UNOBSTRUCTEDNESS AND DIMENSION OF FAMILIES OF CODIMENSION 3 ACM ALGEBRAS

JAN O. KLEPPE, ROSA M. MIRÓ-ROIG^{∗}

Abstract. The goal of this paper is to study irreducible families of codimension 3, Cohen-Macaulay quotients A of a polynomial ring R = k[x_{0}, x_{1}, ..., x_{n}]; mainly, we study families of graded Cohen-Macaulay quotients A of codimension 1 on some codimension 2 Cohen-Macaulay algebra B defined by a regular section σ of (S^{2}K_{B}^{∨})_{λ}. We give lower bounds for the dimension of the irreducible components of the Hilbert scheme which contains Proj(A). The components are generically smooth and the bounds are sharp if λ ≫ 0 and n = 4 and 5.

We also deal with a particular type of codimension 3, Cohen-Macaulay quotients A of R; concretely we restrict our attention to codimension 3 arithmetically Cohen-Macaulay subschemes X ⊂ ℙ^{n} defined by the submaximal minors of a symmetric homogeneous matrix. We prove that such schemes are glicci and we give lower bounds for the dimension of the corresponding component of the Hilbert scheme.

In the last part of the paper, we collect some questions/problems which naturally arise in our context.

Contents

1. Introduction 1 2. Preliminaries 3 3. CM quotients of codimension 1 on a codimension 2 CM algebra 5 4. Ideals generated by submaximal minors of symmetric matrices 15 5. Final remarks and open problems 22 References 23

1. Introduction

The main purpose of this work is to contribute to the classification of codimension r, Cohen-Macaulay graded quotients of a polynomial ring R = k[x_{0}, ⋯, x_{n}] and, in particular, we address the following two problems: (1) to determine the unobstructedness of arithmetically Cohen-Macaulay (briefly ACM) schemes X ⊂ ℙ^{n} of codimension r; and (2) to determine dim_{(X)} Hilb^{p(t)} ℙ^{n} being X ⊂ ℙ^{n} an arithmetically Cohen-Macaulay scheme of codimension r. In codimension 2, the classification of arithmetically Cohen-Macaulay graded quotients of a polynomial ring R = k[x_{0}, ⋯, x_{n}] is well known, it is given by the Hilbert-Burch Theorem; and the unobstructedness of arithmetically Cohen-Macaulay schemes X ⊂ ℙ^{n} of codimension 2 as well as dim_{(X)} Hilb^{p(t)} ℙ^{n} were established

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

∗ Partially supported by BFM2001-3584.

2 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

in 1975 by G. Ellingsrud [8]. There is, in our opinion, little hope of solving the above two problems in full generality and for arbitrary codimension. So, we will restrict our attention to codimension 3, ACM schemes X ⊂ ℙ^{n} which are divisors on some codimension 2, ACM scheme Y ⊂ ℙ^{n}. According to [6]; Theorem 3.12, an effective divisor D ∼ aK_{Y} + bH on Y is ACM if and only if −2 ≤ a ≤ 1. ACM effective divisors X ∼ bH on a codimension 2, ACM scheme Y ⊂ ℙ^{n} were studied in [19] and [20]; ACM effective divisors X ∼ K_{Y} + bH on a codimension 2, ACM scheme Y ⊂ ℙ^{n} were studied in [19]; and ACM effective divisors X ∼ −K_{Y} + bH on a codimension 2, ACM scheme Y ⊂ ℙ^{n} were studied in [21]. In this paper we study the remaining case; namely, we study ACM effective divisors X ∼ −2K_{Y} + bH on a codimension 2 ACM scheme Y ⊂ ℙ^{n} or, equivalently, graded CM quotients A given by

(1.1)  0 ⟶ (S^{2}K_{B})(−λ) ⟶^{σ} B ⟶ A ⟶ 0

where B is a codimension 2 graded generically complete intersection CM quotient of R. In this paper, we deal with divisors on codimension 2 ACM schemes and we refer to [13] for general results about the theory of generalized divisors for schemes satisfying the condition S_{2} of Serre.

Next we outline the structure of the paper. In section 2, we recall the basic facts on deformation theory needed in the sequel. Sections 3 and 4 are the heart of the paper. In section 3, we study families of graded Cohen-Macaulay quotients A of codimension 1 on a codimension 2 Cohen-Macaulay algebra B defined by a regular section σ of (S^{2}K_{B}^{∨})_{λ}; i.e. graded CM quotients A given by (1.1). We determine lower bounds for the dimension of any irreducible component of Hilb^{p(x)}(ℙ^{n}) containing a point (X), X = Proj(A), where A is given by (1.1) (see Theorem 3.6 and Theorem 3.7), and we show that, for n = 5 and 4, they are sharp for λ ≫ 0 (see Corollary 3.9 and Remark 3.11). The lower bounds will be computed in terms of

a := _{0}hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}) − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}),
b := _{0}hom_{B}(I_{A/B}, B) − _{0}ext_{B}^{1}(I_{A/B}, B), and
e := _{0}ext_{B}^{2}(I_{A/B}, I_{A/B}),

where I_{A/B} = S^{2}K_{B}(−λ) and where at least a and b are explicitly given as a sum of binomials involving only the degrees of the generators and first syzygies of I_{B}.

In section 4, we deal with ideals generated by the submaximal minors of a homogeneous symmetric matrix. A classical scheme that can be constructed in this way is the Veronese surface X ⊂ ℙ^{5}. Given rational numbers a_{1}, ..., a_{t} such that a_{i} + a_{j} ∈ ℤ_{+} for all i, j, we denote by S(\underline{a}) = S(a_{1}, ⋯, a_{t}) the irreducible family of codimension 3, ACM schemes X ⊂ ℙ^{n} defined by the submaximal minors of a t × t symmetric homogeneous matrix A = (f_{ji})_{j=1,...,t}^{i=1,...,t} where f_{ji} ∈ k[x_{0}, ..., x_{n}] is a homogeneous polynomial of degree a_{j} + a_{i}. The goal of section 4 is to give lower bounds for the dimension of the irreducible component S(\underline{a}) of Hilb^{p}(ℙ^{n}) containing S(\underline{a}) (cf. Theorem 4.8). We give 2 examples where the first bound turns out to be sharp. Indeed, we guess that, for n = 5, the first bound may be sharp for any scheme defined by linear forms f_{ij}. As a byproduct we also prove that any codimension 3, ACM scheme X ⊂ ℙ^{n} defined by the submaximal minors of a t × t symmetric homogeneous matrix is glicci (see Proposition 4.5). This last result has independently been established by Gorla who has proved that any codimension \binom{m−t+2}{2}, ACM scheme X ⊂ ℙ^{n} defined by the t × t minors of a m × m symmetric homogeneous matrix is glicci ([9]; Corollary 2.7). We end this paper with some questions/problems which naturally arise in our context.

Finally, the authors would like to thank the referee for his/her valuable comments. In particular, we include Remarks 4.1 and 4.2 to answer an imprecision pointed out by the referee.

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

For any closed subscheme X of ℙ^{n}, we denote by I_{X} the ideal sheaf of X and N_{X} its normal sheaf. Let I(X) = H_{∗}^{0}(I_{X}) be the saturated homogeneous ideal of X 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.

For any quotient A of R, we let I_{A} = ker(R ↠ A) and 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. Let H_{J}^{i}(−) be the right derived functor of the functor, Γ_{J}(−), of sections with support in Spec(A/J).

We denote the Hilbert scheme by Hilb^{p}(ℙ^{n}) (cf. [11]). Thus, any point p_{X} ∈ Hilb^{p}(ℙ^{n}) parameterizes a subscheme X ⊂ ℙ^{n} with Hilbert polynomial p ∈ ℚ[s]. By abuse of notation we will write (X) ∈ Hilb^{p}(ℙ^{n}). By definition a scheme X ⊂ ℙ^{n} is unobstructed if Hilb^{p}(ℙ^{n}) is smooth at (X).

2. Preliminaries

This section provides the background and basic results on deformation theory needed later on.

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

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. 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 [12], exp. VI, where the middle form comes from a spectral sequence also treated in [12]. Moreover, we have (cf. [25], Lemma 5)

Lemma 2.1. Let B be Cohen-Macaulay, let r and t be integers and suppose depth_{J} B ≥ r. Then depth_{𝔪} M ≥ dim B − t implies depth_{J} M ≥ r − t.

In dealing with deformations we will need to consider the (co)homology groups of algebras. To define them let

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

be a minimal resolution of B and let H_{1} = H_{1}(I_{B}) be the 1. 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 definition of the 2. algebra homology H_{2}(R, B, B) (cf. [28]), 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 2. 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 know that H^{0}(Y, N_{Y}) is isomorphic to the tangent space of Hilb^{p}(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 _{0}H^{2}(R, B, B) ↪ H^{1}(Y, N_{Y}) is injective in the l.c.i. case, and by [18], Remark 3.7 that _{0}H^{2}(R, B, B) contains the obstructions of deforming Y ⊂ ℙ^{n}. Thus _{0}H^{2}(R, B, B) = 0 suffices for the unobstructedness of a locally complete intersection arithmetically Cohen-Macaulay (ACM) subscheme Y of ℙ^{n} of dim B ≥ 2 (for this conclusion we may even entirely skip “l.c.i.” by slightly extending the argument, as done in [18]).

There are cases where we can conclude that some X = Proj(A) or A is unobstructed without assuming _{0}H^{2}(R, A, A) = 0. One such case which we need in this paper is treated in [19], Theorem 9.4 and extended in [23]. Following [23] we say “(M, B) is unobstructed along any graded deformation of B” if for every small Artin surjection (T, m_{T}) → (S, m_{S}) (i.e. a surjection of local Artinian k-algebras with residue fields k whose kernel 𝔞 satisfies 𝔞 · m_{T} = 0) and every graded deformation (M_{S}, B_{S}) of (M, B), there is a graded deformation of M_{S} to any graded deformation B_{T} of B_{S}. We need the following special case of [23], Proposition 13.

Proposition 2.2. Let _{0}Ext_{B}^{1}(M, M) = 0. Then (M, B) is unobstructed along any graded deformation of B if for every local Artinian k-algebra T with residue field k and for every graded deformation B_{T} of B to T, there exists a graded deformation M_{T} of M to B_{T}.

