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Families of low dimensional determinantal schemes

A scheme X \subset \PP^{n} of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t x t minors of a homogeneous t x (t+c-1) matrix (f_{ij}). Given integers a_0 \le a_1 \le ...\le a_{t+c-2} and b_1 \le ...\le b_t, we denote by W_s(b;a) \subset Hilb(\PP^{n}) the stratum of standard determinantal schemes where f_{ij} are homogeneous polynomials of degrees a_j-b_i and Hilb(\PP^{n}) is the Hilbert scheme (if n-c>0, resp. the postulation Hilbert scheme if n-c = 0). Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of W_s(b;a) in Hilb(\PP^{n}) and we show that Hilb(\PP^{n}) is generically smooth along W_s(b;a) under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of dim W_s(b;a) appearing in [26].


Introduction
The goal of this paper is to study maximal families of determinantal schemes. Recall that a scheme X ⊂ P n of codimension c is called determinantal if its homogeneous saturated ideal can be generated by the r × r minors of a homogeneous p × q matrix (f ij ) with c = (p − r + 1)(q − r + 1). If r = min(p, q), then X is called standard determinantal. X is called good determinantal if it is standard determinantal and a generic complete intersection.
Let Hilb(P n ) be the Hilbert scheme (resp. postulation Hilbert scheme, i.e. the Hilbert scheme of constant Hilbert function) parameterizing closed subschemes of P n of dimension n − c > 0 (resp. n − c = 0). Given integers a 1 ≤ a 2 ≤ ... ≤ a p and b 1 ≤ ... ≤ b q , we denote by W (b; a) (resp. W s (b; a)) the stratum in Hilb(P n ) consisting of good (resp. standard) determinantal schemes where f ij are homogeneous polynomials of degrees a j − b i . Then In this paper we focus, notably for zero dimensional schemes, on the following problems. This paper generalizes and completes several results of [25] and [26] for schemes of dimension 0 or 1. Moreover we announced in [26], Rem. 6.3 that [25], §10 contains inaccurate results in the zero dimensional case, which we fully correct in this paper (Remark 4.26).
By successively deleting columns of the matrix associated to a determinantal scheme X, we get a nest ("flag") of closed subschemes X = X c ⊂ X c−1 ⊂ ... ⊂ X 2 ⊂ P n . We prove our results inductively by considering the smoothness of the Hilbert flag scheme GradAlg(H), or Hilb H (P n ), the representing object of the functor which parameterizes flat families of graded quotients A of R of depth A ≥ min(1, dim A) and with Hilbert function H ( [22], [23]), and we call it "the postulation Hilbert scheme" ( [24], §1.1) even though it may be different from the parameter space studied by Gotzmann, Iarrobino and others ( [12], [20]) who study the "same" scheme with the reduced scheme structure (ours may be non-reduced and is equivalent to the Hilbert scheme of constant postulation considered in [34] in the curve case. They are both special cases of the multigraded Hilbert scheme of Haiman and Sturmfels [17]). Again we let (A), or (X) where X = Proj(A), denote the point of GradAlg(H) which corresponds to A. Note that if depth m A ≥ 1 and 0 Hom R (I X , H 1 m (A)) = 0, then (1. 1) GradAlg(H) ≃ Hilb p (P n ) at (X) , and hence we have an isomorphism 0 Hom(I X , A) ≃ H 0 (N X ) of their tangent spaces (cf. [11] for the case depth m A ≥ 2, and [23], (9) for the general case). If (1.1) holds and X is generically a complete intersection, then 0 Ext 1 A (I X /I 2 X , A) is an obstruction space of GradAlg(H) and hence of Hilb p (P n ) at (X) ( [23], §1.1). When we simply write Hilb(P n ), we interpret it as the Hilbert scheme (resp. postulation Hilbert scheme) if n − c > 0 (resp. n − c = 0). By definition X (resp. A) is unobstructed if Hilb p (P n ) (resp. Hilb H (P n )) is smooth at (X). Note that we called X H-unobstructed in [25] if A was unobstructed.
We say that X is general in some irreducible subset W ⊂ Hilb(P n ) if (X) belongs to a sufficiently small open subset U of W (so any (X) in U has all the openness properties that we want to require).

