = δ ,

(resp. _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, N_{B})).

Here I_{A/B} = ker(B → A) and “X is p_{Y} -generic” if there is an open subset of Hilb^{p(x)}(ℙ^{n}) containing (X) whose members X′ are subschemes of some closed Y′ with Hilbert polynomial p_{Y}. The stratum in Hilb^{p(x)}(ℙ^{n}) of subschemes given by (3.1) around (X) is defined by functorially varying both B, M and the regular section around (B → A) (see [18]; the definition before Theorem 25 for details). Indeed it is proved in [18]; Lemma 29 that pairs of closed subschemes (X′ ⊂ Y′) of ℙ^{n}, X′ = Proj(A′) and Y′ = Proj(B′), obtained as in (3.1) contain an open subset U ∋ (X ⊂ Y) in the Hilbert flag scheme D, and taking such a U small enough, we may define the mentioned stratum to be p(U) where p : D → Hilb^{p(x)}(ℙ^{n}) is the projection morphism induced by (X′ ⊂ Y′) → (X′). Thus “X is p_{Y} -generic” essentially means that the codimension of the stratum of subschemes given by (3.1) around (X) is zero.

Note also that “(X ⊂ Y) is general” means that it is the general member of an irreducible (non-embedded) component of the Hilbert flag scheme D. Since we in the corollary suppose depth_{I(Z)} B ≥ 4 and hence depth_{𝔪} A ≥ 2, this is equivalent to saying that (B → A) is the general member of an irreducible (non-embedded) component of the “Hilbert flag scheme” parameterizing pairs of quotients of R with fixed Hilbert functions. Indeed we can replace the schemes GradAlg(R) of [18] by Hilb^{p(x)}(ℙ^{n}) because we work with algebras of depth at least 2 at 𝔪 ([16]; Remark 3.7 or [8]).

Proof. By the text before (3.4), to prove (i) it suffices to show that _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, N_{B}) = 0. To see it we observe that

Ext_{B}^{1}(I_{B}/I_{B}^{2}, N_{B}) ≃ Ext_{B}^{1}(T_{B}, K_{B}(n + 1))

where T_{B} := Hom_{B}(I_{B}/I_{B}^{2}, I_{B}/I_{B}^{2}) by [18]; Remark 42. We consider the exact sequence (2.8) and we define F := ker(⊕I_{B}/I_{B}^{2}(n_{2,j}) → N_{B}). Since N_{B} is a maximal CM B-module and I_{B}/I_{B}^{2} has codepth 1 (i.e. Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}) = 0 for i ≥ 2) by [2] or (2.9), we get

8 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

Ext_{B}^{2}(F, K_{B}) = 0. It follows that

Ext_{B}^{1}(⊕I_{B}/I_{B}^{2}(n_{1,i}), K_{B}(n + 1)) → Ext_{B}^{1}(T_{B}, K_{B}(n + 1))

is surjective. Since

Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}(n + 1)) ≃ Ext_{R}^{3}(I_{B}/I_{B}^{2}, R) ≃ Ext_{R}^{2}(I_{B}^{2}, R) ,

it suffices to show _{−s}Ext_{R}^{2}(I_{B}^{2}(n_{1,i}), R) = 0 for any i. Looking to (2.9) it is enough to see _{−s}Hom(∧^{2}(⊕R(−n_{2,j}))(n_{1,i}), R) = 0. Since, however, n_{2,j} + n_{2,j′} − n_{1,i} − s < 0 for any i, j, j′, j ≠ j′ by assumption, we easily get this vanishing for any i and hence −sExt_{B}^{1}(I_{B}/I_{B}^{2}, N_{B}) = 0. Finally note that the dimension formula follows from (3.4) and (2.8) since we get (K_{B})_{t−2s} = 0 and δ = 0 from the proof of (ii).

(ii) By (2.5) we have (K_{B})_{t−2s} = 0 provided 2s > max n_{2,j}. By the discussion before Corollary 3.2 we must prove _{t}Ext_{B}^{2}(S^{2}(I_{A/B}(s)), K_{B}) = 0. Using the proof of [18]; Lemma 28 there is an exact sequence

0 → _{t}Ext_{B}^{2}(S^{2}(I_{A/B}(s)), K_{B}) → _{t}Ext_{B}^{2}(S^{2}(N_{B}), K_{B}) → _{s}Ext_{B}^{2}(N_{B}, B)

induced by (3.2) where we have_{t}Ext_{B}^{2}(S^{2}(N_{B}), K_{B}) ≃ _{0}Ext_{B}^{2}(N_{B}, N_{B}) ≃ _{0}Ext_{B}^{2}(N_{B}, I_{A/B}(s)) by (2.1), (3.2) and the fact that N_{B} is a maximal CM B-module. Indeed_{t}Ext_{B}^{2}(S^{2}(N_{B}), K_{B}) ≃

Ext_{O_{U}}^{2} ({ \atop S^{2}(N_{B})}|_{U}, K̃_{B}|_{U}(t)) ≃ Ext_{O_{U}}^{2} (Ñ_{B}|_{U}, Ñ_{B}^{∗} ⊗ K̃_{B}|_{U}(t)) ≃ _{0}Ext_{B}^{2}(N_{B}, N_{B}) by (2.1).

Since Ext_{B}^{1}(N_{B}, B) = 0 by (2.1) and (2.9), it follows that

tExt_{B}^{2}(S^{2}(I_{A/B}(s)), K_{B}) ≃ _{s}Ext_{B}^{1}(N_{B}, A)

which vanishes by assumption.

It remains to prove the final statement. If we apply Hom(−, K_{B}) to the exact sequence (2.2) and we use the exact sequence (2.5), we get _{−2s}Ext_{R}^{i}(I_{B}, K_{B}(t)) = 0 and hence _{−2s}Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}(t)) = 0 for i = 0, 1 provided s > max n_{2,j}. Similarly we use Hom(−, N_{B}) and the exact sequence (2.7) to show that _{−s}Hom(I_{B}, N_{B}) = 0 provided s > max n_{2,j} +max n_{1,i} −min n_{1,i}. We conclude by applying Hom_{B}(I_{B}/I_{B}^{2}, −) to (3.2). □

Remark 3.3. If depth_{I(Z)} B ≥ 4 and char(k) ≠ 2, we showed in [18]; Remark 42 that

0Ext_{B}^{2}(N_{B}, N_{B}) ≃ _{0}Hom_{B}(I_{B}/I_{B}^{2}, H_{I(Z)}^{3}(I_{B}/I_{B}^{2})) ≃ _{0}Hom_{B}(I_{B}/I_{B}^{2}, H_{I(Z)}^{4}(I_{B}^{2})) .

Similarly one shows Ext_{B}^{2}(N_{B}, B) ≃ H_{I(Z)}^{4}(I_{B}^{2}). Hence the group _{s}Ext_{B}^{1}(N_{B}, A) of Corollary 3.2 is isomorphic to the kernel of the natural map _{0}Hom_{B}(I_{B}/I_{B}^{2}, H_{I(Z)}^{4}(I_{B}^{2})) → sH_{I(Z)}^{4}(I_{B}^{2}) induced by the regular section σ. This sometimes allows us to verify _{s}Ext_{B}^{1}(N_{B}, A) = 0.