Example 2.3. Let char(k) ≠ 2, let B ≃ R/I_{B} be a graded CM quotient of R of codimension c and suppose depth_{I(Z)} B ≥ 3 where Y − Z is locally a complete intersection in ℙ^{n}. Then (S^{2}K_{B}(v), B) is unobstructed along any graded deformation of B for every integer v. Indeed the proposition above applies because K_{B_{T}} := Ext_{B_{T}}^{c} (B_{T} , R_{T} (−n − 1)) is flat over T by [16], Proposition A1. Hence K_{B_{T}} ⊗ K_{B_{T}}, as well as S^{2}K_{B_{T}}, are T -flat. It follows that S^{2}K_{B_{T}} (v) is a graded deformation of S^{2}K_{B}(v). Since _{0}Ext_{B}^{1}(S^{2}K_{B}, S^{2}K_{B}) = 0 by (2.1) (explained thouroughly later), we conclude easily. In the next section, e.g. in Corollary 3.9, we use the unobstructedness of (S^{2}K_{B}(v), B) to show that the quotient A of (1.1) is unobstructed provided λ ≫ 0.

Finally in the case B = R/I_{B} is a generically a complete intersection codimension two CM quotient of R, we notice some exact sequences frequently used in this paper. Firstly since F_{2} ↪ F_{1} is injective in (2.2), we get a minimal resolution

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

by taking R-duals. If we apply Hom(−, B) to (2.4), letting K_{B}^{∨} = Hom_{B}(K_{B}, B), we get the exactness to the left in the exact sequence

(2.5)  0 → K_{B}(n + 1)^{∨} → ⊕_{j=1}^{μ−1}B(−n_{2,j}) → ⊕_{i=1}^{μ}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 in this case H_{1} is Cohen-Macaulay by [2] and hence H_{2}(R, B, B) = 0 by (2.3). Moreover since Ext_{R}^{1}(I_{B}, I_{B}) ≃ N_{B} we showed in [24] that there is an exact sequence of the form

(2.6) 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.

Indeed this sequence is deduced from the exact sequence

0 ⟶ R ⟶ ⊕_{i=1}^{μ}I(n_{1,i}) ⟶ ⊕_{j=1}^{μ−1}I(n_{2,j}) ⟶ N_{B} → 0

which we get by applying Hom_{R}(−, I_{B}) to (2.2), (cf. [24],(26)). Then it is straightforward to find Castelnuovo-Mumford regularity of N_{B}, as well as the formula of dim(N_{B})_{0} of [8]. Note also the following frequently used sequences (cf. [7], p. 595 for two of them)

(2.7)  0 → ∧^{2}(F_{1}^{∨}) → F_{1}^{∨} ⊗ F_{2}^{∨} → S^{2}(F_{2}^{∨}) → S^{2}(K_{B})(2n + 2) → 0
(2.8) 0 → ∧^{3}(F_{1}^{∨}) → ∧^{2}(F_{1}^{∨}) ⊗ F_{2}^{∨} → F_{1}^{∨} ⊗ S^{2}(F_{2}^{∨}) → S^{3}(F_{2}^{∨}) → S^{3}(K_{B})(3n + 3) → 0
(2.9)  0 → ∧^{2}(F_{2}) → F_{1} ⊗ F_{2} → S^{2}(F_{1}) → I_{B}^{2} → 0 .

3. CM quotients of codimension 1 on a codimension 2 CM algebra

In this section we study families of graded CM quotients defined by a regular section σ of (S^{2}K_{B}^{∨})_{λ}; i.e. CM quotients A given by

(3.1)  0 ⟶ (S^{2}K_{B})(−λ) ⟶^{σ} B ⟶ A ⟶ 0

where B is a codimension 2 graded generically complete intersection CM quotient of R.

Recall that by [27]; Theorem III.4.2, for any general codimension 2, arithmetically Cohen-Macaulay scheme Y ⊂ ℙ^{n} with degree matrix (u_{ij})_{i=1,··· ,t}^{j=1,··· ,t−1}, u_{ij} > 0, we have P ic(Y) ≅ ℤ.H or P ic(Y) ≅ ℤ^{2}, unless Y ⊂ ℙ^{4} is a Castelnuovo surface and P ic(Y) ≅ ℤ^{9} or Y ⊂ ℙ^{4} is a Bordiga surface and P ic(Y) ≅ ℤ^{11}. In addition, if n = 4 then P ic(Y) ≅ ℤ^{2} = ℤ.H ⊕ ℤ.K with K the canonical divisor on Y and H the hyperplane divisor.

Moreover, by [6] Theorem 3.12, an effective divisor X ∼ aK_{Y} + bH is ACM if and only if −2 ≤ a ≤ 1. ACM effective divisors X ∼ bH on a codimension 2, ACM Y ⊂ ℙ^{n} are determinantal schemes of codimension 3 and they have been studied in [19] and [20]; ACM effective divisors X ∼ −K_{Y} + bH on a codimension 2, ACM Y ⊂ ℙ^{n} are

6 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

arithmetically Gorenstein schemes of codimension 3 and they have been studied in [21]; and ACM effective divisors X ∼ K_{Y} +bH on a codimension 2, ACM Y ⊂ ℙ^{n} were studied in [19]. We will devote this section to the remaining case; namely, to study arithmetically Cohen-Macaulay effective divisors X ∼ −2K_{Y} + bH on a codimension 2, ACM Y ⊂ ℙ^{n}.

Our goal is to determine good lower bounds for the dimension of any irreducible component of Hilb^{p(x)}(ℙ^{n}) containing a point (X), X = Proj(A), where A is given by (3.1), and to show that they are sharp for λ ≫ 0 (when n = 4 and 5). The lower bounds will be computed in terms of

a := _{0}hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}) − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}),
b := _{0}hom_{B}(I_{A/B}, B) − _{0}ext_{B}^{1}(I_{A/B}, B), and
e := _{0}ext_{B}^{2}(I_{A/B}, I_{A/B}),

where I_{A/B} = S^{2}K_{B}(−λ). Indeed since we will use deformation theory related to the flag of surjections

R ⟶ B ⟶ A ≅ B/I_{A/B}

it turns out that the groups

0Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) ( or _{0}Ext_{R}^{1}(I_{B}, I_{A/B}) )
0Ext_{A}^{1}(I_{A/B}/I_{A/B}^{2}, A) ( or _{0}Ext_{B}^{1}(I_{A/B}, A) )

play a central role.

Let Y = Proj(B) and let U = Proj(B) − Z ↪ ℙ^{n} be a local complete intersection (l.c.i.). In the following we almost always assume depth_{I(Z)} B ≥ 3 because our proofs

often use that { \atop K_{B}|U} and { \atop I_{B}/I_{B}^{2}|U} are locally free in a large enough open set. Firstly we will make a more explicit. To do it, we define

s^{i}(ν) := dim(S^{i}K_{B})_{ν}.

Lemma 3.1. If depth_{I(Z)} B ≥ 3, then Ext_{B}^{2}(I_{B}/I_{B}^{2}, S^{2}(K_{B})) = 0, and

a = ∑_{i=1}^{μ} s^{2}(n_{1,i} − λ) − ∑_{i=1}^{μ−1} s^{2}(n_{2,i} − λ) + s^{3}(n + 1 − λ) =

[dim(∧^{2}F_{1}^{∨} ⊗ F_{1}^{∨})_{−v} − dim(∧^{3}F_{1}^{∨})_{−v}] − dim(F_{1}^{∨} ⊗ F_{1}^{∨} ⊗ F_{2}^{∨})_{−v}+

dim(F_{1}^{∨} ⊗ F_{2}^{∨} ⊗ F_{2}^{∨})_{−v} − [dim(S^{2}F_{2}^{∨} ⊗ F_{2}^{∨})_{−v} − dim(S^{3}F_{2}^{∨})_{−v}]

where v = 2n + 2 + λ.

Proof. The exact sequence (2.5) leads to

(3.2)  0 ⟶ H_{1} ⟶ ⊕_{i=1}^{μ}B(−n_{1,i}) ⟶ I_{B}/I_{B}^{2} ⟶ 0, and
(3.3)  0 ⟶ K_{B}(n + 1)^{∨} ⟶ ⊕_{i=1}^{μ−1}B(−n_{2,i}) ≅ F_{2} ⊗ B ⟶ H_{1} ⟶ 0

where H_{1} is the 1.Koszul homology. Since I_{A/B} = S^{2}K_{B}(−λ) we apply Hom_{B}(−, S^{2}K_{B}(−λ)) to (3.2), and we get

(3.4)  0 ⟶ Hom_{B}(I_{B}/I_{B}^{2}, S^{2}K_{B}(−λ)) ⟶ ⊕_{i=1}^{μ}S^{2}K_{B}(−λ + n_{1,i}) ⟶
Hom_{B}(H_{1}, S^{2}K_{B}(−λ)) ⟶ Ext_{B}^{1}(I_{B}/I_{B}^{2}, S^{2}K_{B}(−λ)) ⟶ 0

and

Ext_{B}^{1}(H_{1}, S^{2}K_{B}(−λ)) ≅ Ext_{B}^{2}(I_{B}/I_{B}^{2}, S^{2}K_{B}(−λ)).

Moreover using the exact sequence (3.3) we obtain

(3.5) 0 ⟶ Hom_{B}(H_{1}, S^{2}K_{B}(−λ)) ⟶ ⊕_{i=1}^{μ−1}S^{2}K_{B}(−λ + n_{2,i}) ≅ F_{2}^{∨} ⊗ S^{2}K_{B}(−λ) ⟶

Hom_{B}(K_{B}^{∨}(−n − 1), S^{2}K_{B}(−λ)) ≅ S^{3}K_{B}(n + 1 − λ)) → Ext_{B}^{1}(H_{1}, S^{2}K_{B}(−λ)) → 0.

Note that we have used (2.8) and Lemma 2.1 to see depth_{I(Z)} S^{3}K_{B} ≥ 2 and then (2.1) to get Hom_{B}(K_{B}^{∨}, S^{2}K_{B}) ≅ S^{3}K_{B}. Now since F_{2}^{∨} → K_{B}(n + 1) is surjective, it follows that F_{2}^{∨} ⊗ S^{2}K_{B} → S^{3}K_{B}(n + 1) is surjective and we get Ext_{B}^{2}(I_{B}/I_{B}^{2}, S^{2}(K_{B})) = 0 . We deduce that

a = _{0}hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}) − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B})

= ∑_{i=1}^{μ} s^{2}(n_{1,i} − λ) − ∑_{i=1}^{μ−1} s^{2}(n_{2,i} − λ) + s^{3}(n + 1 − λ)

which proves the first dimension formula. Finally since it is straightforward to write down the minimal resolutions of F_{i}^{∨} ⊗ S^{2}K_{B}(−λ), i = 1, 2 and S^{3}K_{B}(n + 1 − λ) from (2.7) and (2.8), we get the second dimension formula as well and we are done. □

It is also of interest to see the vanishing of the groups _{0}Ext_{B}^{i}(I_{B}/I_{B}^{2}, S^{2}K_{B}(−λ)) above.