Background
In this section we recall some basic results on standard (resp. good) determinantal schemes needed in the sequel, see [3], [10] and [2] for more details. Let be a graded morphism of free R-modules and let A = (f ij ) j=0,...,t+c−2 i=1,...t , deg f ij = a j − b i , be a t × (t + c − 1) homogeneous matrix which represents the dual ϕ * := Hom R (ϕ, R). Let I(A) = I t (A) (or I t (ϕ)) be the ideal of R generated by the maximal minors of A. In the following we always suppose c ≥ 2, t ≥ 2, b 1 ≤ ... ≤ b t and a 0 ≤ a 1 ≤ ... ≤ a t+c−2 .
Recall that a codimension c subscheme X ⊂ P n is standard determinantal if I X = I(A) for some homogeneous t × (t + c − 1) matrix A as above. Moreover X ⊂ P n is a good determinantal scheme if additionally, A contains a (t − 1) × (t + c − 1) submatrix (allowing a change of basis if necessary) whose ideal of maximal minors defines a scheme of codimension c + 1. Note that if X is standard determinantal and a generic complete intersection in P n , then X is good determinantal, and conversely [28], Thm. 3.4. Without loss of generality we assume that A is minimal; i.e., f ij = 0 for all i, j with b i = a j . Let W (b; a) (resp. W s (b; a)) be the stratum in Hilb(P n ) consisting of good (resp. standard) determinantal schemes. By [26], Cor. 2.5 and see the end of p. 2877, we get Let A := R/I t (A) and M := coker(ϕ * ). Using the generalized Koszul complexes associated to a codimension c standard determinantal scheme X, one knows that the Eagon-Northcott complex yields the following minimal free resolution and that the Buchsbaum-Rim complex yields a minimal free resolution of M; [3]; Thm. 2.20 and [10]; Cor. A2.12 and Cor. A2.13. Note that (2.2) show that any standard determinantal scheme is arithmetically Cohen-Macaulay (ACM).
Let B be the matrix obtained deleting the last column of A and let B be the k-algebra given by the maximal minors of B. Let Y = Proj(B). The transpose of B induces a map φ : . Let M B be the cokernel of φ * = Hom R (φ, R) and let M A = M. In this situation we recall that there is an exact sequence is a regular section given by the last column of A. Moreover, is exact by [28] or [25], (3.1), i.e. we may put I X/Y := M B (a t+c−2 ) * . Due to (2.3), M is a maximal Cohen-Macaulay A-module, and so is I X/Y by (2.5). By [10] we have Hence by successively deleting columns from the right hand side of A, and taking maximal minors, one gets a flag of determinantal subschemes and M 2 is a twist of the canonical module of X 2 . In this context we let D i := R/I X i , I D i = I X i and I i : Remark 2.2. Let α be a positive integer. If X is general in W (b; a) and u i−min(α,t),i = a i−min(α,t) − b i ≥ 0 for min(α, t) ≤ i ≤ t, then X j , for all j = 2, · · · , c, is non-singular except for a subset of codimension at least min{2α − 1, j + 2}, i.e.
As observed in Rem. 2.7 of [26], this follows from the Theorem of [6] by arguing as in [6], Example 2.1. See [35] for a special case. In particular, if α ≥ 3, we get that for each i > 0, the closed embeddings X i ⊂ P n and X i+1 ⊂ X i are local complete intersections outside some set Z i of codimension at least min (4 In what follows we always let Z ⊂ X (and similarly for Z i ⊂ X i ) be some closed subset such that U := X − Z ֒→ P n (resp. U i := X i − Z i ֒→ P n ) is a local complete intersection (l.c.i.). Since the 1. Fitting ideal of M is equal to I t−1 (ϕ), we get thatM is locally free of rank one precisely on X − V (I t−1 (ϕ)) [2], Lem. 1.4.8. Since the set of non locally complete intersection points of X ֒→ P n is exactly V (I t−1 (ϕ)) by e.g. [37], Lem. 1.8, we get that U ⊂ X − V (I t−1 (ϕ)) and thatM is locally free on U. Indeed M i and I X i /I 2 Finally notice that there is a close relation between M(a t+c−2 ) and the normal module is exact. Hence we have an injection M A (a t+c−2 ) ֒→ N X/Y , which in the case depth I(Z) B ≥ 3 leads to an isomorphism M A (a t+c−2 ) ≃ N X/Y . Indeed, this follows from the more general fact (by letting L = N = I X/Y ) that if L and N are finitely generated B-modules such that depth I(Z) L ≥ r + 1 andÑ is locally free on U := Y − Z, then the natural map ) is an isomorphism (resp. an injection) for i < r (resp. i = r), and H i * (U, Hom O Y (Ñ,L)) ≃ H i+1 I(Z) (Hom B (N, L)) for i > 0, cf. [15], exp. VI. Note that we interpret I(Z) as m if Z = ∅.

