FAMILIES OF DETERMINANTAL SCHEMES

JAN O. KLEPPE, ROSA M. MIRÓ-ROIG^{∗}

Abstract. Given integers a_{0} ≤ a_{1} ≤ ⋯ ≤ a_{t+c−2} and b_{1} ≤ ⋯ ≤ b_{t}, we denote by W (\underline{b}; \underline{a}) ⊂ Hilb^{p}(ℙ^{n}) the locus of good determinantal schemes X ⊂ ℙ^{n} of codimension c defined by the maximal minors of a t × (t + c − 1) homogeneous matrix with entries homogeneous polynomials of degree a_{j} − b_{i}. The goal of this paper is to extend and complete the results given by the authors in [12] and determine under weakened numerical assumptions the dimension of W (\underline{b}; \underline{a}) as well as whether the closure of W (\underline{b}; \underline{a}) is a generically smooth irreducible component of Hilb^{p}(ℙ^{n}).

1. Introduction

In this paper, we will deal with good and standard determinantal schemes. A scheme X ⊂ ℙ^{n} of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous t×(t+c−1) matrix, and X is said to be good determinantal if it is standard determinantal and a generic complete intersection. We denote the Hilbert scheme by Hilb^{p}(ℙ^{n}). Given integers a_{0} ≤ a_{1} ≤ ⋯ ≤ a_{t+c−2} and b_{1} ≤ ⋯ ≤ b_{t}, we denote by W (\underline{b}; \underline{a}) ⊂ Hilb^{p}(ℙ^{n}) (resp. W_{s}(\underline{b}; \underline{a})) the locus of good (resp. standard) determinantal schemes X ⊂ ℙ^{n} of codimension c defined by the maximal minors of a t × (t + c − 1) homogeneous matrix with entries homogeneous polynomials of degree a_{j} − b_{i}.

In [11] and [12], we addressed the following 3 crucial problems:

(1) To determine the dimension of W (\underline{b}; \underline{a}) in terms of a_{j} and b_{i},

(2) Is the closure of W (\underline{b}; \underline{a}) an irreducible component of Hilb^{p}(ℙ^{n})? and

(3) Is Hilb^{p}(ℙ^{n}) generically smooth along W (\underline{b}; \underline{a})?

In [12] we obtained an upper bound for dim W (\underline{b}; \underline{a}) in terms of a_{j} and b_{i} which was achieved in the cases 2 ≤ c ≤ 5 and n − c > 0 (assuming char(k) = 0 if c = 5), and in codimension c > 5 provided certain numerical conditions are satisfied (See [12], Theorems 3.5 and 4.5; and Corollaries 4.7, 4.10 and 4.14). Concerning problems (2) and (3), we gave in [12] an affirmative answer to both questions in the range 2 ≤ c ≤ 4 and n − c ≥ 2, and in the cases c ≥ 5 and n − c ≥ 1 provided certain numerical assumptions are verified (See [12], Corollaries 5.3, 5.7, 5.9 and 5.10. See also [7], [11] for the cases 2 ≤ c ≤ 3). Note that since every element of W (\underline{b}; \underline{a}) has the same Hilbert function, the assumption n > c is close to being necessary for problem (2). Indeed if n = c the problems (2) and (3) become more natural provided we replace Hilb^{p}(ℙ^{n}) by the postulation Hilbert scheme, see [10].

Date: January 24, 2011.

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

∗ Partially supported by MTM2010-15256 .

2 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

In this work we attempt to extend and complete the results of [11] and [12]. Indeed if a_{t+3} > a_{t−2} we almost solve problem (1) in Theorem 3.2 while Theorem 3.4 and Corollary 3.8, for c > 4, generalize results of [12] for the problems (2) and (3) substantially. To prove these results we use induction on the codimension by successively deleting the columns of the highest degree and the Eagon-Northcott complex associated to a standard determinantal scheme. We also use the theory of Hilbert flag schemes and the depth of certain mixed determinantal schemes (see Theorem 2.7). We end the paper with two conjectures which are supported by our results and by a huge number of examples computed using Macaulay 2.

Notation: Throughout this paper ℙ^{n} is 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}). By Hom_{O_{X}}(F, G) we denote the sheaf of local morphisms between coherent O_{X}-modules while Hom(F, G) denotes the group of morphisms from F to G. Moreover we set hom(F, G) = dim_{k} Hom(F, G) and we correspondingly use small letters for the dimension, as a k-vector space, of similar groups. For any quotient A of R of codimension c, we let K_{A} = Ext_{R}^{c}(A, R)(−n − 1).

In the sequel, _{μ}Hom_{R}(M, N) denotes homomorphisms of degree μ of graded R-modules. Moreover, we denote the Hilbert scheme by Hilb^{p}(ℙ^{n}), p the Hilbert polynomial, and (X) ∈ Hilb^{p}(ℙ^{n}) the point which corresponds to the subscheme X ⊂ ℙ^{n} with Hilbert polynomial p. We denote by I_{X} the saturated homogeneous ideal of X ⊂ ℙ^{n}. We say that X is general in some irreducible subset W ⊂ Hilb^{p}(ℙ^{n}) if (X) belongs to a sufficiently small open subset U of W (small enough so that any (X) ∈ U has all the openness properties that we want to require).

2. Preliminaries

This section provides the background and basic results on standard determinantal ideals, good determinantal ideals and mixed determinantal ideals needed in the sequel. We refer to [3], [6] [12] and [15] for the details.

Let A = (f_{ij})_{i=1,...t}^{j=0,...,t+c−2}, deg f_{ij} = a_{j} − b_{i}, be a t × (t + c − 1) homogeneous matrix and let

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

be the graded morphism of free R-modules represented by the transpose, A^{t}, of A. Let I(A) = I_{t}(A) be the ideal of R generated by the maximal minors of A. A codimension c subscheme X ⊂ ℙ^{n} is said to be standard determinantal if I_{X} = I(A) for some homogeneous t × (t + c − 1) matrix A as above. Moreover X is good determinantal if X is standard determinantal and a generic complete intersection in ℙ^{n} ([13], Theorem 3.4). In this paper we suppose c ≥ 2, t ≥ 2, b_{1} ≤ ... ≤ b_{t} and a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2}. (Note that the case t = 1 for determinantal schemes corresponds to the well-known complete intersections).