Lemma 3.2. With notations as above, let d_{k} = max_{i}{n_{k,i}}. Then, for i = 0 and 1,

0Ext_{B}^{i}(I_{B}/I_{B}^{2}, S^{2}K_{B}(−λ)) = _{0}Ext_{R}^{i}(I_{B}, S^{2}K_{B}(−λ)) = 0 provided λ > 3d_{2} − 2n − 2

and

0Hom_{R}(I_{B}, S^{2}K_{B}(−λ)) = 0 provided λ > d_{1} + 2d_{2} − 2n − 2.

Proof. Using (2.2), we deduce that _{0}Ext_{R}^{1}(I_{B}, S^{2}K_{B}(−λ)) = 0 provided we can show (F_{2}^{∨} ⊗ S^{2}K_{B}(−λ))_{0} = 0, i.e. provided (cf. (2.7))

(F_{2}^{∨} ⊗ S^{2}(F_{2}^{∨})(−2n − 2))_{(−λ)} = 0.

This follows easily from the first assumption of the lemma since F_{2}^{∨} = ⊕R(n_{2,i}). Using F_{1}^{∨} = ⊕R(n_{1,i}) we get the vanishing of the Hom-group from (F_{1}^{∨}⊗S^{2}(F_{2}^{∨})(−2n−2−λ))_{0} = 0. Finally since d_{2} ≥ d_{1}, the first assumption also implies the vanishing of the Ext^{i}-group for i = 0 and we are done. □

Next we will make b more explicit. Firstly note that, if depth_{I(Z)} B ≥ 3, we have

(3.6)  _{0}Ext_{A}^{1}(I_{A/B}/I_{A/B}^{2}, A) ≅ _{0}Ext_{B}^{1}(I_{A/B}, A)

and hence _{0}H^{2}(B, A, A) ≅ _{0}Ext_{B}^{1}(I_{A/B}, A). Indeed by a well-known spectral sequence it suffices to see Hom_{A}(Tor_{1}^{B}(I_{A/B}, A), A) = 0. Since, however, I_{A/B} is invertible in U = Y − Z and the intersection of U with any irreducible component of X is non-empty, we get the assertion.

8 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

Lemma 3.3. If depth_{I(Z)} B ≥ 3, then _{0}Ext_{B}^{2}(I_{A/B}, B) = _{0}Ext_{B}^{2}(S^{3}K_{B}(−λ), K_{B}) = 0, and

b = _{0}hom_{B}(S^{3}K_{B}(−λ), K_{B}) − _{0}ext_{B}^{1}(S^{3}K_{B}(−λ), K_{B}) =
dim(S^{3}F_{2})_{v} − dim(F_{1} ⊗ S^{2}F_{2})_{v} + dim(∧^{2}F_{1} ⊗ F_{2})_{v} − dim(∧^{3}F_{1})_{v}

where v = λ + 2n + 2.

Proof. Since depth_{I(Z)} B ≥ 3 and depth_{I(Z)} K_{B} ≥ 3, we have by (2.1),

Ext_{B}^{1}(I_{A/B}, B) ≅ Ext_{O_{U}}^{1} ({ \atop S^{2}K_{B}}(−λ), B̃) ≅ H_{∗}^{1}(U, Hom({ \atop S^{2}K_{B}}(−λ), B̃))

≅ H_{∗}^{1}(U, Hom({ \atop S^{3}K_{B}}(−λ), K̃_{B})) ≅ Ext_{B}^{1}(S^{3}K_{B}(−λ), K_{B}),

and correspondingly for the Hom groups. Hence we get the first expression of b above.

Moreover Ext_{B}^{2}(I_{A/B}, B) ≅ Ext_{B}^{2}(S^{2}K_{B}(−λ)⊗K_{B}, K_{B}) by the spectral sequence of [14], Satz 2.1 and _{0}Ext_{B}^{2}(S^{3}K_{B}(−λ), K_{B}) = 0 by (2.8). If Λ is the kernel of the surjective map S^{2}K_{B}(−λ)⊗K_{B} → S^{3}K_{B}(−λ), then Λ̃|_{U} = 0 and since Ext_{B}^{2}(Λ, K_{B}) ↪ Ext_{O_{U}}^{2} (Λ̃, K̃_{B}) is injective by (2.1), we get Ext_{B}^{2}(Λ, K_{B}) = 0 from Ext_{O_{U}}^{2} (Λ̃, K̃_{B}) ≅ H_{∗}^{2}(U, Hom(Λ̃, K̃_{B})) = 0. Then the exact sequence

→ Ext_{B}^{2}(S^{3}K_{B}(−λ), K_{B}) → Ext_{B}^{2}(S^{2}K_{B}(−λ) ⊗ K_{B}, K_{B}) → Ext_{B}^{2}(Λ, K_{B}) →

shows that Ext_{B}^{2}(S^{2}K_{B}(−λ)⊗K_{B}, K_{B}) = 0 and we get the vanishing of both Ext^{2}-groups of the lemma.

Finally using Gorenstein duality twice (i.e. over B and over R) we get

0Ext_{R}^{i}(S^{3}K_{B}(−λ), R(−n − 1)) ≅ _{0}Ext_{B}^{i−2}(S^{3}K_{B}(−λ), K_{B})

for i = 2 and 3 and we have _{0}Ext_{R}^{i}(S^{3}K_{B}(−λ), R(−n − 1)) = 0 for i = 0 and 1. Hence if we apply the contravariant functor _{0}Hom(−, R(λ + 2n + 2)) to the exact sequence (2.8), we obtain a complex (where v = λ + 2n + 2);

(3.7)  0 → (S^{3}F_{2})_{v} → (F_{1} ⊗ S^{2}F_{2})_{v} → (∧^{2}F_{1} ⊗ F_{2})_{v} → (∧^{3}F_{1})_{v} → 0 ,

which is exact except at the spots which correspond to _{0}Ext_{R}^{i}(S^{3}K_{B}(−λ), R(−n−1)) ≠ 0. Therefore, the alternating sum of the dimension of this complex must be b and we are done. □

By using (2.1) as above we get the following information about the number e := 0ext_{B}^{2}(I_{A/B}, I_{A/B});

Lemma 3.4. If depth_{I(Z)} B ≥ 3, then _{0}hom_{B}(I_{A/B}, I_{A/B}) = 1, _{0}Ext_{B}^{1}(I_{A/B}, I_{A/B}) = 0 and there is an exact sequence

(3.8) 0 → Ext_{B}^{2}(I_{A/B}, I_{A/B}) → H_{∗}^{2}(U, B̃) → Hom_{B}(I_{A/B}, H_{I(Z)}^{3}(I_{A/B})) → Ext_{B}^{3}(I_{A/B}, I_{A/B}).

In particular (i) if depth_{I(Z)} B ≥ 4 then Ext_{B}^{2}(I_{A/B}, I_{A/B}) = 0, i.e. e = 0, (ii) if depth_{I(Z)} B = 3, I(Z) = 𝔪 and B has a semi-linear resolution, i.e.

(3.9) 0 → R(−s − 2)^{β_{2}} ⊕ R(−s − 1)^{α_{2}} → R(−s − 1)^{β_{1}} ⊕ R(−s)^{α_{1}} → R → B → 0

then

e = β_{2} · \binom{s + 1}{4} + (α_{2} − β_{1}) · \binom{s}{4} − α_{1} · \binom{s − 1}{4}.

Remark 3.5. (a) Suppose depth_{I(Z)} B ≥ 3. Since Ext_{B}^{2}(I_{A/B}, B) = 0 and Ext_{B}^{1}(I_{A/B}, I_{A/B}) = 0 by the lemmas above, the sequence

0 → Ext_{B}^{1}(I_{A/B}, B) → Ext_{B}^{1}(I_{A/B}, A) → Ext_{B}^{2}(I_{A/B}, I_{A/B}) → 0,

deduced from 0 ⟶ I_{A/B} ⟶ B ⟶ A ⟶ 0, is exact. The following sequence related to (3.8) is also useful in computing e;

(3.10) 0 → Ext_{B}^{1}(I_{A/B}, A) → H_{∗}^{1}(U, Hom(Ĩ_{A/B}, Ã)) → Hom_{B}(I_{A/B}, H_{I(Z)}^{2}(A)) →

Note that the corresponding sequence for _{0}Ext_{B}^{1}(I_{A/B}, B) implies

0Ext_{B}^{1}(I_{A/B}, B) ≅ H^{1}(U, Hom(Ĩ_{A/B}, B̃)) ≅ H^{1}(U, Ĩ_{A/B}^{∨}).

(b) Let T → S be a small Artin surjection with kernel 𝔞 and let ψ_{S} : B_{S} → A_{S} (resp. B_{T}) be a graded deformation of B → A to S (resp. of B_{S} to T), see Proposition 2.2 and Example 2.3. By (3.6) and [11] (see [21], Sect. 1.1 for a general introduction) the group _{0}Ext_{B}^{1}(I_{A/B}, A) ⊗_{k} 𝔞 contains the obstruction o(ψ_{S}; B_{T})_{0} of deforming ψ_{S} further to B_{T}. Since (I_{A/B}, B) is unobstructed along any graded deformation of B and since o(ψ_{S}; B_{T})_{0} maps onto the obstruction in _{0}Ext_{B}^{2}(I_{A/B}, I_{A/B}) ⊗_{k} 𝔞 of deforming ker ψ_{S} to B_{T}, it maps to zero!! It follows that o(ψ_{S}; B_{T})_{0} sits in _{0}Ext_{B}^{1}(I_{A/B}, B) ⊗_{k} 𝔞, i.e. we say that _{0}Ext_{B}^{1}(I_{A/B}, B) contains all graded obstructions of deforming B → A to any given deformation of B.

Proof. By (2.7) I_{A/B} = S^{2}K_{B}(−λ) is a maximal CM module and we have depth_{I(Z)} I_{A/B} ≥ 3. Hence, by (2.1),

Ext^{1}(I_{A/B}, I_{A/B}) ≅ Ext_{O_{U}}^{1} (Ĩ_{A/B}, Ĩ_{A/B})
≅ H_{∗}^{1}(U, Hom(Ĩ_{A/B}, Ĩ_{A/B})) ≅ H_{∗}^{1}(U, B̃) ≅ H_{I(Z)}^{2}(B) = 0

and

0Hom(I_{A/B}, I_{A/B}) ≅ H_{∗}^{0}(U, B̃)_{0} ≅ B_{0} ≅ k.