Remark 3.4. The first author takes the opportunity to point out a missing assumption in

[18] as well as in [17]. In these papers there are several theorems involving the codimension of a stratum in which the assumption “(B → A) is general” or “(B) general” is missing. The main result [17]; Theorem 5 (and hence [18]; Theorem 15) uses generic smoothness in its proof and refers to [19]; Proposition 9.14 where the “general” assumption occurs, as it should. In the proof of [17]; Theorem 5 we need “(B → A) general” to compute the dimension of the stratum. It is easily seen from the proof that what we really need is that “(B → A) is general” in the sense that, for a given (B → A), _{0}hom_{R}(I_{B}, I_{A/B}) obtains its least possible value in the irreducible components of GradAlg(H_{B}, H_{A}) to which (B → A) belongs. Thus in [17]; Theorem 5, Proposition 13, Theorem 16 (and hence [18]; Theorem 23), for the codimension statement we should assume “(B) general” or at least that _{−s}hom_{R}(I_{B}, K_{B}) obtains its least possible value in the irreducible component of GradAlg(H_{B}) to which (B) belongs. If we apply our results in a setting where these hom-numbers vanish (this is what we almost always do), we don’t need to assume “(B) or (B → A) general”.

So Remark 3.4 gives the reason for including the assumption “(X ⊂ Y) is general” in Corollary 3.2 (ii) even though this assumption does not occur in the codimension statements of the A)-part of Theorems 1 and 25 of [18].

4. Ideals generated by submaximal minors of square matrices

Let X = Proj(A) ⊂ ℙ^{n} be a codimension 4, determinantal scheme defined by the submaximal minors of a t × t homogeneous matrix. The goal of this section is to compute the dimension of Hilb^{p(x)}(ℙ^{n}) for n ≥ 5 at (X) in terms of the corresponding degree matrix. The proof requires a proposition (valid for n ≥ 3) on how A is determined by a locally regular section of I_{B}/I_{B}^{2}(s) where B = R/I_{B} is a codimension 2 CM quotient. Let us first fix the notation we will use throughout this section.

Given a homogeneous matrix A, i.e. a matrix representing a degree 0 morphism ϕ of free graded R-modules, we denote by I(A) (or I(ϕ)) the ideal of R generated by the maximal minors of A and by I_{j}(A) (or I_{j}(ϕ)) the ideal generated by the j × j minors of A.

Definition 4.1. A codimension c subscheme X ⊂ ℙ^{n} is called a determinantal scheme if there exist integers r, p and q such that c = (p − r + 1)(q − r + 1) and I(X) = I_{r}(A) for some p × q homogeneous matrix A. X ⊂ ℙ^{n} is called a standard determinantal scheme if r = min(p, q). The corresponding rings R/I_{r}(A) are called determinantal (resp. standard determinantal) rings.

Let X ⊂ ℙ^{n} be a codimension 4, determinantal scheme defined by the vanishing of the submaximal minors of a t × t homogeneous matrix A = (f_{ji})_{i,j=1,··· ,t} where f_{ji} ∈ k[x_{0}, ⋯, x_{n}] are homogeneous polynomials of degree a_{j} − b_{i} with b_{1} ≤ b_{2} ≤ ⋯ ≤ b_{t} and a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{t}. We assume without loss of generality that A is minimal; i.e., f_{ji} = 0 for all i, j with b_{i} = a_{j}. If we let u_{ji} = a_{j} − b_{i} for all j = 1, …, t and i = 1, ⋯, t, the matrix U = (u_{ji})_{i,j=1,··· ,t} is called the degree matrix associated to X.

We denote by W_{t,t}^{t−1}(\underline{b}; \underline{a}) ⊂ Hilb^{p(x)}(ℙ^{n}) the locus of determinantal schemes X ⊂ ℙ^{n} of codimension 4 defined by the submaximal minors of a homogeneous square matrix A = (f_{ji})_{i,j=1,··· ,t} as above. Notice that W_{t,t}^{t−1}(\underline{b}; \underline{a}) ≠ ∅ if and only if u_{i−1,i} = a_{i−1} − b_{i} > 0 for i = 2, ..., t.

Let N be the matrix obtained by deleting the last row, let I_{B} = I_{t−1}(N) be the ideal defined by the maximal minors of N and let I_{A} = I_{t−1}(A) be the ideal generated by the submaximal minors of A. Set A = R/I_{A} = R/I(X) and B = R/I_{B}.

10 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

Remark 4.2. If the entries of A and N are sufficiently general polynomials of degree a_{i} − b_{j}, 1 ≤ i, j ≤ t, and a_{i−1} − b_{i} > 0 for 2 ≤ i ≤ t, then B is a graded Cohen-Macaulay quotient of codimension 2 and A is a graded Gorenstein quotient of codimension 4.

The goal of this section is to compute, in terms of a_{j} and b_{i}, the dimension of the determinantal locus W_{t,t}^{t−1}(\underline{b}; \underline{a}) ⊂ Hilb^{p(x)}(ℙ^{n}), where p(x) ∈ ℚ[x] is the Hilbert polynomial of X. Note that the Hilbert polynomial of X can be computed explicitly using the minimal free R-resolution of R/I(X) given by Gulliksen and Negård in [13], see (4.5). We will also analyse whether the closure of W_{t,t}^{t−1}(\underline{b}; \underline{a}) in Hilb^{p(x)}(ℙ^{n}) is a generically smooth, irreducible component of Hilb^{p(x)}(ℙ^{n}). To this end, we consider

F := ⊕_{i=1}^{t}R(b_{i}) ⟶^{ϕ} G := ⊕_{j=1}^{t}R(a_{j})

the morphism induced by the above matrix A and

F ⟶^{ϕ_{t}} G_{t} := ⊕_{j=1}^{t−1}R(a_{j})

the morphism induced by the matrix N obtained by deleting the last row of A. The determinant of A is a homogeneous polynomial of degree

s := deg(det(A)) = ∑_{j=1}^{t} a_{j} − ∑_{i=1}^{t} b_{i},

and the degrees of the maximal minors of N are s + b_{i} − a_{t}, i.e. I_{B} has the following minimal free R-resolution

(4.1)  0 ⟶ G_{t}^{∗}(a_{t} − s) ⟶^{^{t}N} F^{∗}(a_{t} − s) ⟶^{β} I_{B} ⟶ 0.

Proposition 4.3. Suppose char(k) = 0.

(i) Let A = R/I_{t−1}(A) be a determinantal ring of codimension 4 where A is a t × t homogeneous matrix and let B = R/I_{t−1}(N) be the standard determinantal ring associated to N where N is the matrix obtained by deleting the last row of A. Moreover, let Z ⊂ Proj(B) be a closed subset such that Proj(B) − Z ↪ ℙ^{n} is a l.c.i. and suppose depth_{I(Z)} B ≥ 2. Then, there is a regular section σ of (Ĩ_{B}/Ĩ_{B}^{2}(s))_{| Proj(B)−Z} where s = deg(det(A)) whose zero locus precisely defines A as a quotient of B (i.e. σ extends to a map σ : B ⟶ I_{B}/I_{B}^{2}(s) such that A = B/(im σ^{∗})).