Let W (\underline{b}; \underline{a}) (resp. W_{s}(\underline{b}; \underline{a})) be the stratum in Hilb^{p}(ℙ^{n}) consisting of good (resp. standard) determinantal schemes as above. Since our definition does not assume A to be minimal (i.e. f_{ij} = 0 when b_{i} = a_{j}) for X = Proj(R/I_{t}(A)) ∈ W (\underline{b}; \underline{a}) (or W_{s}(\underline{b}; \underline{a})), we

FAMILIES OF DETERMINANTAL SCHEMES 3

must reconsider Corollary 2.6 of [12] where A was supposed minimal in the proof (a slight correction to [11] and [12]!). We may, however, use that proof to see that

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

Indeed if we assume the converse of the condition on the right hand side, then either I_{t}(A) ∋ 1 or one of the maximal minors vanishes, i.e. Proj(R/I_{t}(A)) ∉ W_{s}(\underline{b}; \underline{a}). Conversely assuming the right hand side condition (to simplify notations, assume a_{i−1} ≥ b_{i} for 1 ≤ i ≤ s and a_{i−1} > b_{i} for s < i ≤ t for some integer s < t), then we may take A = ( \begin{matrix}I & O \\ O & A′ \end{matrix}) where I is the s × s identity matrix, O are matrices of zero’s and A′, for t − s > 1, the (t − s) × (t − s + c − 1) matrix used in [12], Corollary 2.6 to define a good determinantal scheme (if t − s = 1 we take the entries of A′ to be a regular sequence). We get Proj(R/I_{t}(A)) ∈ W (\underline{b}; \underline{a}) and we easily deduce (2.2). Note that by [12], end of p. 2877 and [12], Remark 3.7 we still have that the closures of W (\underline{b}; \underline{a}) and W_{s}(\underline{b}; \underline{a}) in Hilb^{p}(ℙ^{n}) are equal and irreducible.

Let A = R/I_{X} be the homogeneous coordinate ring of a standard determinantal scheme. By [3], Theorem 2.20 and [6], Corollaries A2.12 and A2.13 the Eagon-Northcott complex yields a minimal free resolution of A

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

which allows us to deduce that any standard determinantal scheme is arithmetically Cohen-Macaulay (ACM). Moreover if M_{A} := coker(φ^{∗}) then K_{A}(n + 1) ≅ S_{c−1}M_{A}(ℓ_{c}) where

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

Let B be the matrix obtained by deleting the last column of A, let B = R/I_{B} be the kalgebra given by the maximal minors of B and let M_{B} be the cokernel of ϕ^{∗} = Hom_{R}(ϕ, R) where ϕ : F = ⊕_{i=1}^{t}R(b_{i}) → G′ := ⊕_{j=0}^{t+c−3}R(a_{j}) is the graded morphism induced by B^{t}. Recall that if c > 2 there is an exact sequence

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

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

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

is exact by [13] or [11] (e.g. see the text after (3.1) of [11]). Note that the proofs of (2.5)- (2.6) rely heavily on the equality Ann(M_{B}) = I_{B} established in [4]. If c = 2 (codim_{R} B = 1) we have at least Ann(M_{B})I_{t−1}(B) ⊂ I_{B} ⊂ Ann(M_{B}) by [4]; thus the kernel, I_{A/B}, of B → A satisfies I_{A/B} = M_{B}(a_{t+c−2})^{∗} (resp. Ĩ_{A/B}|_{U} = M̃_{B}(a_{t+c−2})^{∗}|_{U} where U = Y − V (I_{t−1}(B))) for c > 2 (resp. c = 2). Due to the Buchsbaum-Rim resolution of M := M_{A}, M is a maximal Cohen-Macaulay A-module, and so is I_{A/B} for c > 2 by (2.6).

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

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

4 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

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

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

such that I_{i}(a_{t+i−1}) is the O_{X_{i}}-dual of M_{i} for 2 ≤ i ≤ c (all this holds also for i = 1 provided we restrict the sheaves to X_{1}−V (I_{t−1}(φ_{1})) where φ_{1} is given by X_{1} = V (I_{t}(φ_{1}))). In this context, we let D_{i} := R/I_{X_{i}}, I_{D_{i}} := I_{X_{i}} and I_{i} := I_{D_{i+1}}/I_{D_{i}}.

Remark 2.1. Assume t ≥ 2, b_{1} ≤ ... ≤ b_{t}, a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and let α ≥ 1 be an integer. If X is general in W (\underline{b}; \underline{a}) and a_{i−min(α,t)} − b_{i} ≥ 0 for min(α, t) ≤ i ≤ t, then

(2.8)  codim_{X_{j}} Sing(X_{j}) ≥ min{2α − 1, j + 2} for all j = 2, ⋯, c.

This follows from the theorem of [5], arguing as in [5], Example 2.1. In particular, if α ≥ 3, we get for each i > 0 that X_{i} ↪ ℙ^{n} and X_{i+1} ↪ X_{i} are local complete intersections (l.c.i.’s) outside some set Z_{i} of codimension at least min(4, i+1) in X_{i+1}, cf. the paragraph below. Finally note that it is easy to show (2.8) for (j, α) = (1, 2) provided a_{i−2} > b_{i} for 2 ≤ i ≤ t, cf. [1], (1.10).

Now let Z ⊂ X (resp. Z_{i} ⊂ X_{i}) be some closed subset such that U := X − Z ↪ ℙ^{n} (resp. U_{i} := X_{i} − Z_{i} ↪ ℙ^{n}) is an l.c.i. By the fact that the 1^{st} Fitting ideal of M is equal to I_{t−1}(φ), we get that M̃ is locally free of rank 1 precisely on X − V (I_{t−1}(φ)) [2], Lemma 1.4.8. Since the set of non-l.c.i points of X ↪ ℙ^{n} is precisely V (I_{t−1}(φ)) by e.g. [17], Lemma 1.8, we get that U ⊂ X − V (I_{t−1}(φ)) and that M̃ is locally free on U. Indeed M_{i} and I_{X_{i}}/I_{X_{i}}^{2} are locally free on U_{i}, as well as on U_{i−1} ∩ X_{i}. Note also that since V (I_{t−1}(B)) ⊂ V (I_{t}(A)), we may suppose Z_{i−1} ⊂ X_{i}.