The dimension of the determinantal locus
In [26] we conjectured the dimension of W (b; a) in terms of the invariant Here the indices belonging to a j (resp. b i ) range over 0 ≤ j ≤ t + c − 2 (resp. 1 ≤ i ≤ t), a n = 0 if a < n and we always suppose W (b; a) = ∅ in the following, cf. Corollary 2.1.
h 1 +b i n and in general For the special case where all the entries of A have the same degree, this means: In [26] we proved that the right hand side in the formula for dim W (b; a) in the Conjectures is always an upper bound for dim W (b; a) ( [26], Thm. 3.5), and moreover, that the Conjectures hold in the range  This contradicts both conjectures for every c ≥ 3.
We have, however, looked at many examples in the range a 0 > b t where we have used Macaulay 2 to compute necessary invariants (cf. (3.3) below), without finding more counterexamples. The counterexample we have is only for zero dimensional schemes. Now we recall a few statements from the proof of (3.2) and a variation which we will need in the next section. In the proof we used induction on c by successively deleting columns of the largest possible degree. Hence we computed the dimension of W (b; a), a = a 0 , a 1 , ..., a t+c−2 in terms of dimension of W (b; a ′ ), where a ′ = a 0 , a 1 , ..., a t+c−3 . As in §2, we let X = Proj(A) belong to W (b; a) and we let Y = Proj(B), (Y ) ∈ W (b; a ′ ), be obtained by deleting the last column of A. We have a j − a t+c−2 + n n , We also get equality in (3.3), as well as Proof. Indeed the proof of Thm. 4.5 of [26] contains the ideas we need, but since the assumptions of Thm. 4.5 are different, we include a proof. First we remark that we have by [26], Prop. 3.13 which combined with the assumption on dim W (b; a ′ ) yields Next by [26], Prop. 4.1 we have the inequality Since K c = 0 hom(coker ϕ, R(a t+c−2 )) by definition (see [26], Prop. 3.12 and (3.14)) we can use (2.3) and (2.4) to get Now looking at (3.1) and noticing that λ c−1 is defined by the analogous expression where Hence all inequalities of displayed formulas in this proof are equalities and we are done. Indeed this is mainly [26], Thm. 4.5, Cor. 4.7, Cor. 4.10, Cor. 4.14 and [11] (c = 2) and [25] (c = 3). Moreover since the proofs of [26] also show (3.3), we get the final dimension formula of Proposition 3.4. Moreover we have (valid also for n = c and chark = 0): 3) for X general, and Conjecture 3.1 hold provided 3 ≤ c ≤ 5 (resp. c > 5) and a t+c−2 > a t−2 (resp. a t+3 > a t−2 ) by [27], Thm 3.2.

the codimension of the determinantal locus
In this section we consider the problem of when the closure of W (b; a) is an irreducible component of Hilb(P n ). If it is not a component, we determine its codimension in Hilb(P n ) under certain assumptions. We also examine when Hilb(P n ) is generically smooth along W (b; a). Moreover we have chosen to introduce the notion "every deformation of X comes from deforming A" because it gives the main reason for why W (b; a) is not always an irreducible component of Hilb(P n ).
In the case the determinantal schemes are of dimension zero or one, then W (b; a) is not necessarily an irreducible component of Hilb(P n ), as the following example shows.
Let B be a general 2 × c matrix of linear entries and let A = [B, v] where the entries of the column v are general polynomials of the same degree 2. The vanishing all 2 × 2 minors of A defines a determinantal scheme X of codimension c in P n .
(i) Let n = c. Then X = Proj(A) is a reduced scheme of 2c + 1 points in P c and with h- for v ≥ 2 and we get 0 Hom R (I X , H 1 m (A)) = 0. By (1.1) the postulation Hilbert scheme is isomorphic to the usual Hilbert scheme at (X), whose dimension is c(2c + 1). Moreover since the dimension of W (b; a) is at most the conjectured value c 2 + 4c − 2, and since it follows that W (0, 0; 1, 1, ..., 1, 2) is not an irreducible component of Hilb H (P c ).
