DEFORMATIONS OF MODULES OF MAXIMAL GRADE AND THE HILBERT SCHEME AT DETERMINANTAL SCHEMES.

JAN O. KLEPPE

Abstract. Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade (which means that the ideal I_{t}(A) generated by the maximal minors of a homogeneous presentation matrix, A, of M has maximal codimension in R). Suppose X := Proj(R/I_{t}(A)) is smooth in a sufficiently large open subset and dim X ≥ 1. Then we prove that the local graded deformation functor of M is isomorphic to the local Hilbert (scheme) functor at X ⊂ Proj(R) under a week assumption which holds if dim X ≥ 2. Under this assumption we get that the Hilbert scheme is smooth at (X), and we give an explicit formula for the dimension of its local ring. As a corollary we prove a conjecture of R.M.Miró-Roig and the author that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component V of the Hilbert scheme. Also their conjecture on the dimension of V is proved for dim X ≥ 1. The cohomology H_{∗}^{i}(N_{X}) of the normal sheaf of X in Proj(R) is shown to vanish for 1 ≤ i ≤ dim X − 2. Finally the mentioned results, slightly adapted, remain true replacing R by any Cohen-Macaulay quotient of a polynomial ring.

1. Introduction

Determinantal objects are central in many areas of mathematics. In algebraic geometry determinantal schemes defined by the vanishing of the p × p -minors of a homogeneous polynomial matrix, may be used to describe classical schemes such as rational normal scrolls and other fibered schemes, Veronese and Segre varieties and Secant schemes to rational normal curves and Segre varieties ([19], [3]). Throughout the years many nice properties are detected for determinantal schemes, e.g. they are arithmetically Cohen-Macaulay with rather well understood free resolutions and singular loci, see [11], [12], [36], [49], and see [6], [5], [13], [15], [30], [35], [39] for history and other important contributions.

In this paper we study the Hilbert scheme along the locus of determinantal schemes. More precisely we study deformations of modules of maximal grade over a polynomial ring R and establish a very strong connection to corresponding deformations of determinantal schemes in ℙ^{n}. Recall that the grade g of a finitely generated graded R-module M is the grade of its annihilator I := ann(M), i.e. g = depth_{I} R = dim R − dim R/I. We say a scheme X ⊂ ℙ^{n} of codimension c is standard determinantal if its homogeneous saturated ideal is equal to the ideal I_{t}(A) generated by the t × t minors of some homogeneous t × (t + c − 1) matrix A = (f_{ij}), f_{ij} ∈ R. If M is the cokernel of the map determined by A, then g = c because the radicals of I and I_{t}(A) are equal. Moreover M has maximal

Date: December 9, 2010.

1991 Mathematics Subject Classification. Primary 14M12, 14C05, 13D07, 13D10; Secondary 14H10, 14J10, 13D03.

2 JAN O. KLEPPE

grade if and only if X = Proj(A), A := R/I_{t}(A) is standard determinantal. In this case ann(M) = I_{t}(A) for c ≥ 2 by [7].

Let Hilb^{p}(ℙ^{n}) be the Hilbert scheme parameterizing closed subschemes of ℙ^{n} of dimension n − c ≥ 0 and with Hilbert polynomial p. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, t ≥ 2, c ≥ 2, we denote by W_{s}(\underline{b}; \underline{a}) ⊂ Hilb^{p}(ℙ^{n}) the stratum of standard determinantal schemes where f_{ij} are homogeneous polynomials of degrees a_{j} − b_{i}. Inside W_{s}(\underline{b}; \underline{a}) we have the open subset W (\underline{b}; \underline{a}) of determinantal schemes which are generically a complete intersection. The elements are called good determinantal schemes. Note that W_{s}(\underline{b}; \underline{a}) is irreducible, and W (\underline{b}; \underline{a}) ≠ ∅ if we suppose a_{i−1} − b_{i} > 0 for i ≥ 1, see (2.2).

In this paper we determine the dimension of a non-empty W (\underline{b}; \underline{a}) provided a_{i−2}−b_{i} ≥ 0 for i ≥ 2 and n − c ≥ 1 (Theorem 5.5, Corollary 5.6). Indeed

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

where λ_{c} and K_{i} are a large sum of binomials only involving a_{j} and b_{i} (see Conjecture 2.2 and (2.10) for the definition of λ_{c} and K_{i}). In terms of the Hilbert function, H_{M}(−), of M, we may alternatively write (1.1) in the form

dim W (\underline{b}; \underline{a}) = ∑_{j=0}^{t+c−2} H_{M}(a_{j}) − ∑_{i=1}^{t} H_{M}(b_{i}) + 1

(Remark 3.9). Moreover we prove that the closure \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of the Hilbert scheme Hilb^{p}(ℙ^{n}) provided _{0}Ext_{A}^{2}(M, M), the degree zero part of the graded module Ext_{A}^{2}(M, M), vanishes for a general X = Proj(A) of W (\underline{b}; \underline{a}) (Theorem 5.8, Corollary 5.9). Indeed

dim_{(X)} Hilb^{p}(ℙ^{n}) − dim W (\underline{b}; \underline{a}) ≤ dim _{0}Ext_{A}^{2}(M, M) ,

and Hilb^{p}(ℙ^{n}) is smooth at (X) if equality holds. We prove that n − c ≥ 2 implies Ext_{A}^{2}(M, M) = 0 (Corollary 4.10), whence \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{n}) in the case n − c ≥ 2 and a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t. This proves Conjecture 4.2 of [32]. Moreover our results hold for every (X) ∈ W (\underline{b}; \underline{a}) provided a depth condition on the singular locus is fulfilled. A general X of W (\underline{b}; \underline{a}) satisfies the condition and we get the mentioned results.

The most remarkable finding in this paper is perhaps the method. Indeed an embedded deformation problem for the determinantal scheme X = Proj(A), A = R/ann(M) is transfered to a deformation problem for the R-module M where it is handled much more easily because every deformation of M comes from deforming the matrix A. The latter is easy to see from the Buchsbaum-Rim complex. In fact it was in [29] we introduced the notion “every deformation of X comes from deforming A” to better understand why \overline{W (\underline{b}; \underline{a})} may fail to be an irreducible component. This led us to study deformations of M because the corresponding property holds for M. Therefore the isomorphism between the graded deformation functors of M and R → A, which we prove under the assumption 0Ext_{A}^{i}(M, M) = 0 for i = 1 and 2, is an important result (Theorem 5.2). Note that the graded deformation functor of R → A is further isomorphic to the local Hilbert (scheme) functor of X provided n − c ≥ 1. Since we also prove that n − c ≥ i ≥ 1 implies Ext_{A}^{i}(M, M) = 0 under mild assumptions (Theorems 4.1 and 4.5), we get our rather algebraic method for studying a geometric object, the Hilbert scheme. Even the corresponding non-graded deformation functors of M and R → A are isomorphic for n − c ≥ 2 (Remark 5.14), which more than indicates that this method holds for local determinantal rings of dimension greater than 2. Hence we expect applications to deformations of determinantal singularities, as well as to multigraded Hilbert schemes. We remark that while the vanishing of Ext_{A}^{i}(M, M) in Theorem 4.1 is mainly known (at least for i = 1, see Remark 4.3), the surprise is Theorem 4.5 which reduces the depth assumption of Theorem 4.1 by 1 in important cases. Note that the local deformation functors of M as an A- as well as an R-module were thoroughly studied by R. Ile in [22],

[24] and in [23] he studies the case of a determinantal hypersurface X (A a square matrix) without proving, to our knowledge, the mentioned results (see Remark 4.4). Ile and his paper [23], and the joint papers [30], [31], [32] have, however, served as an inspiration for this work.

We also get further interesting results, e.g. that arbitrary modules of maximal grade are unobstructed (earlier proved by Ile in [22]), and we show that the dimension of their natural deformation spaces is equal to the right hand side of (1.1) (Theorems 3.1 and 3.8, cf. Remark 3.9). Moreover we prove that the cohomology H^{i}(N_{X}(v)) of the normal sheaf of X ⊂ ℙ^{n} for a X general in W (\underline{b}; \underline{a}) vanishes for 1 ≤ i ≤ dim X − 2 and every v (Theorem 5.11). Even the algebra cohomology groups H^{i}(R, A, A) of André-Quillen vanish for 2 ≤ i ≤ min{dim X − 1, c}. This extends a result from T. Svanes’ thesis [46] proven there for so-called generic determinantal schemes in which the entries of A are the indeterminates of R, see [6], Thm. 15.10 for details. Finally we remark that the assumption that R is a polynomial ring can be weakened. Indeed all theorems and their proofs generalize at least to the case where Proj(R) is any arithmetically Cohen-Macaulay k-scheme (and smooth in Theorem 5.11 (ii)), only replacing all \binom{v+n}{n} in (1.1) with dim R_{v}.

The method of this paper has the power of solving most of the deformation problems the author, together with coauthors (mostly Miró-Roig at Barcelona) has considered in several papers ([30], [31], [32], [29]), mainly:

(1) Determine the dimension of W (\underline{b}; \underline{a}) in terms of a_{j} and b_{i} (see Conjecture 2.2).

(2) Is \overline{W (\underline{b}; \underline{a})} a generically smooth irreducible component of Hilb^{p}(ℙ^{n})?

The main method so far has been to delete columns of the matrix A, to get a “flag” of closed subschemes X = X_{c} ⊂ X_{c−1} ⊂ ... ⊂ X_{2} ⊂ ℙ^{n} and to prove the results by considering the smoothness of the Hilbert flag scheme of pairs and its natural projections into the Hilbert schemes. In fact in [31] we solved problem (1) in the cases 2 ≤ c ≤ 5 and n − c ≥ 1 (assuming char(k) = 0 if c = 5), and recently we almost solved (1) in the remaining cases under the assumption a_{t+3} > a_{t−2} [32]. Concerning problem (2) we gave in [31] an affirmative answer in the range 2 ≤ c ≤ 4 and n−c ≥ 2, (see [14] and [30] for the cases 2 ≤ c ≤ 3). We got further improvements in [32] and conjectured a positive answer to problem (2) provided n − c ≥ 2, but we were not able to solve all technical challenges which increased with the codimension. In this paper we fully prove the conjecture, as well as Conjecture 2.2 for n − c ≥ 1, with the new approach which is much easier than the older one. For the case n − c = 0 we remark that since every element of W (\underline{b}; \underline{a}) has the

4 JAN O. KLEPPE

same Hilbert function, problem (2) becomes more natural provided we replace Hilb^{p}(ℙ^{n}) with GradAlg(H), see [29] for details and the notations below.

We thank R. Ile, R.M. Miró-Roig, J.A. Christophersen, M. Boij, O.A. Laudal, Johannes Kleppe and U. Nagel for interesting discussions on different aspects of this topic.