(ii) Conversely, let B = R/I_{t−1}(N) be a standard determinantal ring of codimension 2, let Z ⊂ Proj(B) be a closed subset such that Proj(B) − Z ↪ ℙ^{n} is a l.c.i. and depth_{I(Z)} B ≥ 2 and let A′ be defined by a regular section σ of (Ĩ_{B}/Ĩ_{B}^{2}(s))_{| Proj(B)−Z}, i.e. given by

(4.2)  0 ⟶ K_{B}(n + 1 − 2s) ⟶ N_{B}(−s) ⟶^{σ^{∗}} B ⟶ A′ ⟶ 0

for some integer s. Then, there is a t × t homogeneous matrix A′ obtained by adding a row to N such that I_{A′} = I_{t−1}(A′).

Proof. First of all, to define σ, we consider the commutative diagram 0 ↓

0 ⟶ G_{t}^{∗}(a_{t} − s) ⟶^{^{t}N} F^{∗}(a_{t} − s) ⟶^{β}  I_{B}  ⟶ 0
↓ α  ∥  ↓
G^{∗}(a_{t} − s) ⟶^{ϕ^{∗}(a_{t}−s)} F^{∗}(a_{t} − s) ⟶ (coker ϕ^{∗})(a_{t} − s) ⟶ 0

↓ R(−s) ↓ 0

where α : G_{t}^{∗}(a_{t}−s) ↪ G^{∗}(a_{t}−s) is the natural inclusion defined by α ( f_{1} ; ⋮ ; f_{t−1} ) = ( f_{1} ; ⋮ ; f_{t−1} ; 0 )

and β is given by multiplication with the maximal minors of the matrix N. The snake Lemma yields the exact sequence

(4.3)  R(−s) ⟶^{· det(ϕ)} I_{B} ⟶ (coker ϕ^{∗})(a_{t} − s) ⟶ 0

and hence

(4.4)  (coker ϕ^{∗})(a_{t}) ≃ I_{B}(s)/ det(ϕ).

If we tensor R(−s) ⟶^{· det(ϕ)} I_{B} with B(s), we get a section σ of I_{B}/I_{B}^{2}(s). Before proving that the zero locus of σ defines precisely A as a quotient of B via im(σ^{∗}) = I_{A/B}, we claim that any locally regular section σ′ of I_{B}/I_{B}^{2}(s) defining A′ via A′ = B/ im(σ′^{∗}) gives rise to a homogeneous matrix A′ and a corresponding map ϕ′ such that (4.3) and (4.4) hold with ϕ′ instead of ϕ. Indeed, given a section σ′ of I_{B}/I_{B}^{2}(s), there exists a map σ′′ fitting into a commutative diagram

\xymatrix{ & F^{∗}(a_{t}) ⊗ B \ar@{->>}[d] \\ B \ar[r]_{σ′} \ar[ur]^{σ′′} & I_{B}/I_{B}^{2}(s)}

and we denote by σ_{R} ∈ Hom_{R}(F, R(a_{t})) the map which corresponds to σ′′(1). Since Hom_{R}(F, R(a_{t})) = Hom(⊕_{i=1}^{t}R(b_{i}), R(a_{t})), the morphism σ_{R} determines a 1 × t row \underline{g} = (g_{1}, ⋯, g_{t}) where g_{i} is a homogeneous form of degree a_{t} − b_{i}, 1 ≤ i ≤ t and we define

A′ = ( N ; \underline{g} ). Since the vertical map in the above diagram is induced by β described

above, we may assume that det(ϕ′) = σ′(1) modulo I_{B}^{2}(s) and we get the claim.

It remains to show that im(σ^{∗}) = I_{A/B} where I_{A} = I_{t−1}(A) and that σ is a locally regular section. Note that this will also show that im(σ^{′∗}) = I_{A′/B} where I_{A′} = I_{t−1}(A′), i.e. we get the converse. Moreover looking at the exact sequence (4.2) with A instead of A′, and recalling that

N_{B} ≃ K_{B}(n + 1) ⊗ I_{B}/I_{B}^{2},

we see that im(σ^{∗}) = coker(σ(−2s) ⊗ id) where id : K_{B}(n + 1) ⟶ K_{B}(n + 1) is the identity map and σ is induced by det(ϕ). Since we get

F (s − a_{t}) ⟶ G_{t}(s − a_{t}) ⟶ K_{B}(n + 1) ⟶ 0

by dualizing the exact sequence (4.1), we see that the cokernel above is the same as the twisted cokernel of the composition

γ : G_{t}(−a_{t}) ⟶ K_{B}(n + 1 − s) ⟶^{σ(−s)⊗id} N_{B}.

Hence, we must prove that coker(γ) = I_{A/B}(s) where I_{A} = I_{t−1}(A).

By [13]; Théorème 2 (see also [15]; Theorem 2), we have an exact sequence:

(4.5)  ker[Hom(F, F) ⊕ Hom(G, G) ⟶^{j} R] ⟶ Hom(F, G) ⟶ I_{A}(s) ⟶ 0

where j(ρ_{0}, ρ_{1}) = tr(ρ_{0}) − tr(ρ_{1}) and tr is the trace map. The map Hom(F, G) ⟶ I_{A}(s) is given by γ ⟶ tr(γψ) where ψ is the matrix of cofactors, i.e. this map is given by the submaximal minors of A while the map Hom(F, F) ⊕ Hom(G, G) ⟶^{η} Hom(F, G) is given as a difference of the obvious compositions with ϕ, i.e., η(ρ_{0}, ρ_{1}) = ρ_{1}ϕ − ϕρ_{0}. Since we have

Hom(F, F) ⊕ Hom(G, G) ⟶^{η} Hom(F, G) ⟶ I_{A}(s)
(id, 0)  ↦  t · det(ϕ)

and since there is a commutative diagram

0 ↓

ker(j)  ⟶ Hom(F, G) ⟶ I_{A}(s) ⟶ 0
↓ ∥ ↓
Hom(F, F) ⊕ Hom(G, G) ⟶^{η} Hom(F, G) ⟶ coker η ⟶ 0

↓ R

we get an exact sequence

R ⟶^{·t det(ϕ)} I_{A}(s) ⟶ coker η ⟶ 0.

Hence, coker(η) ≃ I_{A}(s)/ det(ϕ) (char(k) = 0) and the following sequence is exact:

Hom(F, F) ⊕ Hom(G, G) ⟶^{η} Hom(F, G) ⟶ I_{A}(s)/ det(ϕ) ⟶ 0.