Let us recall the following useful comparison of cohomology groups. If L and N are finitely generated A-modules such that depth_{I(Z)} L ≥ r + 1 and Ñ is locally free on U := X − Z, then the natural map

(2.9)  Ext_{A}^{i}(N, L) ⟶ H_{∗}^{i}(U, Hom_{O_{X}}(Ñ, L̃))

is an isomorphism, (resp. an injection) for i < r (resp. i = r) cf. [9], exp. VI. Note that we interpret I(Z) as m if Z = ∅.

In [12], Conjecture 6.1, we conjectured the dimension of (a non-empty) W (\underline{b}; \underline{a}) in terms of the invariant

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

Here the indices belonging to a_{j} (resp. b_{i}) range over 0 ≤ j ≤ t + c − 2 (resp. 1 ≤ i ≤ t) and \binom{a}{b} = 0 whenever a < b.

Conjecture 2.2. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we set ℓ_{i} := ∑_{j=0}^{t+i−2} a_{j} − ∑_{k=1}^{t} b_{k} and h_{i−3} := 2a_{t+i−2} − ℓ_{i} + n, for i = 3, 4, ..., c. Assume a_{i−min([c/2]+1,t)} ≥ b_{i} for min([c/2] + 1, t) ≤ i ≤ t. Then we have

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

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

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

In [12] we proved that the right hand side in the formula for dim W (\underline{b}; \underline{a}) in Conjecture 2.2 is always an upper bound for dim W (\underline{b}; \underline{a}) ([12], Theorem 3.5) and, moreover, that Conjecture 2.2 holds in the range

(2.11)  2 ≤ c ≤ 5 and n − c > 0 ( supposing char(k) = 0 if c = 5 ) ,

cf. [12], Theorems 3.5 and 4.5; and Corollaries 4.7, 4.10 and 4.14. See also [7], [11] for the cases 2 ≤ c ≤ 3. In [10], however, the first author gave a counterexample to Conjecture 2.2 for zero-dimensional schemes (see the section of conjectures in this paper).

Remark 2.3. Conjecture 2.2 holds in codimension c ≥ 6 provided the numerical conditions of [12], Corollary 4.15 are satisfied. By [12], Remarks 4.16, 4.17 and Corollary 4.18 Conjecture 2.2 holds (without assuming n > c or char(k) = 0) for any W (\underline{b}; \underline{a}) satisfying

a_{t+3} > a_{t−1} + a_{t} + a_{t+1} − a_{0} − a_{1} in codimension c = 5 ,
a_{t+2} > a_{t−1} + a_{t} − a_{0} for n = c = 4 ,
a_{t+1} > a_{t−1} and a_{i−2} > b_{i} for 2 ≤ i ≤ t for n = c = 3 .

Note that in the case n = c = 3 we need a_{i−2} > b_{i} (not a_{i−2} ≥ b_{i} as assumed in [12]) for 2 ≤ i ≤ t for the proof of [12], Corollary 4.18 to hold, see the last sentence of Remark 2.1.

A goal of this paper is to extend the results summarized in (2.11) and in the above remark. To do this we will need the following result where \underline{a} = a_{0}, a_{1}, ..., a_{t+c−2} and \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3}.

Proposition 2.4. Let c ≥ 3, let (X) ∈ W (\underline{b}; \underline{a}) and suppose dim W (\underline{b}; \underline{a′}) ≥ λ_{c−1} + K_{3} + K_{4} + ... + K_{c−1} and depth_{I(Z)} B ≥ 2 for a general Y = Proj(B) ∈ W (\underline{b}; \underline{a′}). If

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

then dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ... + K_{c} and (2.12) turns out to be an equality.

Proof. See [10], Proposition 3.4. □

Let us recall how we proved Remark 2.3 since we want to generalize that approach. Letting a = a_{t+i−2} − a_{t+i−1} we showed in [12] that Hom_{D_{i−1}}(I_{i−1}, I_{i}) ≅ D_{i}(a) for i < c using depth_{I(Z_{i−1})} D_{i} ≥ 2. Indeed this follows from (2.9), i.e. from

H_{∗}^{0}(U_{i−1}, Hom(I_{i−1}, I_{i})) ≅ H_{∗}^{0}(U_{i−1}, Hom_{O_{X_{i}}}(I_{i−1} ⊗_{O_{X_{i−1}}} O_{X_{i}} ⊗ I_{i}^{∗}, O_{X_{i}}))

because M̃_{i}|_{U_{i−1}} ≅ I_{i}^{∗}(−a_{t+i−1})|_{U_{i−1}} is locally free, M̃_{i}|_{U_{i−1}} ≅ M̃_{i−1} ⊗ O_{X_{i}}|_{U_{i−1}} and hence I_{i−1} ⊗_{O_{X_{i−1}}} O_{X_{i}}|_{U_{i−1}} ≅ I_{i}(−a)|_{U_{i−1}}, and note that U_{i−1} ∩ X_{i} ⊂ U_{i}. Then since

(2.13)  0 → Hom_{R}(I_{i−1}, I_{i}) → Hom_{R}(I_{D_{i}}, I_{i}) → Hom_{R}(I_{D_{i−1}}, I_{i})

6 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

is exact we were able to show (2.12) for X = X_{c} ⊂ Y = X_{c−1} by putting _{0}Hom_{R}(I_{D_{c−2}}, I_{c−1}) = 0 (through making the minimal generators of I_{D_{c−2}} and I_{c−1} explicit via (2.3)). Using Proposition 2.4 we obtained Remark 2.3 if the conjecture holds for W (\underline{b}; \underline{a′}) ∋ Y := X_{c−1}.