Notation: In this work R = k[x_{0}, …, x_{n}] will be a polynomial ring over an algebraically closed field, 𝔪 = (x_{0}, …, x_{n}) and deg x_{i} = 1, unless explicitly making other assumptions.

We mainly keep the notations of [31] and [29]. If X ⊂ Y are closed subschemes of ℙ := ℙ^{n} := Proj(R), we denote by I_{X/Y} (resp. N_{X/Y}) the ideal (resp. normal) sheaf of X in Y, and we usually suppress Y when Y = ℙ^{n}. By the codimension, codim_{Y} X, of X in Y we simply mean dim Y − dim X, and we let ω_{X} = Ext_{O_{ℙ^{n}}}^{c}(O_{X}, O_{ℙ^{n}})(−n − 1) if c = codim_{ℙ}X. When we write X = Proj(A) we take A := R/I_{X} and K_{A} = Ext_{R}^{c}(A, R)(−n−1) where I_{X} = H_{∗}^{0}(I_{X}) is the saturated homogeneous ideal of X ⊂ ℙ^{n}. We denote the group of morphisms between coherent O_{X}-modules by Hom_{O_{X}}(F, G) while Hom_{O_{X}}(F, G) denotes the corresponding sheaf. Moreover we set hom(F, G) = dim_{k} Hom(F, G) and we correspondingly use small letters for the dimension, as a k-vector space, of similar groups.

We denote the Hilbert scheme by Hilb^{p}(ℙ^{n}), p the Hilbert polynomial [17], and (X) ∈ Hilb^{p}(ℙ^{n}) the point which corresponds to X ⊂ ℙ^{n}. Let GradAlg(H) be the representing object of the functor which parametrizes flat families of graded quotients A of R of depth_{𝔪} A ≥ 1 and with Hilbert function H; H(i) = dim A_{i} ([27], [28]). We let (A), or (X) where X = Proj(A), denote the point of GradAlg(H) which corresponds to A. Then X (resp. A) is unobstructed if Hilb^{p}(ℙ^{n}) (resp. GradAlg(H)) is smooth at (X). By [14],

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

provided depth A := depth_{𝔪} A ≥ 2. This implies that if dim A ≥ 2 and A is Cohen-Macaulay (CM), then it is equivalent to consider deformations of X ↪ ℙ^{n}, or of R ↠ A, and moreover that their tangent spaces_{0}Hom(I_{X}, A) ≃ H^{0}(N_{X}) are isomorphic where the lower index means the degree zero part of the graded module Hom(I_{X}, A). We also deduce that if X is generically a complete intersection, then _{0}Ext_{A}^{1}(I_{X}/I_{X}^{2}, A) is an obstruction space of Hilb^{p}(ℙ^{n}) at (X) ([28], §1.1). Finally 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 such that any (X) in U has all the openness properties that we want to require.

2. Background

This section recalls basic results on standard and good determinantal schemes needed in the sequel, see [6], [13], [5] and [34] for more details and [12], [8], [11] for background. Let

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

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

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

Recall that a codimension c subscheme X ⊂ ℙ^{n} is standard determinantal if I_{X} = I_{t}(A) for some homogeneous t × (t + c − 1) matrix A as above. Moreover X ⊂ ℙ^{n} is a good determinantal scheme if additionally, I_{t−1}(A) defines a scheme of codimension greater or equal to c + 1 in ℙ^{n}. Note that if X is standard determinantal and a generic complete intersection in ℙ^{n}, then X is good determinantal, and conversely [34], Thm. 3.4. We say that A is minimal if f_{ij} = 0 for all i, j with b_{i} = a_{j}.

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. By [31], see the end of p. 2877, we get that the closures of these strata in Hilb^{p}(ℙ^{n}) are equal and irreducible. Moreover since we will not require A to be minimal for X = Proj(R/I_{t}(A)) to belong to W (\underline{b}; \underline{a}) or W_{s}(\underline{b}; \underline{a}) in their definitions (a slight correction to [30] and [31]!), we must reconsider Cor. 2.6 of [31]. Indeed we may use its proof to see (cf. [32] for details)

(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.

Let A = R/I_{X} (i.e. X) be standard determinantal and let M := coker(φ^{∗}). Then one knows that the Eagon-Northcott complex yields the following free resolution

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

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

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

(the resolutions are minimal if A is minimal), see for instance [6]; Thm. 2.20 and [13]; Cor. A2.12 and Cor. A2.13. Note that (2.3) shows that A is Cohen-Macaulay.

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

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

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

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

is exact by [34] or [30], (3.1), i.e. we may put I_{X/Y} := M_{B}(a_{t+c−2})^{∗}. Due to (2.4), M is a maximal Cohen-Macaulay A-module (depth M = dim A), and so is I_{X/Y} by (2.6). By [13] we have K_{A}(n + 1) ≅ S_{c−1}M_{A}(ℓ_{c}) and hence K_{B}(n + 1) ≅ S_{c−2}M_{B}(ℓ_{c−1}) where

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

Recall that M̃ is locally free of rank one precisely on X − V (I_{t−1}(A)) ([5], Lem. 1.4.8) and that X ↪ ℙ^{n} is a local complete intersection (l.c.i.) by e.g. [48], Lem. 1.8 provided we restrict to X − V (I_{t−1}(A)). By (2.6) it follows that X ↪ Y and Y ↪ ℙ^{n} are l.c.i.’s outside V (I_{t−1}(B)). Note that V (I_{t−1}(B)) ⊂ V (I_{t}(A)) = X.

6 JAN O. KLEPPE

Remark 2.1. Put X_{c} := X and X_{c−1} := Y, let c > 2 and let α be a positive integer. If X is general in W (\underline{b}; \underline{a}) and a_{i−min(α,t)} − b_{i} ≥ 0 for min(α, t) ≤ i ≤ t, then

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

This follows from Rem. 2.7 of [31] (i.e., from [10]). In particular if α ≥ 3, we get that the closed embeddings Y ↪ ℙ^{n} and X ↪ Y are local complete intersections outside some set Z of codimension at least min(4, c). Indeed we may take Z = V (I_{t−1}(B)).

Moreover we recall the following useful general fact that 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), and H_{∗}^{i}(U, Hom_{O_{X}}(Ñ, L̃)) ≃ H_{I(Z)}^{i+1}(Hom_{A}(N, L)) for i > 0, cf. [18], exp. VI. Note that we interpret I(Z) as 𝔪 if Z = ∅.

In [31] we conjectured the dimension of 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 we let \binom{a}{n} = 0 if a is a negative integer. Since [29] discovered that the scheme of c + 1 general points in ℙ^{c} given by the vanishing all 2 × 2 minors of a general 2 × (c + 1) matrix of linear entries was a counterexample to Conjecture 6.1 (and to the special case given in Conjecture 6.2) of [31], we slightly changed Conjecture 6.1 in [32] to

Conjecture 2.2. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, let h_{i−3} := 2a_{t+i−2} − ℓ_{i} + n, for i = 3, 4, ..., c and 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} ,

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

(2.11)  2 ≤ c ≤ 5 and n − c ≥ 1 ( supposing chark = 0 if c = 5 ) .

Indeed this is mainly [31], Thm. 4.5, Cor. 4.7, Cor. 4.10, Cor. 4.14 and [14] (c = 2) and [30] (c = 3). Moreover we have by [32] (valid also for n = c without assuming chark = 0):

Theorem 2.3. Assume a_{0} > b_{t}. Then Conjecture 2.2 holds provided c > 5 (resp. 2 ≤ c ≤ 5) and a_{t+3} > a_{t−2} (resp. a_{t+c−2} > a_{t−2}).

In [32] we stated a Conjecture related to the problem (2) of the Introduction:

Conjecture 2.4. 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}).

By [32], Cor. 3.8 and Thm. 3.4, Conjecture 2.4 holds provided a_{t+3} > a_{t−1} + a_{t} − b_{1} or more generally, if a certain collection of Ext^{1}-groups vanishes. Note that the conclusion of Conjecture 2.4 holds if n − c ≥ 2 and 2 ≤ c ≤ 4 by [14], [30] and [31].

As in [29] we briefly say “T a local ring” (resp. “T artinian”) for a local k-algebra (T, 𝔪_{T}) essentially of finite type over k = T/𝔪_{T} (resp. such that 𝔪_{T}^{r} = 0 for some integer r). The local deformation functors of this paper will be defined over the category \underline{ℓ} of artinian k-algebras. Moreover we say “T ↠ S is a small in \underline{ℓ} ” provided there is a surjection (T, 𝔪_{T}) → (S, 𝔪_{S}) of artinian k-algebras whose kernel 𝔞 satisfies 𝔞 · 𝔪_{T} = 0.

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

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

Lemma 2.5. If X = Proj(A), A = R/I_{t}(A), is a standard determinantal scheme, then A_{T} := R_{T} /I_{t}(A_{T}) and M_{T} := coker φ_{T}^{∗} are (flat) graded deformations of A and M respectively for every choice of A_{T} as above. In particular X_{T} = Proj(A_{T}) ⊂ ℙ_{T}^{n} := Proj(R_{T}) is a deformation of X ⊂ ℙ^{n} to T with constant Hilbert function.

Proof ([29], cf. [44], Rem. to Prop. 1). The Eagon-Northcott and Buchsbaum-Rim complexes are functorial in the sense that, over R_{T}, all free modules and all morphisms in these complexes are determined by A_{T}. Since these complexes become free resolutions of A and M respectively when we tensor with k over T, it follows that A_{T} and M_{T} are flat over T and satisfy A_{T} ⊗_{T} k = A and M_{T} ⊗_{T} k = M. □

Definition 2.6. We say “every deformation of X comes from deforming A” if for every local ring T and every graded deformation R_{T} → A_{T} of R → A to T, then A_{T} is of the form A_{T} = R_{T} /I_{t}(A_{T}) for some A_{T} as above. Note that by (1.2) we can in this definition replace “graded deformations of R → A” by “deformations of X ↪ ℙ^{n}” provided dim X ≥ 1.

Lemma 2.7. Let X = Proj(A) be a standard determinantal scheme, (X) ∈ W (\underline{b}; \underline{a}). If every deformation of X comes from deforming A, then A is unobstructed. Moreover if n − c ≥ 1 then X is unobstructed and \overline{W (\underline{b}; \underline{a})} is an irreducible component of Hilb^{p}(ℙ^{n}).

Proof. Let T ↠ S be a small in \underline{ℓ} and let A_{S} be a deformation of A to S. By assumption, A_{S} = R_{S}/I_{t}(A_{S}) for some matrix A_{S}. We can lift each f_{ij,S} to a polynomial f_{ij,T} with coefficients in T such that f_{ij,T} ⊗_{T} S = f_{ij,S}. By Lemma 2.5, A_{T} := R_{T} /I_{t}(A_{T}) is flat over T. Since A_{T} ⊗_{T} S = A_{S} we get the unobstructedness of A, as well as the unobstructedness of X in the case dim X ≥ 1 by (1.2). For the remaining part of the proof, see [29]. □

Remark 2.8. By these lemmas we get T -flat determinantal schemes by just parameterizing the polynomials of A over a local ring T, see Rem. 4.5 of [29] and Laksov’s papers [36], [35] for somewhat similar results for more general determinantal schemes.