Now we look at the commutative diagram

\xymatrix{Hom(R(−a_{t}), G^{∗}) \ar[d]_{≃} \ar[r] & Hom(R(−a_{t}), F^{∗}) \ar[d]_{≃} \ar[r] & I_{B}(s)/ det(ϕ) \ar[dd] \ar[r] & 0 \\ Hom(G, R(a_{t})) \ar[d]^{(0,·)} \ar[r] & Hom(F, R(a_{t})) \ar[d] \\ Hom(F, F) ⊕ Hom(G, G) \ar[d]^{(id,α_{1}^{∗})} \ar[r]^{η} & Hom(F, G) \ar[d]^{α_{2}^{∗}} \ar[r] & I_{A}(s)/ det(ϕ) \ar[d] \ar[r] & 0 \\ Hom(F, F) ⊕ Hom(G, G_{t}) \ar[r]^{η_{t}} & Hom(F, G_{t}) \ar[r] & coker(η_{t}) \ar[r] & 0}

where α_{1}^{∗} and α_{2}^{∗} are induced by α in a natural way and η_{t} is a difference of the obvious compositions, i.e., η_{t}(ρ_{0}, ρ′_{1}) = ρ′_{1}ϕ − ϕ_{t}ρ_{0}. We see, in particular, that the ideal I_{A/B} = I_{A}/I_{B} is given by an exact sequence

Hom(F, F) ⊕ Hom(G, G_{t}) ⟶^{η_{t}} Hom(F, G_{t}) ⟶ I_{A/B}(s) ⟶ 0

where the rightmost map is given by the submaximal minors of the matrix A which do not belong to I_{B}.

On the other hand, by (2.7), there is an exact sequence

Hom(G_{t}^{∗}, G_{t}^{∗}) ⊕ Hom(F^{∗}, F^{∗}) ⟶ Hom(G_{t}^{∗}, F^{∗}) ⟶ N_{B} ⟶ 0

or, equivalently,

Hom(F, F) ⊕ Hom(G_{t}, G_{t}) ⟶^{η′} Hom(F, G_{t}) ⟶ N_{B} ⟶ 0

where η′ is given by η′(ρ_{0}, ρ_{2}) = ρ_{2}ϕ_{t} − ϕ_{t}ρ_{0}. Using again the exact sequence 0 ⟶ R(a_{t}) ⟶ G ⟶^{α^{∗}} G_{t} ⟶ 0 we get a commutative diagram

[figure: Hom(F, F) ⊕ Hom(G_{t}, G_{t}); (id,α_{3}); η′; Hom(F, G_{t}); N_{B}; 0; Hom(F, F) ⊕ Hom(G, G_{t}); η_{t}; Hom(F, G_{t}); I_{A/B}(s); 0; Hom(R(a_{t}), G_{t})]

where α_{3} is induced by α. Hence we get an exact sequence

Hom(R(a_{t}), G_{t}) ⟶^{γ} N_{B} ⟶ I_{A/B}(s) ⟶ 0.

This proves that coker(γ) = I_{A/B}(s), i.e. im(σ^{∗}) = I_{A/B} as required. Finally note that the above codimension and depth relations imply that σ is a regular section on U := Proj(B) − Z because (im σ̃^{∗})_{|U} must locally on U be generated by two regular elements (to get that (B/ im σ̃^{∗})_{|U} is a codimension 2 Cohen-Macaulay quotient of B̃|U). This completes the proof of Proposition 4.3. □

14 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

The above proposition seems to be known in special cases. We have for instance noticed that Ellingsrud and Peskine claim that the Artinian Gorenstein ring associated to an invertible sheaf O_{S}(C) on a surface S in ℙ^{3}, where C is an arithmetically CM curve, is given by the submaximal minors of a square matrix which extends the Hilbert-Burch matrix associated to C in ℙ^{3} (see the text of [9] before Proposition 6). Since, we get (3.1) with M = N_{B} by applying H_{∗}^{0}(−) to the exact sequence

0 → N_{C/S}(−s) → N_{C}(−s) → N_{S}|_{C}(−s) ≃ O_{C} → 0

of normal sheaves, it is clear that their Gorenstein ring (see their construction 2) is essentially the same as ours in the Artinian case. However, we have given a proof of the above proposition suited to our applications.

As a nice application of Proposition 4.3 we have

Proposition 4.4. Let X ⊂ ℙ^{n}, n ≥ 4, be a codimension 4 scheme defined by the submaximal minors of a t × t homogeneous square matrix A. Then X is in the Gorenstein liaison class of a complete intersection, i.e. X is glicci.

Proof. By [13]; Théorème 2 (see also Proposition 4.3), X is arithmetically Gorenstein and hence glicci ([4]; Theorem 7.1). □

Remark 4.5. The above proposition has been recently generalized by Gorla. In [10]; Theorem 3.1, she has proved that any codimension (t − r + 1)^{2} ACM scheme X ⊂ ℙ^{n} defined by the r × r minors of a t × t homogeneous square matrix A is glicci.

For an introduction to glicciness, see [19].

We are now ready to compute dim W_{t,t}^{t−1}(\underline{b}; \underline{a}) and dim_{(X)} Hilb^{p(x)} ℙ^{n}, n ≥ 5, in terms of a_{1}, ⋯, a_{t} and b_{1}, ⋯, b_{t}. Note that if t = 2 then a general X is a complete intersection in which case these dimensions are well known.

Theorem 4.6. (char(k) = 0) Fix integers a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{t} and b_{1} ≤ b_{2} ≤ ⋯ ≤ b_{t}. Assume t > 2, a_{i} ≥ b_{i+3} for 1 ≤ i ≤ t − 3 (and a_{1} ≥ b_{t} if t = 3), a_{t} > a_{t−1} + a_{t−2} − b_{1} and n ≥ 5. Then W_{t,t}^{t−1}(\underline{b}; \underline{a}) is irreducible. Moreover, if (X) is general in W_{t,t}^{t−1}(\underline{b}; \underline{a}), then X is unobstructed, and

dim W_{t,t}^{t−1}(\underline{b}; \underline{a}) = dim_{(X)} Hilb^{p(x)}(ℙ^{n}) =

∑_{1≤i,j≤t} \binom{a_{j} − b_{i} + n}{n} − ∑_{{1≤i≤t−1 \atop 1≤j≤t}} \binom{a_{j} − a_{i} + n}{n} − ∑_{1≤i,j≤t} \binom{b_{i} − b_{j} + n}{n} + ∑_{{1≤i≤t \atop 1≤j≤t−1}} \binom{b_{i} − a_{j} + n}{n}

− ∑_{{1≤j≤t \atop 1≤i≤k≤t}} \binom{a_{t} − s − b_{i} − b_{k} + a_{j} + n}{n} + ∑_{{1≤i,j≤t \atop 1≤k≤t−1}} \binom{a_{t} − s − b_{i} − a_{k} + a_{j} + n}{n}−

∑_{{1≤i<k≤t−1 \atop 1≤j≤t}} \binom{a_{t} − s − a_{i} − a_{k} + a_{j} + n}{n} + ∑_{2≤i≤t} \binom{a_{t} − s + b_{i} − 2b_{1} + n}{n}.