In the same way, by using (2.1), we get (3.8).

Now we will use (3.8) to show (i) and (ii). (i) is clear. To see (ii) it suffices to show 0Hom_{B}(I_{A/B}, H_{𝔪}^{3}(I_{A/B})) = 0 because, by (2.4) and duality,

h^{2}(Y, B̃) = dim(K_{B})_{0} = ∑_{1≤i≤μ−1} \binom{n_{2,i} − 1}{n} − ∑_{1≤i≤μ} \binom{n_{1,i} − 1}{n}

and n = 4 and we get the expression of e in the lemma. We have I_{A/B} = S^{2}K_{B}(−λ). To see _{0}Hom_{B}(S^{2}K_{B}, H_{𝔪}^{3}(S^{2}K_{B})) = 0 it suffices by (2.7) to show

0Hom_{B}(S^{2}(F_{2}^{∨}), H_{𝔪}^{5}(∧^{2}(F_{1}^{∨}))) = 0,

i.e. to prove _{0}H_{𝔪}^{5}(S^{2}(F_{2}) ⊗ ∧^{2}(F_{1}^{∨})) = 0. Since, however, _{0}H_{𝔪}^{5}(R(t)) = 0 for t ≥ −4, we are done. □

Now we come to the two main theorems of this section. Recall that B = R/I_{B} has normal module N_{B} := Hom_{B}(I_{B}, B) and minimal resolution

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

and that A is defined by (3.1), i.e. A ≅ B/(S^{2}K_{B}(−λ)). We include a criterion of unobstructedness which we may apply if we are able to compute h^{0}(N_{X}) (N_{X} the normal sheaf of X = Proj(A) ⊂ ℙ^{n}).

Theorem 3.6. Let B be a codimension 2 CM quotient of R = k[x_{0}, ..., x_{n}], and let U = Proj(B) − Z ↪ ℙ^{n} be a local complete intersection such that depth_{I(Z)} B ≥ 3. Let X = Proj(A) ⊂ ℙ^{n} be the codimension 3 ACM scheme defined by (3.1). Let h_{A}^{i} := dim _{0}H^{i}(R, A, A) and let v = λ + 2n + 2. Then

b − 1 + dim(N_{B})_{0} − a − e = h^{0}(N_{X}) − h_{A}^{2} ≤ dim_{(X)} Hilb ℙ^{n},

where e := _{0}ext_{B}^{2}(I_{A/B}, I_{A/B}) and

b−1+dim(N_{B})_{0}−a = − ∑_{1≤i<j<k≤μ} \binom{−n_{1,i} − n_{1,j} − n_{1,k} + n + v}{n}+ ∑_{{1≤i<j≤μ \atop 1≤k≤μ−1}} \binom{−n_{1,i} − n_{1,j} − n_{2,k} + n + v}{n}

− ∑_{{1≤i≤μ \atop 1≤j≤k≤μ−1}} \binom{−n_{1,i} − n_{2,j} − n_{2,k} + n + v}{n}+ ∑_{1≤i≤j≤k≤μ−1} \binom{−n_{2,i} − n_{2,j} − n_{2,k} + n + v}{n}+ ∑_{{1≤i≤μ \atop 1≤j≤μ−1}} \binom{n_{2,j} − n_{1,i} + n}{n}

+ ∑_{{1≤i≤μ \atop 1≤j≤μ−1}} \binom{n_{1,i} − n_{2,j} + n}{n} − ∑_{1≤i,j≤μ} \binom{n_{1,j} − n_{1,i} + n}{n} − ∑_{1≤i,j≤μ−1} \binom{n_{2,i} − n_{2,j} + n}{n}

− ∑_{{1≤i<j≤μ \atop k≤j}} \binom{n_{1,i} + n_{1,j} + n_{1,k} + n − v}{n} + ∑_{{1≤i,j≤μ \atop 1≤k≤μ−1}} \binom{n_{1,i} + n_{1,j} + n_{2,k} + n − v}{n}

− ∑_{{1≤i≤μ \atop 1≤j,k≤μ−1}} \binom{n_{1,i} + n_{2,j} + n_{2,k} + n − v}{n} + ∑_{{1≤i,j,k≤μ−1 \atop i≤j>k}} \binom{n_{2,i} + n_{2,j} + n_{2,k} + n − v}{n}

Moreover if h^{0}(N_{X}) + e = b − 1 + dim(N_{B})_{0} − a, then X is unobstructed.

Proof. A general theorem of Laudal, which establishes a lower bound for dimension of the hull of any deformation functor, implies, thanks to Theorem 1.3 and Theorem 2.2 of [18], that h_{A}^{1} − h_{A}^{2} is a lower bound for the dimension of the hull of the graded deformation functor of R → A. Since Theorem 3.6 and Remark 3.7 of [18] (or one may use [8], Proposition 1 since A is CM) implies that the graded deformation functor of R → A is isomorphic to the local Hilbert functor of (X = Proj(A) ⊂ ℙ^{n}), we get

dim_{(X)} Hilb ℙ^{n} ≥ h_{A}^{1} − h_{A}^{2}.

Now to see b − 1 − e + dim(N_{B})_{0} − a = h_{A}^{1} − h_{A}^{2}, we consider the diagram

(3.11)

0 ↓ 0Hom(I_{B}, I_{A/B}) ↓ H^{0}(Ñ_{B}) ↓

0Hom(I_{A/B}, A) ↪ _{0}H^{1} → _{0}Hom_{R}(I_{B}, A) → _{0}H^{2}(B, A, A) → _{0}H^{2} → 0

↓ 0Ext^{1}(I_{B}/I_{B}^{2}, I_{A/B}) ↓ 0

where H^{i} = H^{i}(R, A, A) (cf. [21], Section 1.1). Note that we have used Lemma 3.1 to see that

0H^{2}(R, B, A) ≅ _{0}Ext^{1}(I_{B}/I_{B}^{2}, A) ≅ _{0}Ext^{2}(I_{B}/I_{B}^{2}, I_{A/B}) = 0

and the fact that B is licci and { \atop K_{B}|U} locally free to see

0Ext^{i}(I_{B}/I_{B}^{2}, B) ≅ _{0}Ext^{i}(I_{B}/I_{B}^{2} ⊗ K_{B}, K_{B}) = 0 for 1 ≤ i ≤ 2.

Using the vertical sequence in the diagram, we get _{0}hom_{R}(I_{B}, A) = dim(N_{B})_{0} − a by the definition of a. Hence it suffices to show that

0hom_{B}(I_{A/B}, A) − _{0}h^{2}(B, A, A) = b − 1 − e.

Since one knows that_{0}H^{2}(B, A, A) ≅ _{0}Ext_{B}^{1}(I_{A/B}, A) by (3.6) and we have_{0}h^{2}(B, A, A) = e + _{0}ext_{B}^{1}(I_{A/B}, B) by Remark 3.5(a), we conclude by Lemma 3.4.

It remains to compute b, a, and dim(N_{B})_{0} in terms of n_{j,i}. Thanks to Lemma 3.1, we get

(3.12) a = ∑_{{1≤i<j≤μ \atop k≤j}} \binom{n_{1,i} + n_{1,j} + n_{1,k} + n − v}{n} − ∑_{{1≤i,j≤μ \atop 1≤k≤μ−1}} \binom{n_{1,i} + n_{1,j} + n_{2,k} + n − v}{n}

+ ∑_{{1≤i≤μ \atop 1≤j,k≤μ−1}} \binom{n_{1,i} + n_{2,j} + n_{2,k} + n − v}{n} − ∑_{{1≤i,j,k≤μ−1 \atop i≤j>k}} \binom{n_{2,i} + n_{2,j} + n_{2,k} + n − v}{n}.

Using Lemma 3.3, we get (3.13)

b = − ∑_{1≤i<j<k≤μ} \binom{−n_{1,i} − n_{1,j} − n_{1,k} + n + v}{n} + ∑_{{1≤i<j≤μ \atop 1≤k≤μ−1}} \binom{−n_{1,i} − n_{1,j} − n_{2,k} + n + v}{n}

− ∑_{{1≤i≤μ \atop 1≤j≤k≤μ−1}} \binom{−n_{1,i} − n_{2,j} − n_{2,k} + n + v}{n} + ∑_{1≤i≤j≤k≤μ−1} \binom{−n_{2,i} − n_{2,j} − n_{2,k} + n + v}{n}.

Moreover one knows by [8] that

(3.14)  dim(N_{B})_{0} = ∑_{{1≤i≤μ \atop 1≤j≤μ−1}} \binom{n_{2,j} − n_{1,i} + n}{n} + ∑_{{1≤i≤μ \atop 1≤j≤μ−1}} \binom{n_{1,i} − n_{2,j} + n}{n}−

∑_{1≤i,j≤μ} \binom{n_{1,i} − n_{1,j} + n}{n} − ∑_{1≤i,j≤μ−1} \binom{n_{2,i} − n_{2,j} + n}{n} + 1.

Putting (3.12), (3.13) and (3.14) together we get what we want. □

Looking to the very first part of the proof above, we see that the lower bound is actually a lower bound for the dimension of the moduli space parameterizing graded quotients A of R with fixed Hilbert function. We get the bound because the local ring of this space at (A) is isomorphic to the local ring of Hilb ℙ^{n} at (X). There is also another very natural moduli space to consider, namely the Hilbert-flag scheme D parameterizing graded surjections B → A of R with fixed Hilbert functions. If we write a as a = a^{0} − a^{1} where

(3.15)  a^{i} := _{0}ext_{B}^{i}(I_{B}/I_{B}^{2}, I_{A/B}),

one knows that the local ring of D at (B → A) is isomorphic to (resp. a subring of) the local ring of Hilb ℙ^{n} at (X) provided a^{0} = a^{1} = 0 (resp. a^{0} = 0). Note that Lemma 3.2 gives a simple criterion for the vanishing of a^{0} and a^{i}, i = 0, 1. The main ingredient of our next result is that we can prove that b − 1 + dim(N_{B})_{0} is a lower bound for dim O_{D,(B→A)}.