8 JAN O. KLEPPE

3. deformations of R-modules of maximal grade

Let M be a finitely generated (torsion) R-module with presentation matrix A, i.e. M = coker(φ^{∗}) with φ as in (2.1). Since the grade of M over R is the grade, or codimension, of the annihilator I := ann(M) of M, and since the radicals of I and I_{t}(A) are the same, we get that M has maximal grade if and only if A := R/I_{t}(A) is standard determinantal. In this case I = I_{t}(A), see [7] for details. If M ≅ R/I(−b_{1}) is cyclic (t = 1), we remark that a module of maximal grade is a complete intersection. The main results of this section is variations of the following

Theorem 3.1. Let M be a finitely generated graded module over R of maximal grade. Then M is unobstructed. Moreover if A := R/ann(M) is generically a complete intersection, then

dim _{0}Ext_{R}^{1}(M, M) = λ_{c} + K_{3} + K_{4} + ... + K_{c} and
depth Ext_{R}^{1}(M, M) ≥ dim A − 1 .

Remark 3.2. By deformation theory _{0}Ext_{R}^{1}(M, M) (resp. _{0}Ext_{R}^{2}(M, M)) is the tangent (resp. the natural obstruction) space of the local deformation functor, Def_{M/R}, of M as a graded R-module (e.g. [45]). Since c ≥ 2, _{0}Ext_{R}^{2}(M, M) is in many cases non-vanishing.

Remark 3.3. Note that the assumption on A in Theorem 3.1 is equivalent to assuming A good determinantal. By (2.2) good determinantal schemes exist if standard determinantal schemes exist. Hence if we take the polynomials f_{ij} of degrees a_{j} − b_{i} in a presentation matrix (f_{ij}) of M general enough, then the assumption on A in Theorem 3.1 is satisfied.

Remark 3.4. While distributing a preliminar version of a paper partially containing Theorem 3.1 to specialists in deformations of modules, we learned that the unobstructedness part of Theorem 3.1 (and hence of Theorem 3.8) was proved in R. Ile’s PhD thesis, cf. [22], ch. 6.

Proof. Let T ↠ S be a small in \underline{ℓ} and let M_{S} be any graded deformation of M to the artinian ring S. Let A = (f_{ij}) be a homogeneous matrix which represents φ^{∗}. Since G^{∗} ⟶^{φ^{∗}} F^{∗} → M → 0 is exact (cf. (2.1)), we have M_{S} = coker(φ_{S}^{∗}) where φ_{S}^{∗} corresponds to some matrix A_{S} = (f_{ij,S}), as in (2.12). Since T → S is surjective, we can lift each f_{ij,S} to a polynomial f_{ij,T} with coefficients in T such that f_{ij,T} ⊗_{T} S = f_{ij,S}. By Lemma 2.5, M_{T} := coker(φ_{T}^{∗}) is flat over T and since M_{T} ⊗_{T} S = M_{S} it follows that M is unobstructed.

To see the dimension formula we claim that there is an exact sequence

(3.1) 0 → _{0}Hom_{R}(M, M) → _{0}Hom_{R}(F^{∗}, M) → _{0}Hom_{R}(G^{∗}, M) → _{0}Ext_{R}^{1}(M, M) → 0.

Indeed look at the map d_{1} : ∧^{t+1}G^{∗}⊗S_{0}(F)⊗∧^{t}F → G^{∗} appearing in the Buchsbaum-Rim complex (2.4) and recall that the image of the corresponding map ∧^{t}G^{∗}⊗S_{0}(F)⊗∧^{t}F → R of the Eagon-Northcott complex (2.3) is the ideal I = ann(M) generated by the maximal minors. It follows that im d_{1} ⊂ I ·G^{∗} and hence that the induced map _{0}Hom_{R}(d_{1}, M) = 0. So if we apply _{0}Hom_{R}(−, M) to (2.4), we get (3.1) by the definition of _{0}Ext_{R}^{i}(M, M).

Let E = coker φ. Then we have an exact sequence

0 ⟶ E^{∗} ⟶ G^{∗} ⟶^{φ^{∗}} F^{∗} ⟶ M ⟶ 0 ,

to which we apply the exact functors _{0}Hom_{R}(F^{∗}, −) and _{0}Hom_{R}(G^{∗}, −). We get

0hom_{R}(G^{∗}, M) − _{0}hom_{R}(F^{∗}, M) = λ_{c} − 1 + _{0}hom_{R}(G^{∗}, E^{∗}) − _{0}hom_{R}(F^{∗}, E^{∗})

by using the definition (2.10) of λ_{c}. Note that _{0}hom(M, M) = 1 by [31], Lem. 3.2 since A is good determinantal by assumption. Hence we get the dimension formula of Theorem 3.1 from (3.1) provided we can prove

0hom_{R}(G^{∗}, E^{∗}) − _{0}hom_{R}(F^{∗}, E^{∗}) = K_{3} + ... + K_{c}.

By [31], Prop. 3.12 we have 1 + K_{3} + ... + K_{c} = _{0}hom_{R}(E, E) and by the proof of the same proposition we find _{0}hom_{R}(E, E) = 1 + _{0}hom_{R}(E, G) − _{0}hom_{R}(E, F) and whence we get the dimension formula.

Now we consider the depth of Ext_{R}^{1}(M, M). Firstly observe that it is straightforward to see depth Ext_{R}^{1}(M, M) ≥ dim A − 2. Indeed using Hom_{R}(M, M) ≃ A ([31], Lem. 3.2) and skipping the lower index 0 in (3.1), we get that all three Hom-modules in (3.1) are maximal CM A-modules. Then if D := coker(Hom_{R}(M, M) → Hom_{R}(F^{∗}, M)) we easily conclude since depth D ≥ dim A − 1 and depth Ext_{R}^{1}(M, M) ≥ depth D − 1 by [13], Cor. 18.6.

Looking more carefully at the argument, we can show depth Ext_{R}^{1}(M, M) ≥ dim A − 1. Indeed it suffices to prove depth D = dim A. To see it we use the resolution of A in (2.3) and the resolution of M ⊗ F deduced from (2.4). Let {f_{1}, f_{2}, ..., f_{t}} be the standard basis of F and {y_{1}, y_{2}, ...} the standard basis of G^{∗}. The leftmost free modules in these resolutions are ∧^{t+c−1}G^{∗} ⊗S_{c−1}(F)⊗∧^{t}F and ∧^{t+c−1}G^{∗} ⊗S_{c−2}(F)⊗∧^{t}F ⊗F respectively. We may consider the former as an R-submodule of the latter through the map τ_{c−1} where τ_{k} = id ⊗ τ′_{k} and τ′_{k} : S_{k}(F) → S_{k−1}(F) ⊗ F is induced by sending a symmetric tensor (f_{i_{1}} ⊗_{s}...⊗_{s}f_{i_{k}}) ∈ S_{k}(F) onto the “reduced sum” of ∑_{j=1}^{k}(f_{i_{1}} ⊗_{s}...⊗_{s} f̂_{i_{j}} ⊗_{s}...⊗_{s}f_{i_{k}})⊗f_{i_{j}} ∈ S_{k−1}(F)⊗F. Here “reduced” means sending e.g. (f_{1} ⊗_{s} f_{1} ⊗_{s} f_{1} ⊗_{s} f_{2} ⊗_{s} f_{2} ⊗_{s} f_{3}) ∈ S_{6}(F) onto

a(f_{1} ⊗_{s} f_{1} ⊗_{s} f_{2} ⊗_{s} f_{2} ⊗_{s} f_{3}) ⊗ f_{1} + b(f_{1} ⊗_{s} f_{1} ⊗_{s} f_{1} ⊗_{s} f_{2} ⊗_{s} f_{3}) ⊗ f_{2} + (f_{1} ⊗_{s} f_{1} ⊗_{s} f_{1} ⊗_{s} f_{2} ⊗_{s} f_{2}) ⊗ f_{3}

in S_{5}(F) ⊗ F with (a, b) = (1, 1) (and not (a, b) = (3, 2)!). Then τ_{1} = id and letting τ_{−1} : R → F^{∗} ⊗F be the obvious map and τ_{0} : ∧^{t}G^{∗} ⊗S_{0}(F)⊗∧^{t}F → G^{∗} ⊗F the map induced by sending (y_{i_{1}} ∧...∧y_{i_{t}}) ∈ ∧^{t}G^{∗} onto ∑_{j=1}^{t}(−1)^{j}(φ^{∗}(y_{i_{1}})∧...∧φ̂^{∗}(y_{i_{j}})∧...∧φ^{∗}(y_{i_{t}}))⊗y_{i_{j}} ∈ ∧^{t−1}F^{∗} ⊗ G^{∗} followed by the natural map ∧^{t−1}F^{∗} ⊗ G^{∗} → F ⊗ ∧^{t}F^{∗} ⊗ G^{∗}, one may check that the collection of maps {τ_{i}}_{i≥−1} is actually a map between the free resolutions of A and M ⊗ F. The explicit description in [25] of the differentials in the resolutions (2.3) and (2.4) may be helpful in checking that the diagrams between the resolutions commute. Now using the well known mapping cone construction we find a free resolution of D and since τ_{c−1} ⊗_{R} R/𝔪 is injective, the leftmost term ∧^{t+c−1}G^{∗} ⊗ S_{c−1}(F) ⊗ ∧^{t}F becomes redundant. The minimal R-free resolution of D has therefore the same length as the minimal R-free resolution of A, i.e. D is maximally CM and we are done. □

Remark 3.5. We see from the proof, or Lemma 2.5, that if we arbitrarily lift the polynomials in a presentation matrix of M to polynomials with coefficients in T, we get that M_{T} := coker(φ_{T}^{∗}) is flat over T. This is not true in general, but for modules of maximal grade it is because the Buchsbaum-Rim complex provides us with a resolution of M_{T}.

Remark 3.6. Let E = coker φ, φ = F → G cf. (2.1) and suppose R/I_{t}(A) is good determinantal. It is stated in [31], Rem. 3.14 that dim _{0}Ext_{R}^{1}(E, E) = λ_{c} + K_{3} + K_{4} + ... + K_{c}. Indeed one may use the proof above to see _{0}Ext_{R}^{i}(E, E) ≅ _{0}Ext_{R}^{i}(M, M) for i = 1 while this is not true in general for i ≠ 1.