Proof. Let X ⊂ ℙ^{n} be an arithmetically Gorenstein scheme of codimension 4 defined by the submaximal minors of a homogeneous square matrix A = (f_{ji})_{j=1,...,t}^{i=1,...,t} where f_{ji} ∈ k[x_{0}, ..., x_{n}] is a sufficiently general homogeneous polynomial of degree a_{j} − b_{i} and let Y ⊂ ℙ^{n} be a codimension 2 subscheme defined by the maximal minors of the matrix N obtained deleting the last row of A (see Remark 4.2). So, the homogeneous ideal I_{B} = I(Y) of Y has the following minimal free R-resolution

(4.6)  0 ⟶ F_{2} = ⊕_{j=1}^{t−1}R(a_{t} − s − a_{j}) ⟶^{^{t}N} F_{1} = ⊕_{i=1}^{t}R(a_{t} − s − b_{i}) ⟶ I_{B} ⟶ 0.

By Proposition 4.3, X is the zero locus of a suitable regular section of Ĩ_{B}/Ĩ_{B}^{2}(s) where s = deg(det(A)) and W_{t,t}^{t−1}(\underline{b}; \underline{a}) is irreducible by [18]; Corollary 41. Since the hypothesis a_{t} > a_{t−1} + a_{t−2} − b_{1} is equivalent to

s > s + a_{j_{0}} − a_{t} + max_{{1≤j≤t−1 \atop j≠j_{0}}}(s + a_{j} − a_{t}) − min_{1≤i≤t}(s + b_{i} − a_{t}),

where s + a_{j_{0}} − a_{t} = max_{1≤j≤t−1}(s + a_{j} − a_{t}); and since a_{i} ≥ b_{i+3} for 1 ≤ i ≤ t − 3 (and a_{1} ≥ b_{t} if t = 3) implies that B := R/I_{B} given by (4.6) satisfies depth_{I(Z)} B ≥ 4 (cf. [20]; Remark 2.7), we can apply Corollary 3.2 and we get that X is unobstructed and

dim W_{t,t}^{t−1}(\underline{b}; \underline{a}) = dim_{(X)} Hilb^{p(x)}(ℙ^{n}) = η(s) + ∑_{j=1}^{t−1} η(n_{2,j}) − ∑_{i=1}^{t} η(n_{1,i})

where η(t) = dim(I(Y)/I(Y)^{2})_{t} = dim I(Y)_{t} −dim I(Y)_{t}^{2}, n_{2,j} = s+a_{j} −a_{t}, 1 ≤ j ≤ t−1, and n_{1,i} = s + b_{i} − a_{t}, 1 ≤ i ≤ t. By (2.9), I(Y)^{2} has a minimal free R-resolution of the following type:

(4.7)  0 ⟶ ∧^{2}F_{2} = ⊕_{1≤i<j≤t−1}R(−a_{i} − a_{j} + 2a_{t} − 2s) ⟶
F_{1} ⊗ F_{2} = ⊕_{{1≤i≤t \atop 1≤j≤t−1}} R(−b_{i} − a_{j} + 2a_{t} − 2s) ⟶
S^{2}F_{1} = ⊕_{1≤i≤j≤t}R(−b_{i} − b_{j} + 2a_{t} − 2s) ⟶ I(Y)^{2} ⟶ 0.

Using (4.6) and (4.7), we obtain

η(s) = ∑_{1≤i≤t} \binom{a_{t} − b_{i} + n}{n} − ∑_{1≤i≤t−1} \binom{a_{t} − a_{i} + n}{n} − ∑_{1≤i≤j≤t} \binom{2a_{t} − s − b_{i} − b_{j} + n}{n}+

∑_{{1≤i≤t \atop 1≤j≤t−1}} \binom{2a_{t} − s − b_{i} − a_{j} + n}{n} − ∑_{1≤i<j≤t−1} \binom{2a_{t} − s − a_{i} − a_{j} + n}{n}.

Using again (4.6) and (4.7), we get

∑_{j=1}^{t−1} η(n_{2,j}) − ∑_{i=1}^{t} η(n_{1,i}) = ∑_{{1≤i≤t \atop 1≤j≤t−1}} \binom{a_{j} − b_{i} + n}{n} − ∑_{{1≤i≤t−1 \atop 1≤j≤t−1}} \binom{a_{j} − a_{i} + n}{n}−

∑_{{1≤j≤t−1 \atop 1≤i≤k≤t}} \binom{a_{t} − s − b_{i} − b_{k} + a_{j} + n}{n} + ∑_{{1≤i≤t \atop 1≤j,k≤t−1}} \binom{a_{t} − s − b_{i} − a_{k} + a_{j} + n}{n}−

∑_{{1≤i<k≤t−1 \atop 1≤j≤t−1}} \binom{a_{t} − s − a_{i} − a_{k} + a_{j} + n}{n} − ∑_{{1≤i≤t \atop 1≤j≤t}} \binom{b_{i} − b_{j} + n}{n} + ∑_{{1≤i≤t \atop 1≤j≤t−1}} \binom{b_{i} − a_{j} + n}{n}+

∑_{{1≤i≤t \atop 1≤j≤k≤t}} \binom{a_{t} − s + b_{i} − b_{j} − b_{k} + n}{n} − ∑_{{1≤i,k≤t \atop 1≤j≤t−1}} \binom{a_{t} − s + b_{i} − b_{k} − a_{j} + n}{n}+

∑_{{1≤k<j≤t−1 \atop 1≤i≤t}} \binom{a_{t} − s + b_{i} − a_{k} − a_{j} + n}{n}.

Since a_{i−1} > b_{i} and a_{i} ≥ b_{i+3} for 1 ≤ i ≤ t − 3 (and a_{1} ≥ b_{t} if t = 3), by hypothesis, the last two sums of binomials vanish. Indeed, to see that a_{t} − s + b_{i} − b_{k} − a_{j} < 0 (resp. a_{t}−s+b_{i}−a_{k} −a_{j} < 0) for 1 ≤ i, k ≤ t and 1 ≤ j ≤ t−1 (resp. 1 ≤ i ≤ t and 1 ≤ k < j ≤ t − 1), it suffices to show that b_{t} − b_{1} − a_{1} < s − a_{t} = a_{1} + a_{2} + ⋯ + a_{t−1} − b_{1} − b_{2} − ⋯ − b_{t} (resp. b_{t} − a_{1} − a_{1} < s − a_{t} = a_{1} + a_{2} + ⋯ + a_{t−1} − b_{1} − b_{2} − ⋯ − b_{t}) which is rather straightforward to prove.

Moreover, the same type of argument applies to see that a_{t} − s + b_{i} − b_{j} − b_{k} < 0 for all 1 ≤ i ≤ t and 1 ≤ j < k ≤ t and we can replace the summand ∑_{{1≤i≤t \atop 1≤j≤k≤t}} \binom{a_{t}−s+b_{i}−b_{j}−b_{k}+n}{n}

by ∑_{2≤i≤t} \binom{a_{t}−s+b_{i}−2b_{1}+n}{n}. Putting all together we get

dim W_{t,t}^{t−1}(\underline{b}; \underline{a}) = dim_{(X)} Hilb^{p(x)}(ℙ^{n}) =

∑_{{1≤i≤t \atop 1≤j≤t}} \binom{a_{j} − b_{i} + n}{n} − ∑_{{1≤i≤t−1 \atop 1≤j≤t}} \binom{a_{j} − a_{i} + n}{n} − ∑_{{1≤i≤t \atop 1≤j≤t}} \binom{b_{i} − b_{j} + n}{n}+

∑_{{1≤i≤t \atop 1≤j≤t−1}} \binom{b_{i} − a_{j} + n}{n} − ∑_{{1≤j≤t \atop 1≤i≤k≤t}} \binom{a_{t} − s − b_{i} − b_{k} + a_{j} + n}{n}+

∑_{{1≤i,j≤t \atop 1≤k≤t−1}} \binom{a_{t} − s − b_{i} − a_{k} + a_{j} + n}{n} − ∑_{{1≤i<k≤t−1 \atop 1≤j≤t}} \binom{a_{t} − s − a_{i} − a_{k} + a_{j} + n}{n}+

∑_{2≤i≤t} \binom{a_{t} − s + b_{i} − 2b_{1} + n}{n}.