To this end, let us fix some more notation. We denote by

α : H^{0}(Ñ_{B}) → _{0}Hom(I_{B}, A) → _{0}H^{2}(B, A, A) ≅ _{0}Ext_{B}^{1}(I_{A/B}, A)

the composition of maps appearing in (3.11); see (3.6) for the last isomorphism. If depth_{I(Z)} B ≥ 3, we see by Remark 3.5 that α factors through Ext_{B}^{1}(I_{A/B}, B) giving rise to a map

α′ : H^{0}(Ñ_{B}) ⟶ _{0}Ext_{B}^{1}(I_{A/B}, B).

We have

Theorem 3.7. Let B be a codimension 2 CM quotient of R = k[x_{0}, ..., x_{n}], char(k) ≠ 2, and let U = Y − Z ↪ ℙ^{n}, Y = Proj(B), be a local complete intersection such that depth_{I(Z)} B ≥ 3. Let X = Proj(A) ⊂ ℙ^{n} be the codimension 3 ACM scheme defined by (3.1). Then

b − 1 + dim(N_{B})_{0} − a^{0} ≤ dim_{(X)} Hilb ℙ^{n},

where dim(N_{B})_{0} and a^{0} are given by (3.14) and (3.15) with v = λ + 2n + 2, and b is given by (3.13) , or by b = h^{0}(U, ω_{Y}^{−2}(λ)) − h^{1}(U, ω_{Y}^{−2}(λ)). Moreover suppose that α′ is surjective (e.g. _{0}Ext_{B}^{1}(I_{A/B}, B) = 0). Then,

dim_{(X)} Hilb ℙ^{n} ≤ b − 1 + dim(N_{B})_{0} − a,

and X is unobstructed if this upper bound is sharp. In particular, if a^{0} = a^{1} = 0 and α′ is surjective, then X is unobstructed and

dim_{(X)} Hilb ℙ^{n} = b − 1 + dim(N_{B})_{0}.

Proof. Let k[ϵ] := k[x]/(x^{2}) be the ring of dual numbers. We claim that b−1+dim(N_{B})_{0} is a lower bound for dim O_{D,(B→A)}. Indeed if γ ∈ H^{0}(Ñ_{B}) corresponds to a deformation B_{ϵ} of R → B to k[ϵ], then α(γ) = o(B → A; B_{ϵ})_{0} by [11], see also the proof of Theorem 9.4 in [19] and see Remark 3.5 for notations. Let _{0}A_{B→A}^{1} be defined by the cartesian diagram

\begin{matrix}_{0}A_{B→A}^{1} & ⟶ & H^{0}(Ñ_{B}) \\ ↓ & □ & ↓ \\ _{0}H^{1} & ⟶ & _{0}Hom_{R}(I_{B}, A) \end{matrix}

of morphisms appearing in (3.11). Then _{0}A_{B→A}^{1} is the tangent space of the Hilbert-flag scheme D at (B → A) ([21], Sect. 1.1). Moreover, if B is unobstructed as a graded algebra, then coker α contains all obstructions of deforming B → A as graded algebras by a result of the unpublished thesis of the first author (cf. [22], (1.9) for a closely related result). By Remark 3.5, coker α′ ⊂ coker α contains all obstructions of deforming B → A in the case (I_{A/B}, B) is unobstructed along any graded deformation of B. Then we can again conclude by Laudal’s theorem ([26], Theorem 4.2.4) that

(3.16)  dim _{0}A_{B→A}^{1} − dim coker α′ ≤ dim O_{D,(B→A)}.

Now looking once more to (3.11) and the proof of Theorem 3.6, we easily get

(3.17)  dim _{0}A_{B→A}^{1} − dim coker α′ = b − 1 + dim(N_{B})_{0}

and the claim is proved.

Let p : D → Hilb ℙ^{n} be the second projection morphism, i.e. induced by p((B′ → A′)) = (Proj(A′)). Since the tangent space of the fiber p^{−1}((Proj(A))) is _{0}Hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}) at (B → A) ([18], Theorem 1.6 or [21], Proposition 4) i.e. an a^{0}-dimensional vector space, we get dim O_{D,(B→A)} − a^{0} ≤ dim_{(X)} Hilb ℙ^{n} and hence the lower bound of the theorem.

Finally suppose coker(α′) = 0. Looking at (3.11) and using the definition of a^{i} and 0A_{B→A}^{1}, we get

h^{0}(N_{X}) = dim _{0}H^{1} ≤ dim _{0}A_{B→A}^{1} − a^{0} + a^{1}.

Since dim_{(X)} Hilb ℙ^{n} ≤ h^{0}(N_{X}) we get the upper bound from (3.17). Moreover if the upper bound is sharp, we deduce dim_{(X)} Hilb ℙ^{n} = h^{0}(N_{X}), and it follows that X is unobstructed. Since we get the final conclusions by observing that the upper and lower bounds coincide under the assumption a^{0} = a^{1} = 0, we are done. □

Remark 3.8. (a) Comparing the lower bounds of Theorem 3.6 and Theorem 3.7, we remark that a = a^{0} − a^{1}. Hence if e > a^{1} then the lower bound of Theorem 3.7 is larger and conversely if e < a^{1}.

(b) Moreover note that if α′ is surjective, we see from the proof above that e = dim _{0}H^{2}(R, A, A). In particular if X is licci, then e = 0 ([19], Proposition 6.17).

Corollary 3.9. Let B be a codimension 2 CM quotient of R = k[x_{0}, ..., x_{n}], char(k) ≠ 2 and let Y := Proj(B) ↪ ℙ^{n} be a local complete intersection such that dim Y ≥ 2. Let X = Proj(A) ⊂ ℙ^{n} be the codimension 3 ACM scheme defined by (3.1) with λ ≫ 0. Then X is unobstructed and

dim_{(X)} Hilb ℙ^{n} = b − 1 + dim(N_{B})_{0}.

For the numbers a and e above, we have a = 0. Moreover if dim Y ≥ 3 then e = 0.

14 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

Proof. By Theorem 3.7 it suffices to show a^{0} = a^{1} = 0 and _{0}Ext_{B}^{1}(S^{2}K_{B}(−λ), B) = 0. The first assumption is taken care of by Lemma 3.2. Moreover using the l.c.i. assumption on Y we get that K̃_{B} is locally free, and hence that

(3.18)  _{0}Ext_{B}^{1}(S^{2}K_{B}(−λ), B) ≅ H^{1}(Y, S^{2}K̃_{B}^{∨}(λ)) ≅ H^{dim Y −1}(Y, S^{3}K̃_{B}(−λ))^{∨},

which vanishes for λ ≫ 0. Finally if dim Y = 3, it follows from Lemma 3.4(i) that e = 0, and we are done. □

One may prove the following result directly from Theorem 9.4 of [19] by using the fact that (S^{2}K_{B}(−λ), B) is unobstructed along any graded deformation of B (Example 2.3). We, however, get the corollary by combining Theorems 3.6 and 3.7.

Corollary 3.10. Let Y = Proj(B), U = Y −Z ↪ ℙ^{n} and X = Proj(A) be as in Theorem 3.7 and suppose depth_{I(Z)} B ≥ 3 and _{0}Ext_{B}^{1}(I_{A/B}, A) = 0. Then X is unobstructed and

dim_{(X)} Hilb ℙ^{n} = b − 1 + dim(N_{B})_{0} − a.

Moreover the number b of (3.13) is also equal to b = _{0}hom(I_{A/B}, B) = h^{0}(U, ω_{Y}^{−2}(λ)).

Proof. Using Remark 3.5 (a), we get e = 0 and that α′ is surjective. It follows that the upper bound of Theorem 3.7 and the lower bound of Theorem 3.6 coincide and we conclude by Theorem 3.7. □

Remark 3.11. In this remark we will use (3.18) to make the assumption λ ≫ 0 of Corollary 3.9 into an explicit bound provided the number of minimal generators of I_{B} is μ = 3 (and I(Z) = 𝔪). Firstly we consider the exact sequence of locally free sheaves on Y,

(3.19)  0 ⟶ K̃_{B}(m) ⟶ ⊕_{i=1}^{3}B̃(−n_{1,i}) ⟶ { \atop I_{B}/I_{B}^{2}} ⟶ 0

where m := n+1−∑_{i=1}^{3} n_{1,i}, associated to the exact sequences (3.2) and (3.3). Dualizing it and building the exact sequence of S^{2}(K̃_{B}^{∨})(−2m) in the usual way, we get the sequence

0 → ∧^{2}({ \atop I_{B}/I_{B}^{2}})^{∨} → ({ \atop I_{B}/I_{B}^{2}})^{∨} ⊗ (⊕_{i}B̃(n_{1,i})) → S^{2}(⊕_{i}B̃(n_{1,i})) → S^{2}(K̃_{B}^{∨})(−2m) → 0

which simplifies to the following exact sequence of locally free O_{Y} -sheaves

(3.20) 0 → K̃_{B}(n + 1) → ⊕_{i}Ñ_{B}(n_{1,i}) → ⊕_{i≤j}B̃(n_{1,i} + n_{1,j}) → S^{2}(K̃_{B}^{∨})(−2m) → 0.

If dim Y = 3 and hence n = 5, we have by (3.20) an injection H^{1}(Y, S^{2}(K̃_{B}^{∨})(λ)) ↪ H^{3}(Y, K̃_{B}(n + 1 + 2m + λ)), which vanishes if 6 + 2m + λ > 0, i.e. λ > 2 · ∑_{i=1}^{3} n_{1,i} − 18. Combining with Lemma 3.4(i) and Remark 3.5 (a) we get _{0}Ext_{B}^{1}(I_{A/B}, A) = 0 and Corollary 3.10 applies. Hence X is unobstructed (indeed _{0}H^{2}(R, A, A) = 0) and

(3.21)  dim_{(X)} Hilb ℙ^{n} = b − 1 + dim(N_{B})_{0} − a

provided

λ > 2 ∑_{i=1}^{3} n_{1,i} − 18.

We have (3.21) with a = 0 if λ > max{3d_{2} − 12, 2 ∑_{i=1}^{3} n_{1,i} − 18} by Lemma 3.2.

If dim Y = 2 and n = 4, we can use (3.20) to see that H^{1}(Y, S^{2}(K̃_{B}^{∨})(λ)) = 0 provided H^{2}(Y, Ñ_{B}(n_{1,i} + 2m + λ)) = 0 for any i. Using (2.2) and (2.6) we see that H^{2}(Y, Ñ_{B}(v)) vanishes if H^{4}(ℙ^{4}, F̃_{1}^{∨} ⊗ F̃_{2}(v)) = 0, i.e. for v + s > d_{2} − 5 where s = min{n_{1,i}}. Hence H^{1}(Y, S^{2}(K̃_{B}^{∨})(λ)) = 0 provided 2s+2m+λ > d_{2} −5, i.e. λ > 2·∑_{i=1}^{3} n_{1,i} −2s+d_{2} −15. Combining with Lemma 3.2 and Theorem 3.7, we get that X is unobstructed and that (3.21) holds with a = 0 provided

λ > max{3d_{2} − 10, 2 ∑_{i=1}^{3} n_{1,i} − 2s + d_{2} − 15}.