Remark 3.7. The theorem admits a vast generalization since the assumption that R is a polynomial ring is not necessary. Indeed if R is any commutative graded (resp. local) k-algebra, then a module of maximal grade is unobstructed and the exact sequence (3.1) (resp. where the lower index 0 is removed) holds. In fact all we need for these parts in the proof is the existence and exactness of the Buchsbaum-Rim complex, which hold under almost no assumption on R (cf. [13], Appendix 2).

We will give the details in the graded case of what we claimed in Remark 3.7. This means that we will generalize to arbitrary modules of maximal grade the following wellknown fact for cyclic modules, that a complete intersection R/I is unobstructed and that 0hom_{R}(I, R/I) = ∑_{j=1}^{q} dim(R/I)_{(a_{j})} where ⊕_{j=1}^{q}R(−a_{j}) → I is a minimal surjection.

For the remaining part of this section we let R = ⊕_{v≥0}R_{v} be any graded k-algebra (k = R_{0} a not necessarily algebraically closed field), generated by finitely many elements from 𝔪 := ⊕_{v≥1}R_{v}. Let

(3.2)  G^{∗} := ∑_{j=1}^{q} R(−a_{j}) ⟶^{φ^{∗}} F^{∗} := ∑_{j=1}^{p} R(−b_{i}) ⟶^{π} M → 0

be a minimal presentation of M and suppose M is of maximal grade. Let N = ker π. It is known that the tangent space of the graded deformation functor Def_{M}(F^{∗}) which deforms the surjection π : F^{∗} → M to artinian k-algebras from \underline{ℓ}, using trivial deformations of F^{∗}, is isomorphic to _{0}Hom_{R}(N, M) and that _{0}Ext_{R}^{1}(N, M) contains all the obstructions of the graded deformations (we may deduce it from [37], Thm. 4.1.14 and Lem. 3.1.7, but [17], Prop. 5.1 and Cor. 5.2 and 5.3 is the classical reference since we here deal with the local deformation functor, adapted to graded deformations, of Grothendieck’s Quot scheme). If we apply _{0}Hom_{R}(−, M) to 0 → N → F^{∗} → M → 0, we get the exact sequence

(3.3) 0 → _{0}Hom_{R}(M, M) → _{0}Hom_{R}(F^{∗}, M) → _{0}Hom_{R}(N, M) → _{0}Ext_{R}^{1}(M, M) → 0 ,

and _{0}Ext_{R}^{1}(N, M) ≃ _{0}Ext_{R}^{2}(M, M). We notice that the arguments in the proof of Theorem 3.1 which led to _{0}Hom_{R}(d_{1}, M) = 0, where now d_{1} : ∧^{p+1}G^{∗} ⊗ S_{0}(F) ⊗ ∧^{p}F → G^{∗}, carry over to the general situation we are considering since they relied on how the maps in the Buchsbaum-Rim complex were defined. Hence we get the exact sequence (3.1), and comparing with (3.3), we get that the tangent space of Def_{M}(F^{∗}) is

0Hom_{R}(N, M) ≃ _{0}Hom_{R}(G^{∗}, M) ≃ ⨁_{j=1}^{q} M_{(a_{j})}.

Since the map d_{1} is defined in terms of φ^{∗}, the unobstructedness argument for M in the proof of Theorem 3.1 and the flatness argument of Lemma 2.5 both carry over the general case. Note that also the object π : F^{∗} → M is unobstructed, i.e. Def_{M}(F^{∗}) is formally smooth ([42], [37]) because π is easily deformed once M is deformed. Also the proof of the length of an R-free resolution of Ext_{R}^{1}(M, M) holds and we have

Theorem 3.8. Let M be a finitely generated graded module (as in (3.2)) of maximal grade over a finitely generated graded k-algebra R where R_{0} = k is an arbitrary field.

Let N := ker(F^{∗} → M). Then pd_{R} Ext_{R}^{1}(M, M) ≤ c + 1. Moreover M is unobstructed. Indeed Def_{M}(F^{∗}) is formally smooth and the dimension of the tangent space of Def_{M}(F^{∗}) is

dim _{0}Hom_{R}(N, M) = ∑_{j=1}^{q} dim M_{(a_{j})}.

Remark 3.9. Under the assumptions of Theorem 3.8, we see from the proof that

dim _{0}Ext_{R}^{1}(M, M) − dim _{0}Hom_{R}(M, M) = ∑_{j=1}^{q} dim M_{(a_{j})} − ∑_{i=1}^{p} dim M_{(b_{i})}.

Now suppose R is any graded Cohen-Macaulay quotient of a polynomial ring k[x_{0}, …, x_{n}] with the standard grading where k is any field. This will be the natural setting, having algebraic geometry in mind, to which we can generalize the theorems of this paper (sometimes assuming k = \overline{k} to be algebraically closed). Slightly generalizing [31], Lem. 3.2, we get that Hom_{R}(M, M) ≃ A if depth_{I_{t−1}(A)A} A ≥ 1 (cf. Remark 4.11). Hence 0hom_{R}(M, M) = 1 and the formula above gives an alternative to the formula of Theorem 3.1 for computing _{0}ext_{R}^{1}(M, M). In this general setting one may see that also the formula of Theorem 3.1 holds provided we redefine λ_{c} and K_{i} appearing in (2.10) and Conjecture 2.2 in the obvious way, namely by replacing all \binom{v+n}{n} with dim R_{v}. Indeed this follows from the proof of Theorem 3.1 since the part we use from [31] (Prop. 3.12) also generalize to this setting.

4. the rigidity of modules of maximal grade

In this section we consider a module M of maximal grade as a graded module over A = R/I where I = ann(M). Recall that I = I_{t}(A) where A = (f_{ij})_{i=1,...t}^{j=0,...,t+c−2} is a t × (t + c − 1) homogeneous presentation matrix of M, cf. (2.1), in which case we put M = M_{A}. A main result in this section is the rigidity of M as an A-module (i.e. Ext_{A}^{1}(M, M) = 0) provided X := Proj(A) is smooth of dimension greater or equal to 1. Furthermore if dim X ≥ 2 we also show Ext_{A}^{2}(M, M) = 0. More generally we have the following results.

Theorem 4.1. Let M be a finitely generated graded R-module of maximal grade and let A := R/ann(M). Let j ≥ 1 be an integer and suppose depth_{I_{t−1}(A)A} A ≥ j + 1. Then Hom_{A}(M, M) ≃ A and,

Ext_{A}^{i}(M, M) = 0 for 1 ≤ i ≤ j − 1.

Remark 4.2. Let X = Proj(A), J := I_{t−1}(A) and recall that depth_{JA} A = dim A − dim A/JA and V (JA) ⊂ Sing(X). We may therefore take j as j = codim_{X} Sing(X)−1 = dim X −dim Sing(X)−1 in Theorem 4.1, interpreting dim Sing(X) as −1 if Sing(X) = ∅.

Proof. Since M̃ is locally free of rank one over U := Proj(A) − V (I_{t−1}(A)A) (see the text before Remark 2.1), we can use (2.9) with L = N = M and i = 0. We get Hom_{A}(M, M) ≃ A. It follows, again by (2.9), that

(4.1)  Ext_{A}^{i}(M, M) ≃ H_{∗}^{i}(U, Hom(M̃, M̃)) ≃ H_{JA}^{i+1}(Hom_{A}(M, M)) = 0

for 0 < i < j, whence the result. □

Remark 4.3. We consider the vanishing of Ext_{A}^{i}(M, M) in Theorem 4.1 as mainly known (Schlessinger, see [26], Prop. 2.2.3 for i = 1) since it, as in [43] and [47], is rather clear how to generalize [26], Prop. 2.2.3 to a non-smooth X and to i > 1 (e.g. [47], Rem. 2.5). Our proof is, however, very short and uses more directly Grothendieck’s long exact sequence of Ext-groups appearing in [18], exp. VI, from which (2.9) is deduced.

Remark 4.4. It is clear from the proof that the theorem also holds for c = 1. In this case we can only use the result for j ≤ 2 because the largest possible value of depth_{I_{t−1}(A)A} A is 3. Thus our proof implies the known rigidity of M ([26], Prop. 2.2.3 and [23], Thm. 2). We continue to restrict ourselves to c ≥ 2 and refer to [23] for a nice study when c = 1.

In running some Macaulay 2 computations ([16]) in the situation of Theorem 4.1 we were surprised to see that also Ext_{A}^{i}(M, M) = 0 for i = j in the examples. This observation led us to try to prove Theorem 4.1 under the assumption depth_{I_{t−1}(A)A} A ≥ j. The natural case where this happens and where we succeed is as follows. Let B = R/I_{t}(B) and suppose depth_{JB} B ≥ j + 1 with J = I_{t−1}(B) where B is obtained by deleting a column of A. Then since I_{t}(A) ⊂ I_{t−1}(B) ⊂ I_{t−1}(A), it follows that depth_{JA} A ≥ j. Thus we may take j = codim_{Y} Sing(Y) − 1 = dim X − dim Sing(Y) in the following theorem.

Theorem 4.5. Let B → A be quotients of R defined by the vanishing of the maximal minors of B and A respectively where B is obtained by deleting some column of A. Let M be the finitely generated graded module over R of maximal grade defined by A, i.e. M := M_{A}, cf. (2.5). Let j ≥ 2 be an integer and suppose depth_{I_{t−1}(B)B} B ≥ j + 1. Then Hom_{A}(M, M) ≃ A and

Ext_{A}^{i}(M, M) = 0 for 1 ≤ i ≤ j − 1 .

Proof. Since M has maximal grade we get that A and hence B are standard determinantal rings (by [4] since we may suppose the matrix A is minimal). It follows that N := M_{B} has maximal grade and we can apply Theorem 4.1 to N. We get Ext_{B}^{i}(N, N) = 0 for 1 ≤ i ≤ j − 1. Moreover we have Hom_{A}(M, M) ≃ A by the proof of Theorem 4.1.