□

5. Examples

We will end this work with some examples where we use Theorem 4.6. Moreover, these examples show that the hypothesis a_{t} > a_{t−1} + a_{t−2} − b_{1} cannot be avoided! To handle such cases, we state a proposition which estimates the codimension of the stratum in Hilb^{p(x)}(ℙ^{n}) of subschemes given by the exact sequence (3.1).

Example 5.1. Let R = k[x_{0}, ⋯, x_{5}] and let X = Proj(A) ⊂ ℙ^{5} = Proj(R) be a general arithmetically Gorenstein curve defined by the submaximal minors of a 4×4 matrix whose first 3 rows are linear forms and whose last row are forms of degree s − 3 (s ≥ 4), i.e. b_{i} = 0 for 1 ≤ i ≤ 4, a_{j} = 1 for 1 ≤ j ≤ 3 and a_{4} = s − 3. Then, Theorem 4.6 applies provided s > 5 and we get that X is unobstructed and

dim W_{4,4}^{3}(\underline{0}; 1, 1, 1, s−3) = dim_{(X)} Hilb^{p(x)}(ℙ^{5}) = 12\binom{6}{5}+4\binom{s + 2}{5}−9\binom{5}{5}−3\binom{s + 1}{5}−

16\binom{5}{5} − 10\binom{s − 1}{5} + 12\binom{s − 2}{5} − 3\binom{s − 2}{5} = 2s^{3} − 10s^{2} + 13s + 48.

Moreover deleting the last row and taking maximal minors, we get a threefold Y = Proj(B) with resolution

(5.1)  0 ⟶ R(−4)^{3} ⟶ R(−3)^{4} ⟶ R ⟶ B ⟶ 0,

leading to

H_{B}(ν) = \binom{ν + 3}{3} + 2\binom{ν + 2}{3} + 3\binom{ν + 1}{3} = p_{Y} (ν) for ν ≥ 0.

Since A is given by (3.1) with t = 6 and M = N_{B}, we get O_{X} ≃ ω_{X}(2s − 6). Hence h^{1}(O_{X}(s − 3)) = h^{0}(O_{X}(s − 3)) and the Hilbert polynomial of X must be of the form p_{X}(ν) = dν + 1 − g = d(ν − s + 3). Looking to (5.1) we get

p_{X}(s − 2) = h^{0}(O_{X}(s − 2)) − h^{0}(O_{X}(s − 4)) =
= h^{0}(O_{Y} (s − 2)) − h^{0}(O_{Y} (s − 4)) = 6s^{2} − 28s + 36,

i.e. d = deg(X) = 6s^{2} − 28s + 36 and g = 1 + d(s − 3).

Note that Theorem 4.6 takes care of all cases except for s = 4 and s = 5. For these two values of s, we can, however, use Corollary 3.2 (ii) to find dim_{(X)} Hilb^{p(x)}(ℙ^{5}) because

0Ext_{B}^{2}(N_{B}, N_{B}) ≃ _{0}Hom(I_{B}/I_{B}^{2}, H_{𝔪}^{4}(I_{B}^{2})) = 0

by (5.1) and Remark 3.3. Indeed _{3}H_{𝔪}^{4}(I_{B}^{2}) ↪ _{3}H_{𝔪}^{6}(R(−8)^{3}) = 0 by (2.9). Hence X is unobstructed,

dim_{(X)} Hilb^{p(x)}(ℙ^{5}) = 2s^{3} − 10s^{2} + 13s + 48 + δ

where δ = δ(K_{B})_{6−2s} − δ(N_{B})_{−s}, and moreover, if s = 5, then δ is the codimension of the closure of W_{4,4}^{3} := W_{4,4}^{3}(\underline{0}; 1, 1, 1, s − 3) in Hilb^{p(x)}(ℙ^{5}). We claim that

(δ(K_{B})_{6−2s}, δ(N_{B})_{−s}) = \begin{cases}(−3, −15) & for s = 4 \\ (0, −12) & for s = 5, \end{cases}

i.e., δ = 12 in both cases.

To find δ(K_{B})_{6−2s} we apply Hom_{B}(−, K_{B}(6)) to (2.3) and we get _{−2s}Hom_{B}(I_{B}/I_{B}^{2}, K_{B}(6)) = 0 and _{−2s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}(6)) = _{−2s}Hom(H_{1}, K_{B}(6)). Since the rank of H_{1} is 2, we have

(5.2)  Hom(H_{1}, K_{B}(6)) ≃ H_{1}(∑_{i} n_{1,i}) = H_{1}(12)

by [2] or [22]; Theorem 8, see the isomorphism accompanying (3.1). Using (2.6) or more precisely the exactness of

(5.3)  ∧^{2}(R(−3)^{4}) ⟶ R(−4)^{3} ⟶ H_{1} ⟶ 0

(cf. [2]), we get

δ(K_{B})_{6−2s} = − dim H_{1}(12)_{−2s} = \begin{cases}−3 & for s = 4 \\ 0 & for s = 5. \end{cases}

It remains to compute δ(N_{B})_{−s}. If we dualize the exact sequence (2.3) we get

0 ⟶ N_{B} ⟶ B(3)^{4} ⟶ H_{1}^{∗} ⟶ 0

18 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

to which we apply _{−s}Hom(I_{B}/I_{B}^{2}, −). Combining with

−sHom(I_{B}/I_{B}^{2}, H_{1}^{∗}) ≃ _{−s}Hom(I_{B}/I_{B}^{2} ⊗ K_{B}(6), H_{1}^{∗} ⊗ K_{B}(6)) ≃ _{−s}Hom(N_{B}, H_{1}(12))

where again we have used (5.2), we get

δ(N_{B})_{−s} = 4 dim(N_{B})_{3−s} − dim(_{−s}Hom(N_{B}(−12), H_{1})).