Sometimes the Kodaira vanishing theorem, or related arguments, show H^{1}(Y, ω_{Y}^{−2}(λ)) = 0 more effectively than the arguments of Remark 3.11.

Example 3.12. Let Y = Proj(R/I_{B}) be a smooth Castelnuovo surface in ℙ^{4} and let X ∼ −2K_{Y} + λH, λ ≥ 0, be an effective divisor on Y. In the usual basis of P ic(Y) ≅ ℤ^{9}, then X_{0} := −2K_{Y} corresponds to (6; 2^{8}) and H to (4; 2, 1^{7}). By Kodaira vanishing theorem, or simply by K_{Y} H = −3 < 0, we get

H^{1}(N_{X/Y})^{∨} = H^{0}(Y, O_{Y} (−X + K_{Y})) = 0.

Hence, H^{1}(Y, O_{Y} (X)) = 0 for any λ ≥ 0 while Remark 3.11 implies the vanishing of the same group for λ > 2 ∑ n_{1i} − 2s + d_{2} − 15 = 1, because of

0 ⟶ R(−4)^{2} ⟶ R(−3)^{2} ⊕ R(−2) ⟶ I_{B} ⟶ 0.

In this case we have _{0}Ext^{1}(I_{A/B}, A) = 0 by Remark 3.5(a) and Corollary 3.10 applies for λ ≥ 0. We get that X is unobstructed (indeed _{0}H^{2}(R, A, A) = 0 by Remark 3.9(b) and Lemma 3.4) and

dim_{(X)} Hilb ℙ^{n} = h^{0}(O_{Y} (X)) − 1 + h^{0}(N_{Y}) − a =
X(X − K_{Y})/2 + 32 − a = 35 + 5λ(λ + 3)/2 − a.

By Lemma 3.2, a = 0 provided λ ≥ 3 and we have a = 6, -2, -2 for λ = 0, 1, 2 respectively by (3.12). Hence dim_{(X)} Hilb ℙ^{n} = 29, 47, 62 for λ = 0, 1, 2 respectively and 35 + 5λ(λ + 3)/2 for λ ≥ 3. Note that if d and g is the degree and genus of the curve X, we have h^{0}(N_{X}) − h^{1}(N_{X}) = 5d + 1 − g which is equal to 29, 47 and 60 for λ = 0, 1 and 2 and more generally equal to 29 + (41λ − 5λ^{2})/2 for λ ≥ 0. Thus

h^{1}(N_{X}) = 5λ^{2} − 13λ + 6 − a for λ ≥ 0.

4. Ideals generated by submaximal minors of symmetric matrices

The goal of this section is to write down lower bounds for the dimension of Hilb_{(X)}^{p(t)}(ℙ^{n}) being X ⊂ ℙ^{n} a codimension 3, arithmetically Cohen-Macaulay scheme defined by the submaximal minors of a t × t homogeneous symmetric matrix. We will also analyze when the mentioned bounds are sharp. A classical scheme that can be constructed in this way is the Veronese surface X ⊂ ℙ^{5}. Indeed, the Veronese surface X ⊂ ℙ^{5} can be defined by the 2 × 2 minors of the symmetric matrix

(\begin{matrix}x_{0} & x_{1} & x_{2} \\ x_{1} & x_{3} & x_{4} \\ x_{2} & x_{4} & x_{5} \end{matrix}) .

Let us first fix the notation we will use throughout this section. From now on, X ⊂ ℙ^{n} will be a codimension 3, arithmetically Cohen-Macaulay scheme defined by the submaximal minors of a t × t homogeneous symmetric matrix A = (f_{ji})_{i,j=1,...t} where f_{ji} ∈ k[x_{0}, ..., x_{n}] are homogeneous polynomials of degree a_{i} + a_{j} and let A = R/I(X) be the homogeneous coordinate ring of X. We denote by

U = (\begin{matrix}2a_{1} & a_{1} + a_{2} & ⋯ & a_{1} + a_{t} \\ a_{1} + a_{2} & 2a_{2} & ⋯ & a_{2} + a_{t} \\ ⋮ & ⋮ & & ⋮ \\ a_{1} + a_{t} & a_{2} + a_{t} & ⋯ & 2a_{t} \end{matrix})

the degree matrix of A. The determinant of A is a homogeneous polynomial of degree ℓ = 2(a_{1} + a_{2} + ⋯ + a_{t}). Note that a_{i} + a_{j} is a positive integer for all 1 ≤ i ≤ j ≤ t while a_{i} does not necessarily need to be an integer.

Let B be the matrix obtained by deleting the last row, let I_{B} = I_{t−1}(B) be the ideal defined by the maximal minors of B 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}.

Remark 4.1. Assume char(k) ≠ 2. After a basis change that preserves the symmetry of A, if necessary, we have that the assumption codim_{R} A = 3 implies that codim_{R} B = 2 and I_{B} is Cohen-Macaulay. In fact, we well know that codim_{R} B ≤ 2. Following the approach of the proof of Theorem 2 in [5], and strongly using the fact that A is symmetric and char(k) ≠ 2, we get (see also [9], Theorem 1.22)

ht(I_{A}/I_{B}) = ht(I_{t−1}(A)/I_{t−1}(B)) ≤ 1.

Therefore, we obtain ht(I_{B}) = ht(I_{t−1}(B)) ≥ ht(I_{A}) − 1 = ht(I_{t−1}(A)) − 1 = 2 and we are done.

Remark 4.2. With the above notations, let Y ⊂ ℙ^{n} be the codimension 2, ACM scheme defined by the maximal minors of B. We claim that Y is a generically complete intersection. In fact, consider

0 ⟶ F ⟶^{^{t}B} G ⟶ I_{B} = I(Y) ⟶ 0

the resolution of I(Y) given by Hilbert-Burch theorem. Let P be a minimal associated prime of I(Y) = I_{t−1}(B). We have to see that I(Y)_{P} is a complete intersection. We have ht(P) = ht(I_{t−1}(B)) = 2 < 3 = ht(I(X)) = ht(I_{t−1}(A)) ≤ ht(I_{t−2}(B)). So, P ⊉ I_{t−2}(B). Denote by μ(I(Y)_{P}) the number of minimal generators of I(Y)_{P}. By [4]; Proposition 16.3, μ(I(Y)_{P}) ≤ 2 and we are done.

Remark 4.3. If the entries of A and B are sufficiently general polynomials of degree a_{i} + a_{j} with a_{i} + a_{j} a positive integer for all 1 ≤ i ≤ j ≤ t, then codim_{R} B = 2 and codim_{R} A = 3. In fact, given rational numbers a_{i} ∈ ℚ, 1 ≤ i ≤ t, with a_{i} + a_{j} a positive integer for all 1 ≤ i ≤ j ≤ t, we can consider the t × t symmetric homogeneous matrix

A = (\begin{matrix}0 & 0 & ⋯ & 0 & 0 & 0 & x_{1}^{a_{1}+a_{t−1}} & x_{0}^{a_{1}+a_{t}} \\ 0 & 0 & ⋯ & 0 & 0 & x_{1}^{a_{2}+a_{t−2}} & x_{0}^{a_{2}+a_{t−1}} & x_{2}^{a_{2}+a_{t}} \\ 0 & 0 & ⋯ & 0 & x_{1}^{a_{3}+a_{t−3}} & x_{0}^{a_{3}+a_{t−2}} & x_{2}^{a_{3}+a_{t−1}} & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ x_{0}^{a_{1}+a_{t}} & x_{2}^{a_{2}+a_{t}} & ⋯ & 0 & 0 & 0 & 0 & 0 \end{matrix})

and let B be the matrix obtained by deleting the last row of A. We clearly have codim_{R} B = 2 and codim_{R} A = 3 which proves what we want.

Proposition 4.4. With the above notation and assumptions codim_{R} A = 3 and codim_{R} B = 2, we have an exact sequence

0 ⟶ (S^{2}K_{B})(2n + 2 + p) ⟶ B ⟶ A ⟶ 0

where p = 2a_{t} − 3ℓ and K_{B} = Ext_{B}^{2}(B, R(−n − 1)) is the canonical module of B.

Proof. First of all we observe that the degrees of the minors of B are ℓ − a_{i} − a_{t}, i.e. I_{B} has the following minimal free R-resolution

(4.1)  0 ⟶ ⊕_{i=1}^{t−1}R(−a_{i} + a_{t} − ℓ) ⟶^{^{t}B} ⊕_{i=1}^{t}R(a_{i} + a_{t} − ℓ) ⟶ I_{B} ⟶ 0.

By [17]; Theorem 3.1, I_{A} has a minimal free R-resolution of the following type:

(4.2)  0 ⟶ ⊕_{1≤i<j≤t}R(−a_{i} − a_{j} − ℓ) ⟶ ⊕_{1≤i,j≤t}R(−ℓ − a_{i} + a_{j})/R(−ℓ) ⟶^{A}
⊕_{1≤i≤j≤t}R(a_{i} + a_{j} − ℓ) ⟶ I_{A} ⟶ 0.

The natural injection I_{B} ↪ I_{A} induces a map from the complex (4.1) to the complex (4.2) in the most obvious way, i.e. factors in the complex (4.2) which are not present in the complex (4.1) are mapped to zero’s, otherwise it is mapped by the identity.

Dualizing the exact sequence (4.1) we get

0 ⟶ R ⟶ ⊕_{i=1}^{t}R(−a_{i} − a_{t} + ℓ) ⟶^{B} ⊕_{i=1}^{t−1}R(a_{i} − a_{t} + ℓ) ⟶ K_{B}(n + 1) ⟶ 0

leading to the exact sequence (see [7]; Theorem A2.10)

(4.3)  0 ⟶ ⊕_{1≤i<j≤t}R(−ℓ − a_{i} − a_{j}) ⟶ ⊕_{{1≤i≤t−1 \atop 1≤j≤t}} R(−ℓ + a_{i} − a_{j}) ⟶^{B}
⊕_{1≤i≤j≤t−1}R(−ℓ + a_{i} + a_{j}) ⟶ S^{2}K_{B}(2n + 2 + p) ⟶ 0.