Now we consider the exact sequence

(4.2)  0 ⟶ B(−a_{t+c−2}) ⟶ N ⟶ M ⟶ 0

induced by (2.5) and we put B_{a} := B(−a_{t+c−2}). We claim that Ext_{B}^{1}(M, M) is isomorphic to Hom_{B}(B_{a}, M) ≃ M(a_{t+c−2}) and that Ext_{B}^{i}(M, M) = 0 for 2 ≤ i ≤ j − 1. To see it we apply Hom_{B}(−, M) and Hom_{B}(N, −) to (4.2). Their long exact sequences fit into the following diagram

(4.3)

↓
Ext_{B}^{i}(N, N)
↓
→ Ext_{B}^{i−1}(B_{a}, M) → Ext_{B}^{i}(M, M) → Ext_{B}^{i}(N, M) → Ext_{B}^{i}(B_{a}, M) → Ext_{B}^{i+1}(M, M) →
↓
Ext_{B}^{i+1}(N, B_{a})
↓

where Ext_{B}^{i}(B_{a}, M) = 0 for i > 0. To see that also Ext_{B}^{i+1}(N, B_{a}) = 0 for 1 ≤ i+1 ≤ j, we first notice that Ext_{B}^{i+1}(N, B_{a}) ≃ Ext_{B}^{i+1}(N ⊗K_{B}, K_{B}(−a_{t+c−2})) ([21], Satz 1.2). Moreover since K_{B}(n+1) ≃ S_{c−2}N(ℓ_{c−1}), cf. (2.7), we see that N⊗K_{B} and S_{c−1}N are closely related. Indeed up to twist they are isomorphic if we restrict to U_{B} := Proj(B) − V (I_{t−1}(B)B). Hence if we let Λ be the kernel of the natural surjective map S_{c−2}N ⊗_{B} N → S_{c−1}N, it follows that Supp Λ ⊂ V (I_{t−1}(B)B), e.g. we get Ext_{B}^{i+1}(Λ, K_{B}) = 0 for i + 1 ≤ j by the assumption depth_{I_{t−1}(B)B} B ≥ j + 1. Now we recall that S_{c−1}N is a maximal CM B-module ([13]). It follows that Ext_{B}^{i+1}(S_{c−1}N, K_{B}) = 0 for i + 1 > 0. Since the sequence

→ Ext_{B}^{i+1}(S_{c−1}N, K_{B}) → Ext_{B}^{i+1}(S_{c−2}N ⊗_{B} N, K_{B}) → Ext_{B}^{i+1}(Λ, K_{B}) →

is exact, we deduce that Ext_{B}^{i+1}(N, B_{a}) = 0 for 0 ≤ i ≤ j − 1. Then using the big diagram above we get the claim provided we can prove that the surjective map Hom_{B}(B_{a}, M) → Ext_{B}^{1}(M, M) is an isomorphism. To prove it we continue the horizontal sequence in the big diagram on the left hand side and we get the exact sequence

0 → Hom_{B}(M, M) → Hom_{B}(N, M) → Hom_{B}(B_{a}, M) → Ext_{B}^{1}(M, M) → 0 .

The two leftmost Hom-modules are easily seen to be isomorphic to A, e.g. Hom_{B}(N, M) ≃ H_{∗}^{0}(U_{B}, Hom(Ñ, M̃)) ≃ A because Ñ ⊗ O_{X} ≃ M̃ is invertible over U_{B} ∩ X and the claim is proved.

It remains to compare the groups Ext_{A}^{i}(M, M) and Ext_{B}^{i}(M, M) for which we have a well-known spectral sequence E_{2}^{p,q} := Ext_{A}^{p}(Tor_{q}^{B}(A, M), M), converging to Ext_{B}^{p+q}(M, M), at our disposal. Since E_{2}^{p,0} ≃ Ext_{A}^{p}(M, M) we must show

(4.4)  E_{2}^{p,0} = 0 for 1 ≤ p < j .

Noticing that I_{X/Y} ≃ N(a_{t+c−2})^{∗} by (2.6) in which Ñ(a_{t+c−2})|_{U_{B}} is an invertible O_{Y} - Module over U_{B} ⊂ Y, we get that the sheafification of Tor_{q}^{B}(A, M) ≃ Tor_{q−1}^{B}(I_{X/Y} , M) vanishes over U_{B} ∩ X for q ≥ 2. Hence Supp Tor_{q}^{B}(A, M) ⊂ V (I_{t−1}(B)A), and we get E_{2}^{p,q} = 0 for q ≥ 2 and p < j by the assumption depth_{I_{t−1}(B)B} B ≥ j + 1 which leads to depth_{I_{t−1}(B)A} A ≥ j. Note also that E_{2}^{p,1} = 0 for 0 < p < j − 1 because by (2.9),

E_{2}^{p,1} = Ext_{A}^{p}(I_{X/Y} ⊗_{B} M, M) ≃ H_{∗}^{p}(U, Hom_{O_{X}}(I_{X/Y} ⊗ M̃, M̃)) ≃ H_{I_{t−1}(B)A}^{p+1}M(a_{t+c−2}) = 0

where U := U_{B} ∩ X. Indeed I_{X/Y} ⊗ M̃ ≃ Ñ(a_{t+c−2})^{∗} ⊗_{O_{Y}} O_{X} ⊗ M̃ ≃ O_{X}(−a_{t+c−2}) over U. In the same way

E_{2}^{0,1} = Hom_{A}(I_{X/Y} ⊗_{B} M, M) ≃ H_{∗}^{0}(U, M̃(a_{t+c−2})) ≃ M(a_{t+c−2}) .

The spectral sequence leads therefore to an exact sequence

(4.5)  0 → E_{2}^{1,0} → Ext_{B}^{1}(M, M) → E_{2}^{0,1} → E_{2}^{2,0} → Ext_{B}^{2}(M, M) → 0

and to isomorphisms E_{2}^{p,0} ≃ Ext_{B}^{p}(M, M) for 2 < p < j. We already know Ext_{B}^{p}(M, M) = 0 for 2 ≤ p < j by the proven claim. Hence we get (4.4) provided we can show that the “pushforward morphism” Ext_{B}^{1}(M, M) → E_{2}^{0,1} ≃ M(a_{t+c−2}) of (4.5) is an isomorphism. Since it is clear that this morphism is compatible with the isomorphism Ext_{B}^{1}(M, M) ≃ Hom_{B}(B_{a}, M) ≃ M(a_{t+c−2}) which we proven in the claim (using (4.2)), we get the theorem. □

14 JAN O. KLEPPE

Remark 4.6. The proof of Theorem 4.5 even shows Ext_{B}^{i}(M, M) = 0 for 2 ≤ i ≤ j − 1 .

As we see, the proof of Theorem 4.5 is technically much more complicated that the proof of Theorem 4.1, even though we are only able to weaken the depth assumption on A in some cases (namely when the two algebras in R/I_{t−1}(B) ↠ R/I_{t−1}(A) have the same dimension). This improvement is, however, important in low dimensional cases in which the radical of I_{t−1}(B) often satisfies

(4.6)  𝔪 = \sqrt{I_{t−1}(B)}

and hence \sqrt{I_{t−1}(B)} = \sqrt{I_{t−1}(A)}. For short we say that we get an l.c.i. scheme by deleting some column if (4.6) holds. We immediately get from the theorems

Corollary 4.7. Let X = Proj(A), A = R/I_{t}(A) be a standard determinantal scheme, let M = M_{A} and suppose either depth_{I_{t−1}(A)A} A ≥ 3, or just dim X ≥ 1 provided we get an l.c.i. (e.g. a smooth) scheme by deleting some column of A. Then Hom_{A}(M, M) ≃ A and

Ext_{A}^{1}(M, M) = 0 .

Corollary 4.8. Let X = Proj(A), A = R/I_{t}(A) be a standard determinantal scheme, let M = M_{A} and suppose dim X ≥ 1. Moreover suppose the polynomials f_{ij} of degrees a_{j} − b_{i} in a presentation matrix (f_{ij}) of M are chosen general enough and suppose a_{i−2} ≥ b_{i} for 2 ≤ i ≤ t. Then Hom_{A}(M, M) ≃ A and Ext_{A}^{1}(M, M) = 0 .

Proof. We may suppose that the codimension of X in ℙ^{n} is c ≥ 3 since M is a twist of the canonical module of A if c = 2 in which case the conclusion is well known. Suppose dim X = 1. Then Remark 2.1 with α = 2 shows that both X = X_{c} and Y := X_{c−1} are smooth because X is general. If, however, dim X ≥ 2, then Remark 2.1 still applies to X = X_{c} and we get depth_{I_{t−1}(A)A} A ≥ 3. Hence in any case we conclude by Corollary 4.7. □

In deformation theory it is important to know when Ext_{A}^{2}(M, M) vanishes.

Corollary 4.9. Let X = Proj(A), A = R/I_{t}(A) be a standard determinantal scheme, let M = M_{A} and suppose either depth_{I_{t−1}(A)A} A ≥ 4, or just dim X ≥ 2 provided we get an l.c.i. (e.g. a smooth) scheme by deleting some column of A. Then Hom_{A}(M, M) ≃ A and

Ext_{A}^{i}(M, M) = 0 for i = 1 and 2 .

Proof. This follows immediately from Theorem 4.5 and Theorem 4.1 □

Corollary 4.10. Let X = Proj(A), A = R/I_{t}(A) be a standard determinantal scheme, let M = M_{A} and suppose dim X ≥ 2. Moreover suppose the polynomials f_{ij} of degrees a_{j} − b_{i} in a presentation matrix (f_{ij}) of M are chosen general enough and suppose a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t. Then Hom_{A}(M, M) ≃ A and

Ext_{A}^{i}(M, M) = 0 for i = 1 and 2 .

Proof. We may again suppose that c ≥ 3. Now if dim X = 2, then Remark 2.1 with α = 3 shows that both X = X_{c} and Y := X_{c−1} are smooth. If, however, dim X ≥ 3, then Remark 2.1 still applies to X = X_{c} and we get depth_{I_{t−1}(A)A} A ≥ 4. Thus in any case we conclude by Corollary 4.9. □

Remark 4.11. Also the results of this section admit substantial generalizations since the assumption that R is a polynomial ring is not necessary. For instance let R be any graded quotient of a polynomial ring k[x_{0}, …, x_{n}] with the standard grading where k is any field. In Theorem 4.1 it suffices to have depth_{I_{t−1}(A)A} M = depth_{I_{t−1}(A)A} A and the depth assumption of that theorem to see that the proof works (M̃|_{U} locally free of rank one holds in general by [5], Lem. 1.4.8). Moreover in Theorem 4.5, Corollary 4.7 and Corollary 4.9 we use a few places that R is Cohen-Macaulay in which case we get depth_{I_{t−1}(A)A} M = depth_{I_{t−1}(A)A} A by [13], Cor. A2.13. So all the mentioned results hold if Proj(R) is any ACM-scheme (i.e. R is CM). The remaining corollaries hold as well if Proj(R) is a smooth ACM scheme and k = \overline{k} by Remark 2.1. Indeed Remark 2.1 is really a result for determinantal subschemes of any smooth variety W, not only when W = ℙ^{n}.

5. deformations of modules and determinantal schemes

The main goal of this section is to show a close relationship between the local deformation functor, Def_{M/R}, of the graded R-module M = M_{A} and the corresponding local functor, Def_{A/R}, of deforming the determinantal ring A = R/ann(M) as a graded quotient of R. We will see that these functors are isomorphic (resp. the first is a natural subfunctor of the other) provided dim X ≥ 2 (resp. dim X = 1) and X = Proj(A) is general. If dim X = 1, the mentioned subfunctor is indeed the functor which corresponds to deforming the determinantal k-algebra A as a determinantal quotient of R (Definition 5.1). Combining with results of previous sections and the fact that Def_{A/R} is the same as the local Hilbert (scheme) functor of X if dim X ≥ 1 by (1.2), we get the main results of this paper; the dimension formula for W (\underline{b}; \underline{a}) and the generically smoothness of Hilb^{p}(ℙ^{n}) along W (\underline{b}; \underline{a}). The comparison is mostly to understand well a spectral sequence comparing the tangent and obstruction spaces of the mentioned deformation functors and to use the theorems of the previous sections. This spectral sequence is also important in R. Ile’s PhD thesis [22], and in his papers [23] and [24] (see Remark 5.4). In the following we suppose A is generically a complete intersection (depth_{I_{t−1}(A)A} A ≥ 1), i.e. that X = Proj(A) is a good determinantal scheme.