Let us finish this section by gathering all the results on the depth of mixed determinantal ideals and cogenerated ideals needed in the next section. We start by fixing some notation. Let

(2.14)  A = ( \begin{matrix}f_{1}^{1} & f_{2}^{1} & ⋯ & f_{q}^{1} \\ f_{1}^{2} & f_{2}^{2} & ⋯ & f_{q}^{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ f_{1}^{p} & f_{2}^{p} & ⋯ & f_{q}^{p} \end{matrix} )

be a homogeneous p × q matrix with entries homogeneous polynomials f_{i}^{j} ∈ k[x_{0}, ⋯, x_{n}] of degree a_{j} −b_{i}. For any choice (\underline{α}; \underline{β}) = (α_{1}, ⋯, α_{m}; β_{1}, ⋯, β_{m}) of row indexes 1 ≤ α_{1} < ⋯ < α_{m} ≤ p and of column indexes 1 ≤ β_{1} < ⋯ < β_{m} ≤ q, we denote by I_{(\underline{α};\underline{β})}(A) the ideal cogenerated by (\underline{α}; \underline{β}), i.e. I_{(\underline{α},\underline{β})} is the homogeneous ideal of k[x_{0}, ⋯, x_{n}] generated by all (m + 1) × (m + 1) minors of A, all i × i minors of the rows 1, ⋯, α_{i} − 1 for i = 1, ⋯, m and all i × i minors of the columns 1, ⋯, β_{i} − 1 for i = 1, ⋯, m.

Example 2.5. (1) If (\underline{α}; \underline{β}) = (1, ⋯, t − 1; 1, ⋯, t − 1), then I_{(\underline{α};\underline{β})}(A) is the ideal generated by the t × t minors of A, i.e., I_{(\underline{α};\underline{β})}(A) = I_{t}(A).

(2) Let A be a homogeneous p×q matrix and let A_{i} be the matrix obtained by deleting the last i − 1 columns of A. We assume p ≤ q and we fix an integer m < p and (\underline{α}; \underline{β}) = (1, ⋯, m; β_{1}, ⋯, β_{m}). Set j := min\{i | β_{i} > i\}, c_{1} = 1 and c_{s} = q + 2 − β_{m+2−s} for 2 ≤ s ≤ m + 2 − j. The ideal cogenerated by (\underline{α}; \underline{β}) = (1, ⋯, m; β_{1}, ⋯, β_{m}) can be identified with the following mixed determinantal ideal

I_{(\underline{α};\underline{β})}(A) = I_{m+1}(A_{c_{1}}) + I_{m}(A_{c_{2}}) + ... + I_{j}(A_{c_{m+2−j}}),

where I_{λ}(A_{ϱ}) denotes the ideal generated by all λ × λ minors of A_{ϱ}.

Theorem 2.6. Let A = (x_{i,j}) be a p × q matrix of indeterminates and let (\underline{α}; \underline{β}) = (α_{1}, ⋯, α_{m}; β_{1}, ⋯, β_{m}) be a choice of m row indexes and of m column indexes. Let I_{(\underline{α};\underline{β})} be the ideal cogenerated by (\underline{α}; \underline{β}). Then, I_{(\underline{α};\underline{β})}(A) is a Cohen-Macaulay ideal and

(2.15)  ht(I_{(\underline{α};\underline{β})}(A)) = pq − (p + q + 1)m + ∑_{i=1}^{m}(α_{i} + β_{i}).

Proof. See [3], Corollary 5.12. □

We are now ready to state the main result of this section and to prove that the above formula for the height of a cogenerated ideal associated to a matrix with entries that are indeterminates also works for a general homogeneous matrix with entries thar are homogeneous polynomials of positive degree. Indeed, we have

Theorem 2.7. Fix integers b_{1}, ⋯, b_{q} and a_{1}, ⋯, a_{p}. Let A = (f_{i}^{j})_{i=1,··· ,q}^{j=1,··· ,p} be a general homogeneous p×q matrix A with entries that are homogeneous forms of degree a_{j} −b_{i}, and assume that a_{j} > b_{i} for all j, i. Let (\underline{α}; \underline{β}) = (α_{1}, ⋯, α_{m}; β_{1}, ⋯, β_{m}) be any choice of m row indexes and of m column indexes, and suppose n+1 ≥ pq−(p+q+1)m+∑_{i=1}^{m}(α_{i}+β_{i}). Then I_{(\underline{α};\underline{β})}(A) is a Cohen-Macaulay ideal and

ht(I_{(\underline{α};\underline{β})}(A)) = pq − (p + q + 1)m + ∑_{i=1}^{m}(α_{i} + β_{i}).

Proof. We clearly have ht(I_{(\underline{α};\underline{β})}(A)) ≤ pq − (p + q + 1)m + ∑_{i=1}^{m}(α_{i} + β_{i}) and it will be enough to construct an example of homogeneous p×q matrix M with entries homogeneous forms f_{i}^{j} ∈ k[x_{0}, ⋯, x_{n}] such that the ideal I_{(\underline{α};\underline{β})}(M) cogenerated by (\underline{α}; \underline{β}) is Cohen-Macaulay and ht(I_{(\underline{α};\underline{β})}(M)) = pq − (p + q + 1)m + ∑_{i=1}^{m}(α_{i} + β_{i}). Therefore, let n + 1 ≥ pq − (p + q + 1)m + ∑_{i=1}^{m}(α_{i} + β_{i}). We will distinguish 2 cases:

\underline{Case 1.} Assume a_{j} − b_{i} = 1 for all i, j (i.e. the entries of the matrix A are linear forms). By Theorem 2.6 for any choice of p, q and (\underline{α}; \underline{β}), we have the matrix \overline{A} = (x_{i,j}) of indeterminates and the ideal I_{(\underline{α};\underline{β})}(\overline{A}) ⊂ S := k[x_{1,1} ⋯, x_{p,q}] cogenerated by (\underline{α}; \underline{β}) is Cohen-Macaulay of height

ht(I_{(\underline{α};\underline{β})}(\overline{A})) = pq − (p + q + 1)m + ∑_{i=1}^{m}(α_{i} + β_{i}).