Using (2.6), we see that

0 → _{−s}Hom(N_{B}(−12), K_{B}(6)^{∗}) → _{−s}Hom(N_{B}(−12), B(−4)^{3}) → _{−s}Hom(N_{B}(−12), H_{1}) → 0

is exact because we have Ext_{B}^{1}(I_{B}/I_{B}^{2} ⊗ K_{B}, K_{B}^{∗}) = 0 by [20]; Lemma 4.9. Using [20]; (4.17) we also get the surjectivity of the natural map K_{B}^{∗} ⊗ B(−4)^{4} ⟶ Hom_{B}(I_{B}/I_{B}^{2} ⊗ K_{B}, K_{B}^{∗}). Since we may use (5.3) to see that (H_{1})_{ν} ≃ R(−4)_{ν}^{3} ≃ B(−4)_{ν}^{3} for ν ≤ 5, we get K_{B}(6)_{ν}^{∗} = 0 for ν ≤ 5 by (2.6) and hence

−sHom(N_{B}(−12), K_{B}^{∗}(−6)) ≃ _{−s}Hom(I_{B}/I_{B}^{2} ⊗ K_{B}(6), K_{B}^{∗}(6)) = 0

for s ≥ 4. It follows that

−sHom(N_{B}(−12), H_{1}) ≃ (I_{B}/I_{B}^{2})_{8−s}^{3}

for s ≥ 4 which implies (cf. (2.7) and (2.9)) that

δ(N_{B})_{−s} = \begin{cases}4 dim(N_{B})_{−2} − 3 dim(I_{B})_{3} = −12 & for s = 5 \\ 4 dim(N_{B})_{−1} − 3 dim(I_{B})_{4} = −15 & for s = 4. \end{cases}

Putting all together we get

dim_{(X)} Hilb^{p(x)}(ℙ^{5}) = \begin{cases}2s^{3} − 10s^{2} + 13s + 48 = dim W_{4,4}^{3} & for s > 5 \\ 125 & for s = 5 \\ 80 & for s = 4. \end{cases}

Moreover, applying Corollary 3.2 (ii), we get codim_{Hilb^{p(x)}(ℙ^{5})} W_{4,4}^{3}(\underline{0}; 1, 1, 1, 2) = 12 in the case s = 5. Finally, for s = 4, using Macaulay 2 program [12] we have computed the dimension _{0}hom(I_{B}, I_{A/B}) = 3 for (B → A) general and hence codim_{Hilb^{p(x)}(ℙ^{5})} W_{4,4}^{3}(\underline{0}; \underline{1}) = 0hom(I_{B}, I_{A/B}) + δ = 15.

If a_{t} ≤ a_{t−1} + a_{t−2} − b_{1} we see in the example above that W_{t,t}^{t−1}(\underline{b}; \underline{a}) is a proper closed irreducible subset, i.e. the generic curve of the component of Hilb^{p(x)}(ℙ^{5}) to which W_{t,t}^{t−1}(\underline{b}; \underline{a}) belongs is not defined by submaximal minors of a matrix of forms of degree a_{j} − b_{i}. The converse inequality always implies dim W_{t,t}^{t−1}(\underline{b}; \underline{a}) = dim_{(X)} Hilb^{p(x)}(ℙ^{n}) by Theorem 4.6. The pattern above for small a_{t} may be typical, but is in general rather difficult to prove. We illustrate this by two more examples.

Example 5.2. Let X = Proj(A) ⊂ ℙ^{5} be a general arithmetically Gorenstein curve defined by the submaximal minors of a 3 × 3 matrix whose first 2 rows are linear forms and whose last row are forms of degree s − 2 (s ≥ 3), i.e. b_{i} = 0 for 1 ≤ i ≤ 3, a_{j} = 1 for 1 ≤ j ≤ 2 and a_{3} = s − 2. Thanks to Proposition 4.3 the analysis of [18]; Example 43, immediately transfers to our case. Hence, for s > 4 (i.e., a_{t} > a_{t−1} + a_{t−2} − b_{1}), we see that X is unobstructed and

dim W_{3,3}^{2}(\underline{0}; 1, 1, s − 2) = dim_{(X)} Hilb^{p(x)}(ℙ^{5}) = (s + 1)(s − 1)^{2} + 23.

Since by deleting the last row and taking maximal minors we get a threefold Y = Proj(B) for which _{0}Ext_{B}^{2}(N_{B}, N_{B}) = 0, we have the unobstructedness of X also for s = 3, 4, and

(δ(K_{B})_{6−2s}, δ(N_{B})_{−s}) = \begin{cases}(−1, 2) & for s = 3 \\ (0, −3) & for s = 4 \end{cases} .

That is, δ = −3 when s = 3, and δ = 3 when s = 4. In both cases,

dim_{(X)} Hilb^{p(x)}(ℙ^{5}) = (s + 1)(s − 1)^{2} + 23 + δ.

Thus

dim_{(X)} Hilb^{p(x)}(ℙ^{5}) = \begin{cases}36 & for s = 3 \\ 71 & for s = 4, \end{cases}

see [18]; Example 43 for the computations. Now, applying Corollary 3.2 (ii), we get codim_{Hilb^{p(x)}(ℙ^{5})} W_{3,3}^{2}(\underline{0}; 1, 1, 2) = 3 in the case s = 4. Finally, for s = 3, a Macaulay 2 computation shows _{0}hom(I_{B}, I_{A/B}) = 3 and hence

codim_{Hilb^{p(x)}(ℙ^{5})} W_{3,3}^{2}(\underline{0}; \underline{1}) = _{0}hom(I_{B}, I_{A/B}) + δ = 0 !

In the above examples we were able to analyse the case a_{t} ≤ a_{t−1} + a_{t−2} − b_{1} through Corollary 3.2 (ii) because _{s}Ext_{B}^{1}(N_{B}, A) = 0. Since this vanishing may be rare, we want to improve upon Corollary 3.2 (ii), at least to get estimates of the codimension of the stratum. We prefer to do it in the generality of Theorem 25 of [18] to extend Theorem 25 in this direction. This leads to the proposition below. Indeed with assumptions as in Proposition 5.3, one knows that the projection morphism q : D → Hilb^{p_{Y}} (ℙ^{n}) induced by (X′ ⊂ Y′) → (Y′) is smooth at (X ⊂ Y) ([18]; Theorem 47). Using that the corresponding tangent map is surjective, we get Proposition 5.3 and Remark 5.4 (a). Since we only use these results in Example 5.6 and Remark 5.5, we skip the details of the proof which are rather straightforward once having the results and proofs of [18]. Put

c(I_{A/B}) := _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) − _{t}ext_{B}^{2}(S^{2}(I_{A/B}(s)), K_{B}).

Proposition 5.3. Let B = R/I_{B} be a graded licci quotient of R, let M be a graded maximal Cohen-Macaulay B-module, and suppose M̃ is locally free of rank 2 in U := Proj(B) − Z, that dim B − dim B/I(Z) ≥ 2 and ∧^{2}M̃|_{U} ≃ K̃_{B}(t)|_{U}. Let A be defined by a regular section σ of M̃^{∗}(s) on U, i.e. given by (3.1), let X = Proj(A) and suppose _{s}Ext_{B}^{1}(M, B) = 0 and dim B ≥ 4. Moreover let char(k) = 0, let (B → A) be general and suppose (M, B) is unobstructed along any graded deformation of B and −sExt_{B}^{2}(I_{B}/I_{B}^{2}, M) = 0 . Then the codimension, codi, of the stratum in Hilb^{p(x)}(ℙ^{n}) of subschemes given by (3.1) around (X) satisfies

c(I_{A/B}) ≤ codi ≤ c(I_{A/B}) + _{0}h^{2}(R, A, A) ≤ _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}),

and codi = c(I_{A/B}) + _{0}h^{2}(R, A, A) if and only if X is unobstructed.