Consider the well-known spectral sequence

E_{2}^{p,q} := Ext_{A}^{p}(Tor_{q}^{R}(A, M), M) ⇒ Ext_{R}^{p+q}(M, M) ,

and note that E_{2}^{p,0} ≃ Ext_{A}^{p}(M, M) and Tor_{q}^{R}(A, M) ≃ Tor_{q−1}^{R}(I_{X}, M) for q ≥ 1. The spectral sequence leads to the following exact sequence

(5.1) 0 → Ext_{A}^{1}(M, M) → Ext_{R}^{1}(M, M) → E_{2}^{0,1} → Ext_{A}^{2}(M, M) → Ext_{R}^{2}(M, M) → E_{2}^{1,1} → .

Indeed E_{2}^{0,2} = Hom(Tor_{2}^{R}(A, M), M) = 0 because Tor_{2}^{R}(A, M) is supported in V (I_{t−1}(A)A). Moreover

E_{2}^{0,1} ≃ Hom_{A}(I_{X} ⊗_{R} M, M) ≃ Hom_{R}(I_{X}, Hom_{R}(M, M)) ,

and see [24], Def. 3 for an explicit description of Ext_{R}^{1}(M, M) → E_{2}^{0,1}. In our situation we recall that depth_{I_{t−1}(A)A} A ≥ 1 lead to Hom_{A}(M, M) ≃ A by [31], Lem. 3.2. It follows that the edge homomorphism Ext_{R}^{1}(M, M) → E_{2}^{0,1} of the spectral sequence above induces a natural map

(5.2)  _{0}Ext_{R}^{1}(M, M) ⟶ (E_{2}^{0,2})_{0} ≃ _{0}Hom_{R}(I_{X}, A)

16 JAN O. KLEPPE

between the tangent spaces of the two deformation functors Def_{M/R} and Def_{A/R} respectively. Even though we only partially use the spectral sequence in the proof below, Theorem 5.2 is fully motivated by the spectral sequence.

Definition 5.1. Let X = Proj(A), A = R/I_{t}(A), be a good determinantal scheme and let \underline{ℓ} be the category of artinian k-algebras (cf. the text before (2.12)). Then the local deformation functor Def_{A∈W (\underline{b};\underline{a})}, defined on \underline{ℓ}, is the subfunctor of Def_{A/R} given by:

Def_{A∈W (\underline{b};\underline{a})}(T) = \{A_{T} ∈ Def_{A/R}(T)|A_{T} = R_{T} /I_{t}(A_{T}) for some matrix A_{T} lifting A to T \} .

Note that there is a natural map Def_{M/R} → Def_{A∈W (\underline{b};\underline{a})} because for every graded deformation M_{T} of M to T there exists a matrix A_{T} whose induced morphism has M_{T} as cokernel (see the first part of the proof of Theorem 3.1) and because different matrices inducing the same M_{T} define the same ideal of maximal minors (Fittings lemma, [13], Cor. 20.4). The map is surjective since we can use the matrix A_{T} in Definition 5.1 to define M_{T} ∈ Def_{M/R}(T).

The phrase “for some matrix A_{T} lifting A to T ” which means that there exists a homogeneous matrix A_{T} lifting A to T, may be insufficient for forcing Def_{A∈W (\underline{b};\underline{a})} to have nice properties. For instance we do not know whether Def_{A∈W (\underline{b};\underline{a})} is pro-representable, or even has a hull, since we have no proof for the surjectivity of

(5.3)  Def_{A∈W (\underline{b};\underline{a})}(T_{1} ×_{S} T_{2}) ⟶ Def_{A∈W (\underline{b};\underline{a})}(T_{1}) ×_{Def_{A∈W (\underline{b};\underline{a})}(S)} Def_{A∈W (\underline{b};\underline{a})}(T_{2})

for every pair of morphisms T_{i} → S, i = 1, 2, in \underline{ℓ} with T_{2} ↠ S small (see Schlessinger’s main theorem in [42]). In [44] Schaps solves a related problem by assuming that A has the unique lifting property and she gets some results on the existence of a hull for determinantal non-embedded deformations. In our context, assuming _{0}Ext_{A}^{1}(M, M) = 0, then we shall see that Def_{A∈W (\underline{b};\underline{a})} behaves well because for every element of Def_{A∈W (\underline{b};\underline{a})}(T) there exists a unique module M_{A_{T}} even though A_{T} is not unique.

Indeed let D := k[ϵ]/(ϵ^{2}) be the dual numbers and let

λ := dim _{0}Ext_{R}^{1}(M, M) = λ_{c} + K_{3} + K_{4} + ... + K_{c} ,

cf. Theorem 3.1. Recalling that W (\underline{b}; \underline{a}) is a certain quotient of an open irreducible set in the affine scheme 𝕍 = Hom_{O_{ℙ^{n}}}(G^{∗}, F^{∗}) whose rational points correspond to t × (t + c − 1) matrices and that dim W (\underline{b}; \underline{a}) ≤ λ ([31], p. 2877 and Thm. 3.5), we get

Theorem 5.2. Let X = Proj(A), A = R/I_{t}(A) be a good determinantal scheme. If 0Ext_{A}^{1}(M, M) = 0 then the functor Def_{A∈W (\underline{b};\underline{a})} is pro-representable, the pro-representing object has dimension dim W (\underline{b}; \underline{a}) and

Def_{A∈W (\underline{b};\underline{a})} ≃ Def_{M/R} .

Hence Def_{A∈W (\underline{b};\underline{a})} is formally smooth. Moreover the tangent space of Def_{A∈W (\underline{b};\underline{a})} is the subvector space of _{0}Hom_{R}(I_{X}, A) which corresponds to graded deformation R_{D} → A_{D} of R → A to D of the form A_{D} = R_{D}/I_{t}(A_{D}) for some matrix A_{D} which lifts A to D.

If in addition _{0}Ext_{A}^{2}(M, M) = 0, then Def_{M/R} ≃ Def_{A∈W (\underline{b};\underline{a})} ≃ Def_{A/R} and Def_{A/R} is formally smooth. Moreover every deformation of X comes from deforming A (cf. Definition 2.6).

Proof. We already know that Def_{M/R}(T) → Def_{A∈W (\underline{b};\underline{a})}(T) is well defined and surjective. To see that it is injective, we will construct an inverse. Suppose therefore that there are two matrices (A_{T})_{1} and (A_{T})_{2} lifting A to T and such that I_{t}((A_{T})_{1}) = I_{T} ((A_{T})_{2}). The two matrices define two graded deformations M_{1} and M_{2} of the R-modules M to R_{T} by Lemma 2.5. Since, however, the two matrices define the same graded deformation A_{T} := R_{T} /I_{T} ((A_{T})_{1}) of A to T, we get that M_{1} and M_{2} are two graded deformations of the A-module M to A_{T}! Due to _{0}Ext_{A}^{1}(M, M) = 0, Hom_{A}(M, M) ≃ A and deformation theory, we conclude that M_{1} = M_{2} up to multiplication with a unit of T, i.e. we get a well defined map which clearly is an inverse.

Since we have Def_{A∈W (\underline{b};\underline{a})} ≃ Def_{M/R} and we know that Def_{M/R} has a hull ([45]), it follows that Def_{A∈W (\underline{b};\underline{a})} has a hull (or one may easily show the surjectivity of (5.3) directly by using the uniqueness of M_{A_{T}}). Note that the injectivity of (5.3) follows from Def_{A∈W (\underline{b};\underline{a})} being a subfunctor of the pro-representable functor Def_{A/R} ([28], Prop. 9), whence Def_{A∈W (\underline{b};\underline{a})} is pro-representable by [42]. Moreover using Def_{A∈W (\underline{b};\underline{a})} ≃ Def_{M/R} and Theorem 3.1 we get that Def_{A∈W (\underline{b};\underline{a})} is formally smooth and that dim H = λ where H is the pro-representing object of Def_{A∈W (\underline{b};\underline{a})}. The description of its tangent space follows from Definition 5.1 and (5.1) -(5.2) since _{0}Hom_{R}(I_{X}, A) is the tangent space of Def_{A/R}.

So far we know dim W (\underline{b}; \underline{a}) ≤ λ = dim H. To see that dim H = dim W (\underline{b}; \underline{a}), it suffices to see that the family of determinantal rings over H, corresponding to the “universal object” of Def_{A∈W (\underline{b};\underline{a})}, is algebraizable. This is clear in our context, (see the explicit description of H in the proof of [37], Thm. 4.2.4). Indeed take λ independent elements of 0Ext_{R}^{1}(M, M) ≃ Def_{A∈W (\underline{b};\underline{a})}(D), let A + ϵA_{1}, ..., A + ϵA_{λ} be corresponding presentation matrices of the elements (i.e. modules), and let A_{T} := A + t_{1}A_{1} + ... + t_{λ}A_{λ} (linear combination in the parameters t_{k}) where T be the polynomial ring T = k[t_{1}, ..., t_{λ}]. Then the algebraic family A_{T} := R_{T} /I_{t}(A_{T}) is T -flat at (0, ..., 0) ∈ Spec(T) (Lemma 2.5) and hence flat in a neighborhood and we get what we want.