We choose pq −n−1 general linear forms ℓ_{1}, ⋯, ℓ_{pq−n−1} ∈ S = k[x_{1,1} ⋯, x_{p,q}] and we set S/(ℓ_{1}, ⋯, ℓ_{pq−n−1}) ≅ k[x_{0}, x_{1}, ⋯, x_{n}] =: R. Let us call I ⊂ R the ideal of R isomorphic to the ideal I_{(\underline{α};\underline{β})}(\overline{A})/(ℓ_{1}, ⋯, ℓ_{pq−n−1}) of S/(ℓ_{1}, ⋯, ℓ_{pq−n−1}). Note that I is a Cohen-Macaulay ideal of height ht(I) = pq − (p + q + 1)m + ∑_{i=1}^{m}(α_{i} + β_{i}) and, in addition, I is nothing but the ideal I_{(\underline{α};\underline{β})}(M) where M = (m_{i}^{j}) is a p × q homogeneous matrix with entries linear forms in k[x_{0}, x_{1}, ⋯, x_{n}] obtained from \overline{A} = (x_{i,j}) by substituting using the equations ℓ_{1}, ⋯, ℓ_{pq−n−1}, which proves what we want.

\underline{Case 2.} Assume a_{j} − b_{i} ≥ 1 for all i, j. In this case, it is enough to raise the entry m_{i}^{j} of the above matrix M to the power a_{j} − b_{i}. □

Remark 2.8. Fix integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}. Let X ⊂ ℙ^{n}, (X) ∈ W (\underline{b}; \underline{a}) be a general determinantal ideal associated to a t × (t + c − 1) matrix A represented by a graded morphism as in (2.1). Let

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

be the flag of standard determinantal subschemes that we obtain by successively deleting columns from the right hand side of A and let φ_{i} be the graded morphism associated to the matrix which defines X_{i}. Assume a_{0} > b_{t} and c ≥ 3. Applying Theorem 2.7, we get the following formula

(2.16)  dim R/(I_{t−1}(φ_{1}) + I_{t}(φ_{c−1})) = dim D_{c−1} − 2 .

Even more, if dim D_{c−1} ≥ 3 and c ≥ 4 we have the equalities

(2.17)  dim R/(I_{t−1}(φ_{i}) + I_{t}(φ_{c−1})) = dim D_{c−1} − i − 1

for 1 ≤ i ≤ 2 which will play an important role in the next section.

8 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

Remark 2.9. If t = 2 we can prove (2.16) directly as follows. Let A = [C, D, v] where C is a 2 by 2 general enough matrix in the variables x_{0}, x_{1}, x_{2}, x_{3} (e.g. with rows (x_{0}^{a_{0}−b_{1}}, x_{1}^{a_{1}−b_{1}}) and (x_{2}^{a_{0}−b_{2}}, x_{3}^{a_{1}−b_{2}}) ), v is some column and D is a 2 by c − 2 matrix whose first row is (x_{4}^{a_{2}−b_{1}}, x_{5}^{a_{3}−b_{1}}, ..., x_{c}^{a_{c−2}−b_{1}}, 0) and whose second row is (0, x_{4}^{a_{3}−b_{2}}, x_{5}^{a_{4}−b_{2}}, ..., x_{c}^{a_{c−1}−b_{2}}). Note that φ_{1} corresponds to C and φ_{c−1} to [C, D]. Since C is general, we get codim_{R} R/I_{t−1}(φ_{1}) = 4, i.e. the radical of I_{t−1}(φ_{1}) is (x_{0}, x_{1}, x_{2}, x_{3}). Since it is clear that the radical of the ideal generated by the maximal minors of D is (x_{4}, x_{5}, ..., x_{c}), it follows that the scheme X_{c−1} = Proj(D_{c−1}) ∈ W (\underline{b}; \underline{a′}) given by the maximal minors of [C, D] satisfies (2.16). The general element X of W (\underline{b}; \underline{a}) will therefore have a flag where X_{c−1} satisfies (2.16).

3. Smoothness and dimension of the determinantal locus

This section is the heart of the paper and contains the results which generalize quite a lot of our previous contributions to problems (1)-(3) stated in the introduction.

Proposition 3.1. With notation as in Remark 2.8, let c ≥ 3 and suppose (2.16) (this holds if a_{0} > b_{t}). If a_{t+c−2} > a_{t−2} then (2.12) holds for X := X_{c} ⊂ Y := X_{c−1}, i.e.

0hom_{R}(I_{D_{c−1}}, I_{D_{c}/D_{c−1}}) ≤ ∑_{j=0}^{t+c−3} \binom{a_{j} − a_{t+c−2} + n}{n}.

In particular we get dim W (\underline{b}; \underline{a}) = λ_{c} + K_{3} + K_{4} + ... + K_{c} provided dim W (\underline{b}; \underline{a′}) = λ_{c−1} + K_{3} + K_{4} + ... + K_{c−1} where \underline{a′} = a_{0}, a_{1}, ..., a_{t+c−3}.

Proof. We claim that depth_{I(Z_{j})} D_{c−1} ≥ 2 for all j satisfying 0 < j < c − 1. Indeed by the discussion right after Remark 2.1 we see that we may take I(Z_{j}) to be the ideal I_{t−1}(φ_{j}) ⊂ R of submaximal minors of the matrix which defines X_{j}, and I(Z_{j})D_{c−1} to be (I_{t−1}(φ_{j}) + I_{t}(φ_{c−1}))D_{c−1}. It follows that depth_{I_{t−1}(φ_{j})} D_{c−1} ≥ 2 for j = 1, i.e. dim D_{c−1} −dim D_{c−1}/(I_{t−1}(φ_{1})+I_{t}(φ_{c−1}))D_{c−1} ≥ 2, implies the claim. Hence we conclude the proof of the claim by (2.16).

For every j, 0 < j < c − 1, put a := a_{t+j−1} − a_{t+c−2}. We claim that

Hom_{D_{j}}(I_{j}, I_{c−1}) ≅ D_{c−1}(a) .

To prove this claim we remark that depth_{I(Z_{j})} I_{c−1} ≥ 2 since I_{c−1} is maximally CM. Using that I_{j} is locally free on U_{j} and the arguments in the text before (2.13), see the text accompanying (2.7) for j = 1, we get

I_{j} ⊗_{O_{X_{j}}} O_{X_{c−1}}|_{U_{j}} ≅ I_{j} ⊗_{O_{X_{j}}} O_{X_{j+1}} ⊗ ... ⊗_{O_{X_{c−2}}} O_{X_{c−1}}|_{U_{j}} ≅ I_{c−1}(−a)|_{U_{j}}.