(ii) Let n = c + 1. Then X is a smooth connected curve in P c+1 of degree d = 2c + 1 and genus g = c. Since the dimension of W (b; a) is at most the conjectured value, which is c 2 +7c+2, and since the dimension of the Hilbert scheme is at least In what follows we briefly say "T a local ring" (resp. "T artinian") for a local k-algebra (T, m T ) essentially of finite type over k = T /m T (resp. such that m r T = 0 for some integer r). Moreover we say "T → S is a small artinian surjection" provided there is a morphism (T, m T ) → (S, m S ) of local artinian k-algebras whose kernel a satisfies a · m T = 0.
Let A = R/I t (A). If T is a local ring, we denote by A T = (f ij,T ) a matrix of homogeneous polynomials belonging to the graded polynomial algebra Note that all elements from T are considered to be of degree zero.
Once having such a matrix A T , we get an induced morphism , is a standard determinantal scheme, then A T := R T /I t (A T ) and M T are (flat) graded deformations of A and M respectively for every choice of A T as above. In particular Proof. Consulting (2.2) and (2.3) we see that the Eagon-Northcott and Buchsbaum-Rim complexes are functorial in the sense that, over R T , all free modules and all morphisms in these complexes are induced by ϕ T , i.e. they are determined by A T . Since these complexes become free resolutions of A and M respectively when we tensor with k over T , it follows that A T and M T are flat over T and satisfy A T ⊗ T k = A and M T ⊗ T k = M.
, be a standard determinantal scheme. We say "every deformation of X comes from deforming A" if for every local ring T and every graded deformation R T → A T of R → A to T , then A T is of the form A T = R T /I t (A T ) for some A T as above. Note that by (1.1) we can in this definition replace "graded deformations of R → A" by "deformations of X ֒→ P n " provided dim X ≥ 1.
Proof. Let T → S be a small artinian surjection and let A S be a deformation of A to S. By assumption, the unobstructedness of A, as well as the unobstructedness of X in the case dim X ≥ 1 by (1.1).
Finally let T be the local ring of Hilb(P n ) at (X) and let A T , or Proj(A T ) if dim X ≥ 1, be the pullback of the universal object of Hilb(P n ) to Spec(T ). Then there is a matrix is an open neighborhood of (X) for which the Eagon-Northcott complex associated to the matrix A D = (f ij,D ) is exact at any (X ′ ) ∈ Spec(D) (cf. [33], Lem. 6.3 or [11], proof of Thm. 1; in our case the existence of Spec(D) is quite easy since the Eagon-Northcott complex of the homogeneous coordinate ring of a standard determinantal scheme is always exact). It follows that Spec(D) ⊂ W (b; a), and since Spec(D) is open in Hilb(P n ) we are done.
Remark 4.5. The arguments of these lemmas, which rely on the fact that we get T -flat schemes by just parameterizing the polynomials of A over a local ring T , seem to be known, see e.g. Laksov's papers [30], [29] where he looks to flat families of determinantal schemes and their singular loci for arbitrary determinantal schemes. Indeed we may expect from Laksov's papers (or prove by other arguments, as we remember Laksov did in a talk at the university of Oslo in the 70's) that families of arbitrary determinantal schemes obtained by parameterizing polynomials as above are T -flat; thus he mainly shows the unobstructedness part of Lemma 4.4. Since we in this paper only look at determinantal schemes defined through maximal minors, our proofs are rather easy. Surprisingly enough the corresponding unobstructedness result of the R-module M (of maximal grade [4]) seems less known. Indeed one may prove the unobstructedness of M as we did for A in Lemma 4.4 because we from the Buchsbaum-Rim complex may see that every deformation of M to T comes from deforming A. We have learned, by distributing a preliminary version of this paper to specialists in deformations of modules, that the unobstructedness of M was proved in Runar Ile's PhD thesis, cf. [21], ch. 6 (Lem. 6.1.2 or Cor. 6.1.4).
The following result is a key proposition to our work in this section. Here the morphisms of the Ext 1 -groups are induced by the inclusion I X/Y ֒→ B, e.g.