One easily checks that the natural maps from the complex of I_{A} onto the complex of S^{2}K_{B}(2n + 2 + p) are a morphism of complexes, and hence we get the exact sequence

0 ⟶ I_{B} ⟶ I_{A} ⟶ (S^{2}K_{B})(2n + 2 + p) ⟶ 0

and thus there is an exact sequence

0 ⟶ (S^{2}K_{B})(2n + 2 + p) ⟶ B ⟶ A ⟶ 0

which proves what we want. □

As an interesting consequence of the above result we have

Proposition 4.5. Assume char(k) ≠ 2. Let X ⊂ ℙ^{n} be a codimension 3, ACM scheme defined by the submaximal minors of a t × t homogeneous symmetric matrix A. Then X is glicci.

Proof. Let Y be the codimension 2, ACM subscheme of ℙ^{n} defined by the maximal minors of the t × (t − 1) matrix B obtained deleting the last row of A, after a basis change that preserves the symmetry of A, if necessary. By Remark 4.2, Y is a generically complete intersection. Hence Y satisfies the property G_{0} and we have the concept of generalized divisors on Y. By Proposition 4.4, X ∈ |−2K_{Y} −(2n+2+p)H| where K_{Y} is the canonical divisor on Y and H is a hyperplane divisor. By [19]; Corollary 5.5, for m ≫ 0, G ∼ mH − K_{Y} is arithmetically Gorenstein and G-links X to an element of the linear system |K_{Y} +βH| for a suitable β ∈ ℤ. So, it suffices to check that an element of the linear system |K_{Y} + βH| is glicci. To this end, we consider a codimension 3 standard determinantal scheme D ⊂ Y ⊂ ℙ^{n} defined by the maximal minors of the matrix [B, L] obtained adding to B a sufficiently general column L such that [B, L] is again homogeneous. By [19]; Theorem 3.6, D ∈ |K_{Y} + tH| for some t ∈ ℤ and D is glicci. Moreover, by [19]; Corollary 5.13, K_{Y} + tH and D are G-bilinked, so any effective divisor of type K_{Y} + dH is ACM and glicci and we are done. □

Remark 4.6. The above proposition has also been proved by Gorla. Our proofs are independent except possibly for Remark 4.1, which for some time has been known to experts in the field. Note, however, that Remark 4.3 implies Remark 4.1 in the generic case. In [9]; Corollary 2.7, she has proved that any codimension \binom{m−t+2}{2}, ACM scheme X ⊂ ℙ^{n} defined by the t × t minors of a m × m symmetric homogeneous matrix is glicci.

Lemma 4.7. With the above notation and assumptions codim_{R} A = 3 and codim_{R} B = 2, suppose in addition a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{t}. Then, for i = 1 and 0,

0Ext_{B}^{i}(I_{B}/I_{B}^{2}, S^{2}K_{B})(2n+2+p) = _{0}Ext_{R}^{i}(I_{B}, (S^{2}K_{B})(2n+2+p)) = 0 provided a_{t} > 3a_{t−1}

and

0Hom_{R}(I_{B}, S^{2}K_{B})(2n + 2 + p) = 0 provided a_{t} > 2a_{t−1} − a_{1}.

Proof. It follows from Lemma 3.2. □

Given rational numbers a_{1}, ..., a_{t} such that a_{i} + a_{j} ∈ ℤ_{+} for all i, j we denote by S(\underline{a}) = S(a_{1}, ⋯, a_{t}) the irreducible family of codimension 3, ACM schemes X ⊂ ℙ^{n} defined by the submaximal minors of a t × t symmetric homogeneous matrix A = (f_{ji})_{i,j=1,...,t} where f_{ji} ∈ k[x_{0}, ..., x_{n}] is a homogeneous polynomial of degree a_{j} + a_{i}. Our next goal is to determine a lower bound for the dimension of the irreducible component S(\underline{a}) of Hilb^{p}(ℙ^{n}) containing S(\underline{a}).

Set h_{A}^{i} := dim _{0}H^{i}(R, A, A) where A = R/I(X). Applying Theorem 3.6, we get

b − 1 − e + dim(N_{B})_{0} − a = h_{A}^{1} − h_{A}^{2} = h^{0}(N_{X}) − h_{A}^{2} ≤ dim_{(X)} Hilb ℙ^{n}.

Using this and Theorem 3.7, we are ready to write down bounds for dim_{(X)} Hilb^{p(t)} ℙ^{n} in terms of a_{1}, ⋯, a_{t}, and e := _{0}ext_{B}^{2}(I_{A/B}, I_{A/B}), resp. a^{0} := _{0}hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}). Note that a^{0} (resp. a) vanishes if a_{t} > 2a_{t−1} − a_{1} (resp. a_{t} > 3a_{t−1}) by the previous lemma.

Theorem 4.8. Let X ⊂ ℙ^{n} be a codimension 3, ACM scheme defined by the submaximal minors of a t × t symmetric matrix A = (f_{ji})_{i,j=1,...,t} where f_{ji} ∈ k[x_{0}, ..., x_{n}] is a homogeneous polynomial of degree a_{j} + a_{i}. Let B be the matrix obtained by deleting the last row, let Y = Proj(B) ⊂ ℙ^{n} be the codimension 2 ACM scheme defined by the maximal minors of B, and suppose U = Y −Z ↪ ℙ^{n} is a local complete intersection for some closed subset Z such that depth_{I(Z)} B ≥ 3. Then,

dim_{(X)} Hilb^{p(t)}(ℙ^{n}) ≥ b − 1 + dim(N_{B})_{0} − a − e

where e := _{0}ext_{B}^{2}(S^{2}K_{B}, S^{2}K_{B}),

b − 1 + dim(N_{B})_{0} = ∑_{1≤i≤j≤k≤t−1} \binom{a_{t} − a_{i} − a_{j} − a_{k} + n}{n} − ∑_{{1≤i≤t \atop 1≤j≤k≤t−1}} \binom{a_{t} + a_{i} − a_{k} − a_{j} + n}{n}+

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

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

and

a = − ∑_{{1≤i,j,k≤t−1 \atop i≤j>k}} \binom{a_{i} + a_{j} + a_{k} − a_{t} + n}{n}− ∑_{{1≤k,j≤t \atop 1≤i≤t−1}} \binom{a_{i} − a_{j} − a_{k} − a_{t} + n}{n}+ ∑_{{1≤k,i≤t−1 \atop 1≤j≤t}} \binom{a_{i} − a_{j} + a_{k} − a_{t} + n}{n}.

Moreover if char(k) ≠ 2, then

dim_{(X)} Hilb ℙ^{n} ≥ b − 1 + dim(N_{B})_{0} − a^{0}.

Proof. It follows from Theorem 3.6 and Theorem 3.7 taking into account that, by (4.1) n_{1i} = ℓ − a_{i} − a_{t} for 1 ≤ i ≤ t and n_{2j} = ℓ + a_{j} − a_{t} for 1 ≤ j ≤ t − 1. □

We are led to pose the following question.

Question 4.9. Under which extra hypothesis the bounds given in Theorem 4.8 are sharp?

We will now give an example.

Example 4.10. Set R = k[x_{0}, x_{1}, ⋯, x_{5}]. Let X ⊂ ℙ^{5} = Proj(R) be the Veronese surface defined by the 2 × 2 minors of the symmetric matrix

A = (\begin{matrix}x_{0} & x_{1} & x_{2} \\ x_{1} & x_{3} & x_{4} \\ x_{2} & x_{4} & x_{5} \end{matrix}) .

Let I_{A} be the ideal generated by the 2 × 2 minors of A and I_{B} the ideal generated by the

2 × 2 minors of B = (\begin{matrix}x_{0} & x_{1} & x_{2} \\ x_{1} & x_{3} & x_{4} \end{matrix}) . Set A = R/I_{A} and B = R/I_{B}. It is well known that

dim_{(X)} Hilb(ℙ^{5}) = 27. If we apply Theorem 4.8 we obtain dim_{(X)} Hilb(ℙ^{5}) ≥ 29 − e where e = _{0}ext_{B}^{2}(I_{A/B}, I_{A/B}). Using Macaulay program [3] we have computed the dimension of 0Ext_{B}^{2}(I_{A/B}, I_{A/B}) and we have got e = 2. Thus, dim_{(X)} Hilb(ℙ^{5}) ≥ 27 and hence the first bound given in Theorem 4.8 is sharp.

Example 4.11. Set R = k[x_{0}, x_{1}, ⋯, x_{5}]. Let X ⊂ ℙ^{5} = Proj(R) be the surface defined by the 3 × 3 minors of the symmetric matrix

A = (\begin{matrix}x_{0} & x_{1} & x_{2} & L_{1} \\ x_{1} & x_{3} & x_{4} & L_{2} \\ x_{2} & x_{4} & x_{5} & L_{3} \\ L_{1} & L_{2} & L_{3} & L_{4} \end{matrix})

where L_{i} are general linear forms. Let I_{A} be the ideal generated by the 3 × 3 minors of A

and I_{B} the ideal generated by the 3 × 3 minors of B = (\begin{matrix}x_{0} & x_{1} & x_{2} & L_{1} \\ x_{1} & x_{3} & x_{4} & L_{2} \\ x_{2} & x_{4} & x_{5} & L_{3} \end{matrix}) . Set A = R/I_{A}

and B = R/I_{B}. If we apply Theorem 4.8 we obtain dim_{(X)} Hilb(ℙ^{5}) ≥ 59 − e where e = _{0}ext_{B}^{2}(I_{A/B}, I_{A/B}). Using the Macaulay program [3] we have computed the dimension of _{0}Ext_{B}^{2}(I_{A/B}, I_{A/B}) and we have got e = 14. Thus, dim_{(X)} Hilb(ℙ^{5}) ≥ 45. Using again the Macaulay program we have computed h^{0}(N_{X}) and we have got h^{0}(N_{X}) = 45. Hence the first bound given in Theorem 4.8 is sharp.

Example 4.12. Let R = k[x_{0}, x_{1}, ⋯, x_{5}] and let I_{A} be the ideal generated by the 2 × 2 minors of the symmetric matrix

A = (\begin{matrix}x_{0} & x_{1} & x_{2}^{2} \\ x_{1} & x_{3} & x_{4}^{2} \\ x_{2}^{2} & x_{4}^{2} & x_{5}^{3} \end{matrix}) .