It follows that Hom_{O_{X_{j}}} (I_{j}, I_{c−1})(−a) ≅ Hom_{O_{X_{c−1}}}(I_{c−1}, I_{c−1}) ≅ O_{X_{c−1}} are isomorphic as sheaves on U_{j} ∩ X_{c−1}, i.e. we get that H_{∗}^{0}(U_{j}, Hom(I_{j}, I_{c−1})) ≅ H_{∗}^{0}(U_{j}, O_{X_{c−1}})(a) and hence the claim from depth_{I(Z_{j})} D_{c−1} = depth_{I(Z_{j})} I_{c−1} ≥ 2 and (2.9).

Now we repeatedly use the exact sequence

(3.1) 0 → D_{c−1}(a) ≅ Hom_{D_{j}}(I_{j}, I_{c−1}) → Hom_{R}(I_{D_{j+1}}, I_{c−1}) → Hom_{R}(I_{D_{j}}, I_{c−1}) →

for j = c − 2, c − 3, ..., 1. Since a_{t+j−1} ≤ a_{t+c−2}, we have dim D_{c−1}(a)_{0} = \binom{a+n}{n}. It follows that

0hom(I_{D_{c−1}}, I_{c−1}) ≤ _{0}hom(I_{D_{1}}, I_{c−1}) + ∑_{i=t}^{t+c−3} \binom{a_{i} − a_{t+c−2} + n}{n}

where we have replaced a_{t+j−1} by a_{i} in which case 1 ≤ j ≤ c − 2 corresponds to t ≤ i ≤ t + c − 3.

It remains to prove that _{0}hom(I_{D_{1}}, I_{c−1}) ≤ ∑_{i=0}^{t−1} \binom{a_{i}−a_{t+c−2}+n}{n} since we then by Proposition 2.4 get the dimension formula. Using that X_{1} = Proj(D_{1}) is a hypersurface of degree ℓ_{1}, we find _{0}hom(I_{D_{1}}, I_{c−1}) ≅ dim I_{c−1}(ℓ_{1})_{0} where ℓ_{k} = ∑_{j=0}^{t+k−2} a_{j} − ∑_{i=1}^{t} b_{i}. Now we have to make the degrees of the minimal generators of I_{c−1} explicit. Taking a close look at (2.3), we see that a minimal generator f of I_{c−1} ≅ I_{D_{c}}/I_{D_{c−1}} of the smallest possible degree has degree s(I_{c−1}) := ℓ_{c} − ∑_{j=t−1}^{t+c−3} a_{j} because a_{0} ≤ a_{1} ≤ ⋯ ≤ a_{t+c−2}. Since ℓ_{1} − s(I_{c−1}) = a_{t−1} − a_{t+c−2} ≤ 0 by the definition of ℓ_{k}, we get either dim I_{c−1}(ℓ_{1})_{0} = 0 or a_{t−1} = a_{t+c−2}. In the latter case the assumption a_{t+c−2} > a_{t−2} implies that the degrees of all minimal generators, except for f, are strictly greater than s(I_{c−1}), i.e. we get dim I_{c−1}(ℓ_{1})_{0} = \binom{a_{t−1}−a_{t+c−2}+n}{n} and we are done. □

By repeatedly using Proposition 3.1 we get the

Theorem 3.2. Let X ⊂ ℙ^{n}, (X) ∈ W (\underline{b}; \underline{a}), be a general determinantal scheme and suppose a_{0} > b_{t}. Moreover if c ≥ 6 we suppose a_{t+3} > a_{t−2} (or a_{t+4} > a_{t−2} provided chark = 0) and if 3 ≤ c ≤ 5 we suppose a_{t+c−2} > a_{t−2}. Then we have

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

Proof. If 3 ≤ c ≤ 6 (chark = 0 if c = 6) and a_{t+c−2} > a_{t−2} we use (2.11) to find dim W (\underline{b}; \underline{a′}) and we conclude the proof by Proposition 3.1.

If c ≥ 6 and a_{t+3} > a_{t−2} (resp. a_{t+4} > a_{t−2} if c ≥ 7) we repeatedly use Proposition 3.1 to reduce to the case c = 5 (resp. c = 6) and we conclude by the first part of the proof (note that the assumption a_{t+c−2} > a_{t−2} of Proposition 3.1 is satisfied in this induction). □

Remark 3.3. We expect that the assumption a_{0} > b_{t} can be weakened in Theorem 3.2, as well as in (2.16). At least it does for c = 3 provided we assume a_{i−2} > b_{i} for 2 ≤ i ≤ t. Indeed since I_{t}(φ_{2}) ⊂ I_{t−1}(φ_{1}) we first show (2.16) using Remark 2.1. Then the proof above applies to conclude as in Theorem 3.2 provided a_{t+1} > a_{t−2}, cf. Remark 2.3.

If the condition (2.17) is satisfied, then we can prove the following result for \overline{W (\underline{b}; \underline{a})} to be a generically smooth irreducible component.

Theorem 3.4. Let X ⊂ ℙ^{n}, (X) ∈ W (\underline{b}; \underline{a}), be a general determinantal scheme of dimension n − c ≥ 1, let c > 2 and let X = X_{c} ⊂ X_{c−1} ⊂ ... ⊂ X_{2} ⊂ ℙ^{n}, X_{i} = Proj(D_{i}), be the flag obtained by successively deleting columns from the right hand side. If a_{0} > b_{t},

0Ext_{D_{2}}^{1}(I_{D_{2}}/I_{D_{2}}^{2}, I_{2}) = 0 and _{0}Ext_{D_{3}}^{1}(I_{D_{3}}/I_{D_{3}}^{2}, I_{i}) = 0 for i = 3, ..., c − 1 ,

then \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of the Hilbert scheme Hilb^{p}(ℙ^{n}).