Finally we suppose _{0}Ext_{A}^{2}(M, M) = 0. Using Def_{A∈W (\underline{b};\underline{a})}(D) ≃ _{0}Ext_{R}^{1}(M, M) and the spectral sequence (5.1) we get isomorphisms Def_{A∈W (\underline{b};\underline{a})}(D) ≃ _{0}Hom_{R}(I_{X}, A) ≃ Def_{A/R}(D) of tangent spaces. To show Def_{A∈W (\underline{b};\underline{a})}(T) ≃ Def_{A/R}(T) for any (T, 𝔪_{T}) in \underline{ℓ}, we may by induction suppose 𝔪_{T}^{r+1} = 0 and Def_{A∈W (\underline{b};\underline{a})}(T/𝔪_{T}^{r}) ≃ Def_{A/R}(T/𝔪_{T}^{r}). Consider the commutative diagram

(5.4)

Def_{A∈W (\underline{b};\underline{a})}(T) ↪ Def_{A/R}(T)
↓  ↓
Def_{A∈W (\underline{b};\underline{a})}(T/𝔪_{T}^{r}) ≃ Def_{A/R}(T/𝔪_{T}^{r})

and notice that the leftmost vertical map is surjective since Def_{A∈W (\underline{b};\underline{a})} is formally smooth. Hence for a given A_{T} ∈ Def_{A/R}(T) there exists A′_{T} ∈ Def_{A/R}(T) such that A′_{T} ≃ R_{T} /I_{t}(A_{T}) for some matrix A_{T} which lifts a matrix A_{T/𝔪_{T}^{r}} defining A_{T} ⊗_{T} T/𝔪_{T}^{r} to T. The difference of A_{T} and A′_{T} belongs to Def_{A/R}(D) ⊗_{k} 𝔪_{T}^{r} ≃ Def_{A∈W (\underline{b};\underline{a})}(D) ⊗_{k} 𝔪_{T}^{r}, and “adding” it to A′_{T} we get that A_{T} ∈ Def_{A∈W (\underline{b};\underline{a})}(T), whence Def_{A∈W (\underline{b};\underline{a})}(T) ≃ Def_{A/R}(T). It follows that the completion of the local ring O_{Hilb,(X)} of Hilb^{p}(ℙ^{n}) at (X) is isomorphic to H. Since we in the preceding paragraph explicitly constructed an algebraic determinantal family over some neighborhood of (0, ..., 0) in Spec(k[t_{1}, ..., t_{λ}]) (thinking about it, we must have O_{Hilb,(X)} ≃ k[t_{1}, ..., t_{λ}]_{(t_{1},...,t_{λ})} since k = \overline{k}), we get that “every deformation of X comes from deforming A“ and we are done. □

18 JAN O. KLEPPE

Remark 5.3. Let us endow the closed subset \overline{W (\underline{b}; \underline{a})} of Hilb^{p}(ℙ^{n}) with the reduced scheme structure (this is natural since “the part W (\underline{b}; \underline{a}) of Hilb^{p}(ℙ^{n}) is unobstructed” by the proof of Lemma 2.7). Let X = Proj(A), A = R/I_{t}(A) belong to W (\underline{b}; \underline{a}). Then the proof related to dim W (\underline{b}; \underline{a}) = λ above imply that the Zariski tangent space, (𝔪_{W} /𝔪_{W}^{2} )^{∨}, of W (\underline{b}; \underline{a}) satisfies

(5.5)  (𝔪_{W} /𝔪_{W}^{2} )^{∨} = Def_{A∈W (\underline{b};\underline{a})}(D) .

In the proof we used dim W (\underline{b}; \underline{a}) ≤ λ ([31], Thm. 3.5) to show dim W (\underline{b}; \underline{a}) = λ. We will now explain this inequality by a direct argument. Indeed take any (X′) ∈ W (\underline{b}; \underline{a}). Then there is a matrix t×(t+c−1) matrix A′ whose maximal minors define X′. By Lemma 2.5 the matrix A + x(A′ − A), x a parameter, defines a flat family of good determinantal schemes over some open set U ⊂ Spec(k[x]) ≃ 𝔸^{1} containing x = 0 and x = 1. Thus to any (X′) ∈ W (\underline{b}; \underline{a}) there is a tangent direction, i.e. an element t_{X} of (𝔪_{W} /𝔪_{W}^{2} )^{∨} ⊂ 0Hom_{R}(I_{X}, A) = Def_{A/R}(D), given by the matrix A′ − A. By Definition 5.1, t_{X} ∈ Def_{A∈W (\underline{b};\underline{a})}(D), thus (𝔪_{W} /𝔪_{W}^{2} )^{∨} ⊂ Def_{A∈W (\underline{b};\underline{a})}(D) by the relationship between W (\underline{b}; \underline{a}) and its Zariski tangent space. Taking dimensions we have shown dim W (\underline{b}; \underline{a}) ≤ λ. Then the proof of Theorem 5.2 implies dim W (\underline{b}; \underline{a}) = λ and hence we get (5.5).

Remark 5.4. If the assumption _{0}Ext_{A}^{1}(M, M) = 0 of Theorem 5.2 is not satisfied, then the local deformation functor Def_{M/A} of deforming M as a graded A-module and its connection to Def_{M/R} may be quite complicated, see [24] which compares the corresponding non-graded functors using (5.1). However, by the results of the preceding section, Ext_{A}^{1}(M, M) = 0 and Hom_{A}(M, M) ≃ A are week assumptions for modules of maximal grade.

We now deduce the main theorems of the paper. In the first theorem we let

ext^{2}(M, M) := dim ker( _{0}Ext_{A}^{2}(M, M) → _{0}Ext_{R}^{2}(M, M) ) , cf. (5.1),

and notice that we write Hilb(ℙ^{n}) for Hilb^{p}(ℙ^{n}) (resp. GradAlg(H)) if n − c ≥ 1 (resp. n − c = 0), cf. the text accompanying (1.2) for explanations and notations.

Theorem 5.5. Let X = Proj(A) ⊂ ℙ^{n}, A = R/I_{t}(A) be a good determinantal scheme of W (\underline{b}; \underline{a}) of dimension n − c ≥ 0, let M = M_{A} and suppose _{0}Ext_{A}^{1}(M, M) = 0. Then

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

Moreover, for the codimension of W (\underline{b}; \underline{a}) in Hilb(ℙ^{n}) in a neighborhood of (X) we have

dim_{(X)} Hilb(ℙ^{n}) − dim W (\underline{b}; \underline{a}) ≤ ext^{2}(M, M) ,

with equality if and only if Hilb(ℙ^{n}) is smooth at (X). In particular these conclusions hold if depth_{I_{t−1}(A)A} A ≥ 3, \underline{or} if n − c ≥ 1 and we get an l.c.i. (e.g. a smooth) scheme by deleting some column of A.

Proof. This follows from Theorem 5.2, Theorem 3.1, (5.1) -(5.2) and Corollary 4.7. □

Corollary 5.6. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we suppose n − c ≥ 1 and a_{i−2} − b_{i} ≥ 0 for 2 ≤ i ≤ t. Then

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

provided dim W (\underline{b}; \underline{a}) ≠ ∅. In particular Conjecture 4.1 of [32] holds in the case n − c ≥ 1.

Proof. This follows from Theorem 5.5 and Corollary 4.8 since Conjecture 4.1 of [32] is Conjecture 2.2 of this paper (and remember that we always suppose c ≥ 2 and t ≥ 2). □

Remark 5.7. Even for zero-dimensional determinantal schemes (n − c = 0) the assumption _{0}Ext_{A}^{1}(M, M) = 0 seems very week, and hence we almost always have the conjectured value of dim W (\underline{b}; \underline{a}). Thus Theorem 5.5 completes Theorem 4.19 of [29] in the zero-dimensional case. Indeed in computing many examples by Macaulay 2 we have so far only found _{0}Ext_{A}^{1}(M, M) ≠ 0 for examples outside the range of Conjecture 2.2.

Note that \overline{W (\underline{b}; \underline{a})} is not always an irreducible component of Hilb(ℙ^{n}). An example showing this was given in [30], Ex. 10.5, and many more were found in [29], Ex. 4.1, in which there are examples for every c ≥ 3 (the matrix is linear except for the last column). All examples satisfy n − c ≤ 1. Indeed [29] contains exact formulas for the codimension of \overline{W (\underline{b}; \underline{a})} in Hilb(ℙ^{n}) under some assumptions. Further investigations in [32] led us to conjecture that \overline{W (\underline{b}; \underline{a})} is an irreducible component provided n − c ≥ 2. Now we can prove it!

Theorem 5.8. Let X = Proj(A) ⊂ ℙ^{n}, A = R/I_{t}(A) be a good determinantal scheme of W (\underline{b}; \underline{a}) of dimension n − c ≥ 1, let M = M_{A} and suppose _{0}Ext_{A}^{i}(M, M) = 0 for i = 1 and 2. Then the Hilbert scheme Hilb^{p}(ℙ^{n}) is smooth at (X),

dim_{(X)} Hilb^{p}(ℙ^{n}) = λ_{c} + K_{3} + K_{4} + ... + K_{c} ,

and every deformation of X comes from deforming A. In particular this conclusion holds if depth_{I_{t−1}(A)A} A ≥ 4, \underline{or} if n − c ≥ 2 and we get an l.c.i. (e.g. a smooth) scheme by deleting some column of A.

Proof. This follows immediately from Theorem 5.2, Theorem 3.1 and Corollary 4.9. □

Corollary 5.9. Given integers a_{0} ≤ a_{1} ≤ ... ≤ a_{t+c−2} and b_{1} ≤ ... ≤ b_{t}, we suppose n − c ≥ 1, a_{i−2} − b_{i} ≥ 0 for 2 ≤ i ≤ t and _{0}Ext_{A}^{2}(M, M) = 0 for a general X = Proj(A) of W (\underline{b}; \underline{a}). Then the closure \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of the Hilbert scheme Hilb^{p}(ℙ^{n}) of dimension

λ_{c} + K_{3} + K_{4} + ... + K_{c} .

In particular this conclusion holds if n − c ≥ 2, a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t and dim W (\underline{b}; \underline{a}) ≠ ∅. It follows that Conjecture 4.2 of [32] holds.

Proof. This follows from Corollary 4.8, Theorem 5.8, Lemma 2.7 and Corollary 4.10, and note that Conjecture 4.2 of [32] is the same as Conjecture 2.4 of this paper. □

Even in the one-dimensional case (n − c = 1) the assumption _{0}Ext_{A}^{2}(M, M) = 0 seems rather week, and we can often conclude as in Corollary 5.9. Note that if _{0}Ext_{A}^{2}(M, M) = 0 for a general X of W (\underline{b}; \underline{a}) and a_{i−2} − b_{i} ≥ 0 for 2 ≤ i ≤ t, we get

(5.6)  _{0}hom_{R}(I_{X}, A) = λ_{c} + K_{3} + K_{4} + ... + K_{c}

by Corollary 4.8 and (5.1) -(5.2). So one may alternatively compute _{0}hom_{R}(I_{X}, A) and check if (5.6) holds, to conclude as in Corollary 5.9 (cf. Theorem 5.2). In [29], Prop. 4.15 (which generalizes [30], Cor. 10.15) we gave several criteria for determining \overline{W (\underline{b}; \underline{a})} in the one-dimensional case. None of them apply in the following example.

20 JAN O. KLEPPE

Example 5.10 (determinantal curves in ℙ^{4}, i.e. with c = 3).

Let A = (f_{ij}) be a 2 × 4 matrix whose entries are general polynomials of the same degree f_{ij} = 2. The vanishing of all 2 × 2 minors of A defines a smooth curve X of degree 32 and genus 65 in ℙ^{4}. A Macaulay 2 computation shows _{0}Ext_{A}^{2}(M, M) = 0. It follows from Corollary 5.9 that \overline{W (\underline{b}; \underline{a})} is a generically smooth irreducible component of Hilb^{p}(ℙ^{4}) of dimension λ_{3} + K_{3} = 101.