Let I_{B} the ideal generated by the 2×2 minors of B = (\begin{matrix}x_{0} & x_{1} & x_{2}^{2} \\ x_{1} & x_{3} & x_{4}^{2} \end{matrix}). Set X = Proj(R/I_{A})

and I_{A/B} = I_{A}/I_{B}. If we apply Theorem 4.8 with a_{1} = a_{2} = 1/2 and a_{3} = 3/2 we obtain dim_{(X)} Hilb(ℙ^{5}) ≥ 92 − e where e = _{0}ext_{B}^{2}(I_{A/B}, I_{A/B}). Indeed b − 1 + h^{0}(Ñ_{B}) = 90, a = −2 and we find dim_{(X)} Hilb(ℙ^{5}) ≥ b − 1 + h^{0}(Ñ_{B}) − a − e = 92 − e = 84 since we got e = 8 by Macaulay 2. In this case, however, _{0}hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}) = 0 by Lemma 4.5 and the second bound of Theorem 4.6 applies. We get dim_{(X)} Hilb(ℙ^{5}) ≥ b − 1 + h^{0}(Ñ_{B}) = 90 which is sharp or 1 from being sharp because we have computed h^{0}(N_{X}) by Macaulay 2 and got 91.

Remark 4.13. We want to point out that b − 1 + h^{0}(N_{Y}) − a − e is a correct formula for h^{0}(N_{X}) − h^{1}(N_{X}) if X is a l.c.i. and dim(X) ≥ 2. Moreover, examples computed using Macaulay 2 show that the dimension of the corresponding component might be significantly different from our lower bounds if h^{1}(N_{X}) is large. If dim(X) = 1 then b − 1 + h^{0}(N_{Y}) − a − e is a correct formula for h^{0}(N_{X}) − _{0}h^{2}(R, A, A).

Let CM_{c}^{H}(R) (resp. Gor_{c}^{H}(R)) be the open subscheme of GradAlg^{H}(R) parameterizing Cohen-Macaulay (resp. Gorenstein) quotients of codimension c in R. If c ≤ dim R−2, one may alternatively define CM_{c}^{H}(R) (resp. Gor_{c}^{H}(R)) as the open subscheme of Hilb^{p}(ℙ^{n}) parameterizing arithmetically Cohen-Macaulay (resp. arithmetically Gorenstein) subschemes of codimension c in ℙ^{n}. In [21], the first author established a strong connection (an incidence correspondence) between Gor_{c+1}^{H′}(R) and CM_{c}^{H}(R) for some H′, in which Gorenstein quotients (G) ∈ Gor_{c+1}^{H′}(R) were defined by some exact sequence

(4.4)  0 ⟶ K_{A}(−s) ⟶ A ⟶ G ⟶ 0

with (A) ∈ CM_{c}^{H}(R) and s a fixed integer. In particular, he proved that if s is large enough (see [21] for explicit bounds) then this connection essentially determines a well defined injective map from the set of irreducible components of CM_{c}^{H}(R) to the set of irreducible components of Gor_{c+1}^{H′}(R). Note that if Proj(G) is obtained as the intersection of two geometrically G-linked schemes, then G is given by (4.4) for some s. In the next example, we will prove the existence of at least two irreducible components C_{1} and C_{2} of CM_{3}^{H}(R) mapping to the same irreducible component of Gor_{4}^{H′}(R) in the correspondence above, showing that we can not skip the assumption that s is sufficiently large. Moreover, we will see that the generic graded Cohen-Macaulay quotients A_{1} ∈ C_{1} and A_{2} ∈ C_{2} have both a linear resolution with the same graded Betti numbers.

Example 4.14. We keep the notation introduced in Example 4.10. B = R/I_{B} is a codimension 2 Cohen-Macaulay graded quotient and the minimal R-free resolution of B is

0 ⟶ R(−3)^{2} ⟶ R(−2)^{3} ⟶ R ⟶ B ⟶ 0.

So Y = Proj(B) ⊂ ℙ^{5} is a codimension 2, arithmetically Cohen-Macaulay scheme. A = R/I_{A} is the codimension 3 Veronese surface and the minimal free R-resolution of A is

0 ⟶ R(−4)^{3} ⟶ R(−3)^{8} ⟶ R(−2)^{6} ⟶ R ⟶ A ⟶ 0.

So, X = Proj(A) ⊂ ℙ^{5} is a codimension 3, arithmetically Cohen-Macaulay scheme.

Any effective divisor of the linear system | − K_{Y} | is an arithmetically Gorenstein, codimension 3 subscheme X_{0} ⊂ ℙ^{5} and its homogeneous coordinate ring A_{0} has a minimal R-free resolution (X_{0} = Proj(A_{0}), A_{0} = R/I_{A_{0}}):

0 ⟶ R(−6) ⟶ R(−4)^{3} ⊕ R(−3)^{2} ⟶ R(−3)^{2} ⊕ R(−2)^{3} ⟶ R ⟶ A_{0} ⟶ 0.

We G-link A to A′ using some codimension 3, graded Gorenstein quotient A_{0}. A′ is a codimension 3, graded Cohen-Macaulay quotient with a R-free resolution:

0 ⟶ R(−4)^{3} ⟶ R(−3)^{10} ⟶ R(−2)^{6} ⊕ R(−3)^{2} ⟶ R ⟶ A′ ⟶ 0.

Let us check that X′ = Proj(A′) and X = Proj(A) belong to different components of Hilb^{p(t)}(ℙ^{5}).

It is well known that h^{0}N_{X} = 27 and H^{i}N_{X} = 0 for i > 0. Therefore, X belongs to a generically smooth irreducible component C_{1} of Hilb^{p(t)} ℙ^{5} of dimension 27. On the other hand, X′ ∈ |K_{Y} +4H| and a general element X_{2} = Proj(A_{2}) of |K_{Y} +4H| is a codimension 3 determinantal scheme defined by the maximal minors of a 2 × 4 matrix obtained by adding to B a column of linear forms. Hence A_{2} has a linear resolution. Moreover, by [19]; §10, X_{2} belongs to a generically smooth irreducible component of Hilb^{p(t)} ℙ^{5} of dimension 29.

So, we conclude that the Veronese surface X ⊂ ℙ^{5} and a determinantal surface defined by the maximal minors of a 2 × 4 matrix with linear entries have a linear resolution with the same graded Betti numbers but they belong to different components of Hilb^{p(t)} ℙ^{5}.

22 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

Finally, since X ∈ | − 2K_{Y} − 4H| and X′ ∈ |K_{Y} + 4H| are G-linked by X_{0} ∈ | − K_{Y} | we have that Z = X ∩ X′ = Proj(R/I_{A} + I_{A′}) ⊂ ℙ^{5} is an arithmetically Gorenstein scheme of codimension 4 and I_{C} := I_{A} + I_{A′} has a minimal free R-resolution

0 ⟶ R(−6) ⟶ R(−4)^{9} ⟶ R(−3)^{16} ⟶ R(−2)^{9} ⟶ I_{C} ⟶ 0.

Thus, we conclude that there are two generically smooth irreducible components of CM_{3}^{H_{A}}(R) mapping to the same irreducible component of Gor_{4}^{H_{C}}(R) in the correspondence described in (4.4).

5. Final remarks and open problems

The results proved in this paper give rise to a number of quite interesting questions and possible generalizations that we gather together in this last section. In fact, Examples 4.10 and 4.11 suggest - and prove for t ≤ 4 and n = 5 - the following question:

Question 5.1. Let A = (f_{ji})_{i,j=1,...,t} be a t × t symmetric matrix where f_{ji} ∈ k[x_{0}, ..., x_{5}] are general linear forms. Let X ⊂ ℙ^{5} be a codimension 3, ACM scheme defined by the submaximal minors of A. Is it true that

dim_{(X)} Hilb^{p(t)} ℙ^{5} = 6t[(t − 1)^{2} + (t − 1)] − t^{2} − (t − 1)(t^{2} − 1) − t\binom{t}{2}

−\binom{7}{2}\binom{t}{3} − 6\binom{t + 1}{3} − e ?

More generally, we would like to know if under certain numerical conditions on a_{i} the bounds given in Theorem 4.8 are sharp. So, we are led to pose the following problem:

Problem 5.2. Find numerical conditions on the rational numbers a_{i} which allow us to assure that the bounds given in Theorem 4.8 are sharp.

We consider a t × t homogeneous symmetric matrix A = (f_{ji})_{i,j=1,...t} where f_{ji} ∈ k[x_{0}, ..., x_{n}] are homogeneous polynomials of degree a_{i} + a_{j} ∈ ℤ_{+} and let X ⊂ ℙ^{n} be the codimension 3, arithmetically Cohen-Macaulay scheme defined by the submaximal minors of A. Let B be the matrix obtained by deleting the last row of A and let Y ⊂ ℙ^{n} be the codimension 2, arithmetically Cohen-Macaulay scheme defined by the maximal minors of B.

If dim(X) ≥ 2 then Theorem 4.8 only gives a lower bound for dim_{(X)} Hilb^{p(t)} ℙ^{n} ≥ h^{0}(N_{X}) − h^{1}(N_{X}) and, using Macaulay 2, we have checked that h^{1}(N_{X}) may be large in many examples. So, we are lead to pose the following problems

Problem 5.3. Find an explicit formula for dim_{(X)} Hilb^{p(t)} ℙ^{n} and for h^{0}(N_{X}).

Keeping the notation introduced in §4, we denote by S(a_{1}, ⋯, a_{t}) the irreducible family of codimension 3, ACM schemes X ⊂ ℙ^{n} defined by the submaximal minors of a t × t homogeneous symmetric matrix A = (f_{ji})_{i,j=1,...t} where f_{ji} ∈ k[x_{0}, ..., x_{n}] are homogeneous polynomials of degree a_{i} + a_{j} ∈ ℤ_{+} and let S(B) be the part of Hilb ℙ^{n} consisting of those Y′ obtained by deforming B as a matrix with symmetric ”left” (t − 1) × (t − 1) matrix.

Problem 5.4. Find an explicit formula for dim S(a_{1}, ⋯, a_{t}) and dim S(B).

Problem 5.5. Is dim_{(X)} Hilb^{p(t)} ℙ^{n} = dim_{(X)} S(a_{1}, ⋯, a_{t})?

The computations we have made, using Macaulay 2 [10], suggest the following conjecture

Conjecture 5.6. If t = 3 and 2a_{i} = p for all i, then

dim_{(X)} Hilb^{p(t)} ℙ^{n} = dim_{(X)} S(a_{1}, ⋯ a_{t}) = h^{0}(N_{X}) = 6\binom{p + n}{n} − 9.

We have checked that the right hand equality of the conjecture is true for n = 5 and 1 ≤ p ≤ 20, for some choices of X.

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24 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

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Oslo University College,Faculty of Engineering, Pb. 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