Remark 3.5. If n − c ≥ 2 then we have _{0}Ext_{D_{2}}^{1}(I_{D_{2}}/I_{D_{2}}^{2}, I_{2}) = 0 provided c = 3, and 0Ext_{D_{2}}^{1}(I_{D_{2}}/I_{D_{2}}^{2}, I_{2}) = _{0}Ext_{D_{3}}^{1}(I_{D_{3}}/I_{D_{3}}^{2}, I_{3}) = 0 provided c = 4 by [12], (5.4)-(5.8). Hence

10 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

the conclusion of Theorem 3.4 holds provided n − c ≥ 2 and 3 ≤ c ≤ 4. Moreover if n − c ≥ 1 and c > 4 then both Ext-groups above still vanish by [12] and we may in this case replace the assumption of Theorem 3.4 given by the displayed formula with

0Ext_{D_{3}}^{1}(I_{D_{3}}/I_{D_{3}}^{2}, I_{i}) = 0 for i = 4, ..., c − 1 .

Note that in the case n − c ≥ 1 and c = 2, the conclusion of Theorem 3.4 holds and moreover, Hilb^{p}(ℙ^{n}) is smooth at any (X) ∈ W (\underline{b}; \underline{a}) by [7].

Remark 3.6. For c > 2 one knows that Hilb^{p}(ℙ^{n}) is not always smooth at any (X) ∈ W (\underline{b}; \underline{a}) [14]. Indeed, since W (\underline{b}; \underline{a}) is irreducible, it is not difficult to find singular points of Hilb^{p}(ℙ^{n}) by first computing its tangent space dimension, h^{0}(N_{X}), at a general (X) ∈ W (\underline{b}; \underline{a}), using Macaulay 2. Then by experimenting with special choices of (X_{0}) ∈ W (\underline{b}; \underline{a}) one may find h^{0}(N_{X}) < h^{0}(N_{X_{0}}) which means that Hilb^{p}(ℙ^{n}) is singular at (X_{0}), see [16] which even computes the obstructions of deformations, using Singular, in a related case.

Proof. Due to Theorem 5.1 of [12] we must show that

(3.2)  _{0}Ext_{D_{i}}^{1}(I_{D_{i}}/I_{D_{i}}^{2}, I_{i}) = 0 for i = 2, ..., c − 1.

By assumption we need to prove the vanishing (3.2) for i = 4, ..., c − 1 and c > 4. By induction on c it suffices to show it for i = c − 1, c ≥ 5. Hence it suffices to see that there exist injections

(3.3)  _{0}Ext_{D_{j+1}}^{1}(I_{D_{j+1}}/I_{D_{j+1}}^{2}, I_{c−1}) ↪ _{0}Ext_{D_{j}}^{1}(I_{D_{j}}/I_{D_{j}}^{2}, I_{c−1}) for j = 2, ..., c − 2.

By (2.17) and the arguments in the first paragraph of the proof of Proposition 3.1 we may suppose depth_{I(Z_{j})} D_{c−1} ≥ 3 for all j satisfying 1 < j < c − 1.

We claim that the left-exact sequence (3.1) is also right-exact, i.e. that the rightmost map of the Hom-groups is surjective for 1 < j < c − 1. To show this, it certainly suffices to prove Ext_{R}^{1}(I_{j}, I_{c−1}) = 0. However, by paying closer attention to the modules of (3.1) we shall see that also Ext_{D_{j}}^{1}(I_{j}, I_{c−1}) = 0 for 1 < j < c − 1 suffices for proving the claim. Indeed if we apply (−) ⊗_{R} D_{j} to 0 → I_{D_{j}} → I_{D_{j+1}} → I_{j} → 0 we get the right-exact sequence I_{D_{j}} ⊗_{R} D_{j} → I_{D_{j+1}} ⊗_{R} D_{j} → I_{j} → 0 where I_{D_{j+1}} ⊗_{R} D_{j} ≅ I_{D_{j+1}}/I_{D_{j}} · I_{D_{j+1}} and since I_{D_{j}} ⊗_{R} D_{j+1} ≅ I_{D_{j}}/I_{D_{j}} · I_{D_{j+1}} = ker(I_{D_{j+1}}/I_{D_{j}} · I_{D_{j+1}} → I_{j}) we obtain the exact sequence

0 ⟶ I_{D_{j}} ⊗_{R} D_{j+1} ⟶ I_{D_{j+1}} ⊗_{R} D_{j} ⟶ I_{j} ⟶ 0

to which we apply Hom_{D_{j}}(−, I_{c−1}). Then we get exactly (3.1) continued to the right by Ext_{D_{j}}^{1}(I_{j}, I_{c−1}). Thus the vanishing of Ext_{D_{j}}^{1}(I_{j}, I_{c−1}) implies that (3.1) is right-exact.

To see that Ext_{D_{j}}^{1}(I_{j}, I_{c−1}) = 0 we use the isomorphism Hom_{O_{X_{j}}} (I_{j}, I_{c−1})(−a)|_{U_{j}∩X_{c−1}} ≅ O_{X_{c−1}}|_{U_{j}∩X_{c−1}} which we obtained in the proof of Proposition 3.1. By (2.9) it follows that

Ext_{D_{j}}^{1}(I_{j}, I_{c−1}) ≅ H_{∗}^{1}(U_{j}, Hom(I_{j}, I_{c−1})) ≅ H_{∗}^{1}(U_{j}, O_{X_{c−1}}(a)) = 0

because we have depth_{I(Z_{j})} D_{c−1} ≥ 3 and hence depth_{I(Z_{j})} I_{c−1} ≥ 3 (I_{c−1} is maximally CM). This proves the claim.

Now we can rewrite the exact sequence (3.1) as

(3.4) 0 → D_{c−1}(a) → Hom_{D_{j+1}}(I_{D_{j+1}}/I_{D_{j+1}}^{2}, I_{c−1}) → Hom_{D_{j}}(I_{D_{j}}/I_{D_{j}}^{2}, I_{c−1}) → 0 .