Here “(M, B) unobstructed along any graded deformation of B” means that for every graded deformation (M_{S}, B_{S}) of (M, B), S local and Artinian with residue field k, there is a graded deformation of M_{S} to any graded deformation B_{T} of B_{S} for any small Artin surjection T → S ([18]; Definition 11). The important remark for our application M = N_{B}

20 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

where the codimension 2 CM quotient B satisfies depth_{I(Z)} B ≥ 4, is that all assumptions of the proposition are satisfied provided char(k) = 0 and (B → A) is general (see proof of Corollary 41 and Remark 42 of [18]).

Moreover recall that if we put δ := −δ(I_{A/B})_{0} = _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B})− _{0}hom_{R}(I_{B}, I_{A/B}) and we use the exact sequence (3.2), we get δ = δ(K_{B})_{t−2s} − δ(N_{B})_{−s} , as previously.

Remark 5.4. a) With assumptions as in Proposition 5.3, except for (B → A) being general we can also show

δ − _{t}ext_{B}^{2}(S^{2}(I_{A/B}(s)), K_{B}) ≤ codi .

b) Moreover if depth_{I(Z)} B ≥ 4, then we show

tExt_{B}^{2}(S^{2}(I_{A/B}(s)), K_{B}) ≃ _{s}Ext_{B}^{1}(M, A)

exactly as we did for M = N_{B} in the proof of Corollary 3.2. Hence if _{s}Ext_{B}^{1}(M, A) = 0, then the lower bound c(I_{A/B}) of Proposition 5.3 is equal to the upper bound and we essentially get Corollary 3.2 (ii)! Moreover since codi ≥ 0, Corollary 3.2 (i) corresponds to the case where the upper bound is zero!

Remark 5.5. In the case s > max n_{2,j}/2, depth_{I(Z)} B ≥ 4 and char(k) = 0, the inequalities of Proposition 5.3 lead to

ϵ + δ − _{0}ext_{B}^{1}(N_{B}, A) ≤ dim_{(X)} Hilb^{p(x)}(ℙ^{n}) ≤ ϵ + δ

with ϵ as in Corollary 3.2.

Example 5.6. Now let X = Proj(A) ⊂ ℙ^{5} be a general arithmetically Gorenstein curve defined by the submaximal minors of a 3×3 matrix whose first (resp. second) row consists of linear (resp. quadratic) forms and whose last row are forms of degree s − 3 (s ≥ 5), i.e. b_{i} = 0 for 1 ≤ i ≤ 3, a_{1} = 1, a_{2} = 2 and a_{3} = s − 3. In the following we skip a few details which we leave to the reader. Note that the case a_{t} > a_{t−1} + a_{t−2} − b_{1} or equivalently s > 6, is taken care of by Theorem 4.6. So we concentrate on the cases s = 5 and 6, which we analyse by using Proposition 5.3 and Remark 5.4. First, we use Remark 3.3 to compute _{0}ext_{B}^{2}(N_{B}, N_{B}) where B is obtained by deleting the last row and taking maximal minors. We easily get _{0}ext_{B}^{2}(N_{B}, N_{B}) = _{s}ext_{B}^{1}(N_{B}, A) = 3 by using

0 ⟶ R(−5) ⊕ R(−4) ⟶ R(−3)^{3} ⟶ R ⟶ B ⟶ 0,

(2.9) and _{0}Ext_{B}^{2}(N_{B}, N_{B}) ≃ _{0}Hom_{B}(I_{B}/I_{B}^{2}, H_{𝔪}^{4}(I_{B}^{2})). Moreover dim(K_{B})_{6−2s} = 0 by (2.5). Now if we apply _{−2s}Hom(−, K_{B}(6)) to (2.3) we get δ(K_{B})_{6−2s} = 0 and −2sExt_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}(6)) = 0 for s ≥ 5 provided we can show _{−2s}Hom(H_{1}, K_{B}(6)) = 0. Using (2.3) we get that H_{1} has rank 1 and H_{1} ≃ K_{B}(−3). Hence _{−2s}Hom(H_{1}, K_{B}(6)) ≃ B(9)_{−2s} = 0 for s ≥ 5.

It remains to compute δ(N_{B})_{−s}. We claim that δ(N_{B})_{−s} = −8 (resp. δ(N_{B})_{−s} = −3) for s = 5 (resp. s = 6). Indeed dualizing the exact sequence (2.3) we get

0 ⟶ N_{B} ⟶ B(3)^{3} ⟶ H_{1}^{∗} ⟶ 0 .

If we apply _{−s}Hom(I_{B}/I_{B}^{2}, −) to this sequence, recalling H_{1} ≃ K_{B}(−3) and hence −sHom(I_{B}/I_{B}^{2}, H_{1}^{∗}) ≃ (I_{B}/I_{B}^{2})_{9−s}, we get an exact sequence which rather easily proves the claim. It follows that the numbers δ = δ(K_{B})_{6−2s} −δ(N_{B})_{−s} and _{s}ext_{B}^{1}(N_{B}, A) appearing in Remark 5.4 are computed. We conclude, for s = 5, that the codimension, codi, of the stratum in Hilb^{p(x)}(ℙ^{5}) of subschemes given by (3.1) around (X) is at least 5-dimensional. In fact a Macaulay 2 computation shows _{0}h^{2}(R, A, A) = 0 and _{0}hom(I_{B}, I_{A/B}) = 1 and hence we have codi = c(I_{A/B}) + _{0}h^{2}(R, A, A) = 6 by Proposition 5.3. For s = 6 the lower bound for codi of Remark 5.4 (a) is 0. Since a Macaulay 2 computation shows 0hom(I_{B}, I_{A/B}) = 0 the better lower bound of Proposition 5.3 is also 0 while the smallest upper bound of Proposition 5.3 is 3. The latter is the correct bound for the codimension of the stratum provided X is unobstructed. In conclusion if X belongs to a reduced component V of Hilb^{p(x)}(ℙ^{5}), then codi = 3, but codi = 0 is possible in which case V is non-reduced. We have not been able to fully tell what happens, but we expect V to be reduced and codi = 3.

The last case of the preceding example illustrates how difficult the analysis of when codi is positive could be. Especially cases where a_{t} is close to a_{t−1} + a_{t−2} − b_{1} seem difficult to handle. Since it turns out that the lower bounds of Proposition 5.3 and Remark 5.4 (a) are often negative (also in the case a_{t} > a_{t−1} + a_{t−2} − b_{1} treated in Theorem 4.6), they are not very helpful. This, however, also indicates that the conclusions of Theorem 4.6 are rather strong.

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22 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

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Faculty of Engineering, Oslo University College, Postboks 4, St. Olavs Plass, N-0130 Oslo, Norway

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

Facultat de Matemàtiques, Departament d’Algebra i Geometria, Gran Via de les Corts Catalanes 585, 08007 Barcelona, SPAIN

E-mail address: miro@ub.edu