Note that our previous method was to delete a column to get a matrix B and an algebra B := R/J, J := I_{t}(B) and to verify _{0}Ext_{B}^{1}(J/J^{2}, I/J) = 0 with I = I_{t}(A). However, by Macaulay 2, _{0}Ext_{B}^{1}(J/J^{2}, I/J) as well as _{0}Ext_{A}^{1}(I/I^{2}, A), are 5-dimensional and the approach of using Prop. 4.15 (i) does not apply (since _{0}Ext_{B}^{1}(I/J, A) ≠ 0), neither do Prop. 4.15 (ii) nor (iii).

It is known that the vanishing of the cohomology group H^{1}(N_{X}) (resp. Ext_{A}^{1}(I_{X}/I_{X}^{2}, A)) of a locally (resp. generically) complete intersection X ↪ ℙ implies that X is unobstructed, and that the converse is not true, e.g. we may have H^{1}(N_{X}) ≠ 0 for X unobstructed. Since we by Theorem 5.8 get that X is unobstructed by mainly assuming n − c ≥ 2, one may wonder if we can prove a little more, namely H^{1}(N_{X}) = 0. Indeed we can if n − c ≥ 3. More precisely recalling depth_{J} A = dim A − dim A/J we have

Theorem 5.11. Let X = Proj(A) ⊂ ℙ^{n}, A = R/I_{t}(A) be a standard determinantal scheme.

(i) If depth_{I_{t−1}(A)A} A ≥ 4 or equivalently, dim X ≥ 3 + dim R/I_{t−1}(A), then

Ext_{A}^{i}(I_{X}/I_{X}^{2}, A) = 0 for 1 ≤ i ≤ dim X − 2 − dim R/I_{t−1}(A), and

H^{i}(N_{X}(v)) = 0 for 1 ≤ i ≤ dim X − 2 and every v.

(ii) In particular if dim X ≥ 3, a_{i−min(3,t)} ≥ b_{i} for min(3, t) ≤ i ≤ t and X is general in W (\underline{b}; \underline{a}), then conclusions of (i) hold. If furthermore a_{j} ≥ b_{i} for every j and i, then

Ext_{A}^{i}(I_{X}/I_{X}^{2}, A) = 0 for 1 ≤ i ≤ min{dim X − 2, c − 1} .

Proof. (i) Using (5.1) -(5.2) and Corollary 4.9 we get Ext_{R}^{1}(M, M) ≃ Hom_{R}(I_{X}, A). It follows that

(5.7)  depth Hom_{R}(I_{X}, A) ≥ dim X

by Theorem 3.1. Thus the local cohomology group H_{𝔪}^{i}(Hom_{R}(I_{X}, A)) vanishes for i < dim X. Recalling that the sheafification of Hom_{R}(I_{X}, A) ≃ Hom_{A}(I_{X}/I_{X}^{2}, A) is N_{X}, we get H_{∗}^{i}(N_{X}) = 0 for 1 ≤ i < dim X − 1, whence we have the second vanishing of (i).

Next let r := depth_{J} A − 1 where J is the ideal I_{t−1}(A)A of A. It is known that (5.7) also implies depth_{J} Hom_{R}(I_{X}, A) ≥ r (e.g. [33], Lem. 28). Thus the local cohomology group H_{J}^{i}(Hom_{A}(I_{X}/I_{X}^{2}, A)) vanishes for i < r and we get the first vanishing of (i) by (2.9) (letting N = I_{X}/I_{X}^{2} and L = A).

(ii) Finally we use Remark 2.1 with α = 3 (resp. α ≥ (c+3)/2) to see that depth_{J} A ≥ 4 (resp. codim_{X} V (J) ≥ c + 2). In particular (i) applies to get the first statement. For the final statement, we recall the well known fact that c + 2 is the largest possible value of the height of J in A, whence codim_{X} V (J) = c + 2 with the usual interpretation that c + 2 = codim_{X} V (J) > dim X implies V (J) = ∅. This implies the theorem. □

For the algebra cohomology groups H^{i}(R, A, A) of André-Quillen (cf. [1]) we deduce

Corollary 5.12. Let A = R/I_{t}(A) be a standard determinantal graded k-algebra. ( i) If depth_{I_{t−1}(A)A} A ≥ 4 then

H^{i}(R, A, A) = 0 for 2 ≤ i ≤ depth_{I_{t−1}(A)A} A − 2 .

( ii) If dim A ≥ 4, a_{j} ≥ b_{i} for every j, i and Proj(A) is general in W (\underline{b}; \underline{a}), then

H^{i}(R, A, A) = 0 for 2 ≤ i ≤ min{dim A − 2, c} .

Proof. The spectral sequence relating algebra cohomology to algebra homology ([1], Prop 16.1 or [37]), implies, under the sole assumption depth_{I_{t−1}(A)A} A ≥ 1, that

Ext_{A}^{i}(I_{X}/I_{X}^{2}, A) ≃ H^{i+1}(R, A, A) .
□

Remark 5.13. (i) The vanishing of H_{∗}^{i}(N_{X}) of Theorem 5.11 is known if c = 3 ([33], Lem. 35) or c = 4 ([31], Cor. 5.5). It c = 2 even more is true by [9] (or see [30], Cor. 6.5).

(ii) Note that Corollary 5.12 for so-called generic determinantal schemes is proved by Svanes (see [6], Thm. 15.10) while [2], (1.4.3) shows the corollary for some non-generic determinantal schemes as well.

(iii) As for c = 2 one may hope that H_{∗}^{i}(N_{X}) = 0 also for i = dim X − 1. This is not true, as one may see through examples, using e.g. Macaulay 2. We have checked it for some surfaces in the range 3 ≤ c < 6 and always found it to be non-zero (cf. [6], 15.11).

Remark 5.14. In proving Theorem 5.11 we used Corollary 4.10 to see that not only 0Ext_{A}^{i}(M, M) vanishes for i = 1, 2, but in fact that the whole Ext_{A}^{i}(M, M)-group vanishes for i = 1 and 2. Arguing as in Theorem 5.2 and using the vanishing of the whole Ext_{A}^{i}(M, M)-group for i = 1 and 2, we may see that the non-graded deformation functor;

Def_{A/R}^{non−gr}(T) = {R_{T} → A_{T} |A_{T} is T −flat and A_{T} ⊗_{T} k ≃ A}

in which a deformation A_{T} of R → A to an artinian T in \underline{ℓ} is possibly non-graded, is formally smooth provided depth_{I_{t−1}(A)A} A ≥ 4, \underline{or} dim X ≥ 2 and we get an l.c.i. scheme by deleting some column of A. This result is the best possible with regard to dim X ≥ 2 because one knows that Def_{A/R}^{non−gr} is non-smooth for a one-dimensional rational normal scroll Proj(A) ⊂ ℙ^{n} for n ≥ 4 ([41]). Note also that we may deduce the result above for generic determinantal schemes satisfying dim X ≥ 3 by works of Svanes ([6], Thm. 15.10)

Remark 5.15. The results so far of this section admit substantial generalizations with respect to R being a polynomial ring. Indeed we may let R be any graded CM quotient of a polynomial ring k[x_{0}, …, x_{n}], k = \overline{k}, with the standard grading provided we in all results replace ℙ^{n} by Proj(R) and interpret the assumption “A good determinantal” by “A standard determinantal satisfying depth_{I_{t−1}(A)A} A ≥ 1” ([30], Prop. 3.2). Then the proof of Theorem 5.2 works (we need Remark 5.3) since we have Hom_{R}(M, M) ≃ A by Remark 3.9. Using Remark 4.11 we get that Theorem 5.5, Theorem 5.8 and Theorem 5.11(i) are valid in this generality while it for the corollaries and Theorem 5.11 (ii) suffices to suppose that Proj(R) is a smooth ACM-scheme (in the case c > 2, see the next theorem for c = 2). Note that the assumption k = \overline{k} allows us to keep the definition W (\underline{b}; \underline{a}) as a certain locus in Hilb^{p}(ℙ^{n}).

Finally we will illustrate the results mentioned in the last remark to see that, in addition to reproving and generalizing Ellingsrud’s codimension 2 result ([14]) a little, we can enlighten the differences between the cases c = 2 and c > 2. Indeed the main ingredient is that if c = 2 and X = Proj(A) is standard determinantal in an ACM scheme Y = Proj(R), then M ≃ K_{A}(s) for some integer s where K_{A} is the canonical module of A (cf. the line before (2.7)). It follows that we do not need the results of section 4 at all to conclude that _{0}Ext_{A}^{i}(M, M) = 0 for i > 0 because this is well known. Moreover in section 5 we needed the weak assumption depth_{I_{t−1}(A)A} A ≥ 1 to get Hom_{A}(M, M) ≃ A which was central in (5.1) -(5.2) and hence in the proof of Theorem 5.2. Now this isomorphism always holds, again by M ≃ K_{A}(s), and we get Def_{M/R} ≃ Def_{A/R} without requiring depth_{I_{t−1}(A)A} A ≥ 1. These functor are formally smooth (Theorem 3.1, Remarks 3.2 and 3.9) and we deduce the theorem below where we interpret Hilb(Y) as Hilb^{p}(Y) (resp. GradAlg(H)) if dim X ≥ 1 (resp. dim X = 0) as in Theorem 5.5. Notice that we now deal with standard determinantal schemes X of codimension 2 in Y = Proj(R) (they are usually not determinantal schemes in ℙ^{n}). With \underline{b}, \underline{a} as in (3.2) and X ∈ W_{s}(\underline{b}; \underline{a}) we get

Theorem 5.16. Let Y = Proj(R) ⊂ ℙ_{k}^{n} be an ACM scheme where k is any field and let X = Proj(A) ⊂ Y, A = R/I_{t}(A), be any standard determinantal scheme of codimension 2 in Y. Then Hilb(Y) is smooth at (X) and dim_{(X)} Hilb(Y) = λ(R)_{2} where

λ(R)_{2} := ∑_{i,j} dim R_{(a_{i}−b_{j})} + ∑_{i,j} dim R_{(b_{j}−a_{i})} − ∑_{i,j} dim R_{(a_{i}−a_{j})} − ∑_{i,j} dim R_{(b_{i}−b_{j})} + 1.

Moreover every deformation of X comes from deforming A. In particular if k = \overline{k}, then Hilb(Y) is smooth along W_{s}(\underline{b}; \underline{a}) and the closure \overline{W_{s}(\underline{b}; \underline{a})} in Hilb(Y) is an irreducible component of dimension λ(R)_{2}.

Indeed there are no singular points (X) of Hilb(Y), (X) ∈ W_{s}(\underline{b}; \underline{a}) while singular points of Hilb(Y) for c > 2 at (X) ∈ W_{s}(\underline{b}; \underline{a}) are quite common (see [38] and Rem. 3.6 of [32]).

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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