Sheafifying, restricting to U_{j} ∩ X_{c−1} (note that I_{D_{j+1}} is also locally free on U_{j} ∩ X_{c−1}) and taking cohomology, we get

→ H^{1}(U_{j}, O_{X_{c−1}}(a)) → H^{1}(U_{j}, Hom(I_{X_{j+1}}/I_{X_{j+1}}^{2}, I_{c−1})) → H^{1}(U_{j}, Hom(I_{X_{j}}/I_{X_{j}}^{2}, I_{c−1})) →

Since depth_{I(Z_{j})} D_{c−1} ≥ 3, the two latter H^{1}-groups are by (2.9) isomorphic to the _{0}Ext^{1}- groups quoted in (3.3) and since H^{1}(U_{j}, O_{X_{c−1}}(a)) = 0 we are done. □

Remark 3.7. Since (2.17) holds also for i = 3 provided dim D_{c−1} ≥ 4, c ≥ 5 and a_{0} > b_{t} by Theorem 2.7, we may continue the proof above to see that the injections (3.3) are isomorphisms for j ≥ 3. Hence if X is general and n − c ≥ 2, then

0Ext_{D_{3}}^{1}(I_{D_{3}}/I_{D_{3}}^{2}, I_{c−1}) ≅ _{0}Ext_{D_{c−1}}^{1}(I_{D_{c−1}}/I_{D_{c−1}}^{2}, I_{c−1}) .

Here the leftmost Ext^{1}-group is computed much faster by Macaulay 2 than the rightmost one. We also get an injection in (3.3) for j = 2, but now it is not necessarily an isomorphism.

Corollary 3.8. Let n − c ≥ 1, c ≥ 5 and suppose a_{0} > b_{t} and a_{t+3} > a_{t−1} + a_{t} − b_{1}. Then \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{n}) of dimension λ_{c} + K_{3} + ... + K_{c}.

Proof. We get dim \overline{W (\underline{b}; \underline{a})} from Theorem 3.2. Hence by (3.3) and Remark 3.5 it suffices to show that _{0}Ext_{R}^{1}(I_{D_{2}}, I_{i}) = 0 for 4 ≤ i ≤ c − 1 since _{0}Ext_{D_{2}}^{1}(I_{D_{2}}/I_{D_{2}}^{2}, I_{i}) is a subgroup of _{0}Ext_{R}^{1}(I_{D_{2}}, I_{i}). First let i = c − 1. By the Eagon-Northcott resolution (2.3) we see that the largest possible degree of a relation for I_{D_{2}} is ℓ_{2} − b_{1} and the smallest possible degree of a generator of I_{c−1} ≅ I_{D_{c}}/I_{D_{c−1}} is ℓ_{c} − ∑_{j=t−1}^{t+c−3} a_{j}. Since ℓ_{c} = ℓ_{2} + ∑_{j=t+1}^{t+c−2} a_{j}, we get _{0}Ext_{R}^{1}(I_{D_{2}}, I_{c−1}) = 0 from

ℓ_{2} − b_{1} < ℓ_{2} + ∑_{j=t+1}^{t+c−2} a_{j} − ∑_{j=t−1}^{t+c−3} a_{j} = ℓ_{2} − a_{t−1} − a_{t} + a_{t+c−2} ,

i.e. from a_{t+c−2} > a_{t−1} + a_{t} − b_{1}. Since we need the vanishing of _{0}Ext_{R}^{1}(I_{D_{2}}, I_{i}) for any i = 4, 5, ..., c−1, we must suppose a_{t+3} > a_{t−1} +a_{t} −b_{1} and hence we get the corollary. □

Remark 3.9. Note that if c = 3 (resp. c = 4) we can argue as above to see that the conclusions of Corollary 3.8 hold provided a_{t+1} > a_{t−1}+a_{t}−b_{1} (resp. a_{t+2} > a_{t−1}+a_{t}−b_{1}). This is, however, proved in [12], Corollary 5.10. For c ≥ 5, Corollary 3.8 generalizes the corresponding result [12], Corollary 5.9 quite a lot.

4. Conjectures

In [10] the first author discovered a counterexample to Conjecture 2.2 for every c in the range n = c ≥ 3. Indeed the vanishing all 2 × 2 minors of a general 2 × (c + 1) matrix of linear entries defines a reduced scheme of c + 1 different points in ℙ^{c}. The conjectured dimension of W (0, 0; 1, 1, ..., 1) is c(c + 1) + c − 2 while its actual dimension is at most c(c + 1).

12 JAN O. KLEPPE, ROSA M. MIRÓ-ROIG

On the other hand Theorem 3.2 is quite close to proving Conjecture 2.2. The crucial assumption in Theorem 3.2 is the inequality a_{t+c−2} > a_{t−2} (or a_{t+3} > a_{t−2} if c > 5). Since we, in addition to proving Theorem 3.2, have computed quite a lot of examples where we have a_{t+c−2} = a_{t−2} and a_{i−min([c/2]+1,t)} > b_{i}, and each time, except for the counterexample, obtained (2.12) and hence the conjecture, we now want to slightly change Conjecture 2.2 to

Conjecture 4.1. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we assume a_{i−min([c/2]+1,t)} ≥ b_{i} provided n > c and a_{i−min([c/2]+1,t)} > b_{i} provided n = c for min([c/2] + 1, t) ≤ i ≤ t. Except for the family W (0, 0; 1, 1, ..., 1) of zero-dimensional schemes above we have, for W (\underline{b}; \underline{a}) ≠ ∅, that

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

Indeed in the situation of Proposition 2.4 we even expect (2.12) to hold! This will imply Conjecture 4.1 provided the conjecture holds for W (\underline{b}; \underline{a′}). Note that the conclusion of the conjecture is true provided n − c ≥ 1 and 2 ≤ c ≤ 5 (char(k) = 0 if c = 5) by [12].

Finally we will state a conjecture related to the problems (2) and (3) of the Introduction:

Conjecture 4.2. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we suppose n−c ≥ 2, c ≥ 5 and a_{0} > b_{t}. Then \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of the Hilbert scheme Hilb^{p}(ℙ^{n}).

Indeed due to the results of this paper and many examples computed by Macaulay 2 in the range a_{0} > b_{t} we even expect the groups _{0}Ext_{D_{3}}^{1}(I_{D_{3}}/I_{D_{3}}^{2}, I_{i}) for i = 4, ..., c − 1 of Theorem 3.4 to vanish! This will imply Conjecture 4.2. The conclusion of the conjecture may even be true for 0 ≤ n − c ≤ 1 (we have no counterexample), but in this range we have verified that the Ext^{1}-groups above do not always vanish. Note that the conclusion of Conjecture 4.2 is true provided n − c ≥ 2 and 2 ≤ c ≤ 4 ([7], [11], [12]).

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Faculty of Engineering, Oslo University College, 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