Then every deformation of X comes from deforming A. Moreover where the lower isomorphism follows from the fact that M B is maximally CM, i.e. from depth I(Z) M B = depth I(Z) B ≥ 2. Note that since an M B -regular sequence lifts to an M B T -regular sequence, we also have sufficient depth to get the upper isomorphism. Hence we get a section B T → M B T (a t+c−2 ) and an induced matrix A T , as required. Let . Let T r := T /m r T , B Tr := B T ⊗ T T r , I X Tr /Y Tr := ker(B Tr → A T ⊗ T T r ), I ′ Tr = ker(B Tr → A ′ ⊗ T T r ) and S := T r−1 . To prove the claim we first show that I ′ Tr = I X Tr /Y Tr for every integer r > 0. To see that this follows from the assumption (1), we suppose by induction that I ′ S = I X S /Y S as ideals of B S . Then I ′ Tr and I X Tr /Y Tr are two deformations of the same ideal I X S /Y S ֒→ B S to T r and their difference, as graded B Tr modules, corresponds to an element, diff, of 0 Ext 1 where o i are the following obstructions. One of them, say o 1 (resp. the other o 2 ) is the obstruction of deforming the graded morphism I ′ S ֒→ B S (i.e. the ideal) to a graded morphism between I ′ T and B T (resp. between I X Tr /Y Tr and B Tr ), cf. [23], Rem. 3 for a similar situation. Since I ′ T and I X Tr /Y Tr are ideals in B Tr , such graded morphisms exist. Hence o i = 0 for i = 1, 2, whence diff = 0 by the injectivity of ρ 1 , and we conclude that I ′ Tr = I X Tr /Y Tr . To get the claim let A ′′ := B T /(I ′ T + I X T /Y T ). It suffices to show that the natural maps A ′ → A ′′ and A T → A ′′ are isomorphisms. Note that A ′′ ⊗ T k ≃ A ′′ ⊗ R T R ≃ A and that we have similar isomorphisms for A ′ and A T . Notice also that every maximal ideal of R T lies over m T . Hence we get both isomorphisms by the lemma of Nakayama, Azumaya and Krull if we can show that A ′′ is T -flat. But by the proof in the paragraph above the induced maps A ′ ⊗ T T r → A ′′ ⊗ T T r are isomorphism for every r > 0. It follows that

we get that
A ′′ is T -flat by the local criterion of flatness (see Thm. 20.C of [32]).
It remains to prove the dimension formula. Recall that there is a "standard" diagram (whose square is cartesian) which defines the tangent space A 1 of the Hilbert flag scheme GradAlg(H 2 , H 1 ) at (B → A) and where the morphisms T pr 1 and δ are natural maps (cf. [23], (10) and note that the algebra cohomology group 0 H 2 (B, A, A) ≃ 0 Ext 1 B (I X/Y , A) (cf. [23], §1.1)). Under the assumption (2) the vertical sequence is exact. We claim that δ = 0. To see it, it suffices to prove that T pr 1 is surjective. The cartesian diagram is, however, well understood in terms of the deformation theory of the Hilbert flag scheme. Indeed if we take an arbitrary deformation B S of B to the dual numbers S := k[t]/(t 2 ), then T pr 1 is surjective provided we can prove that there is a deformation (B S → A S ) of (B → A) to S. The latter follows from the first part of the proof, or simply, from the assumption (3) because we by (3) get B S = R S /I t (B S ) for some matrix B S and we can take A S = [B S , v S ] where v S is any lifting of the last column of A to S. Letting A S := R S /I t (A S ) we get the claim.
Since we have dim (X) Hilb(P n ) = 0 hom(I X , A) and dim (Y ) Hilb(P n ) = 0 hom(I Y , B) by Lemma 4.4, we get the dimension formula from the big diagram in which δ = 0 provided we can prove that 0 hom(I X/Y , A) = dim M B (a t+c−2 ) 0 −1. To see it we apply Hom(I X/Y , −) onto 0 → I X/Y → B → A → 0. If we use that Hom(I X/Y , B) ≃ M B (a t+c−2 ), see (2.10), we get the exact sequence , and we conclude by the assumption (1). Remark 4.9. Suppose τ X/Y is not injective. Then the vertical sequence in the diagram (4.2) is not exact, and δ may be non-zero. It is, however, easy to enlarge the diagram (4.2) to a diagram of exact horizontal and vertical sequences by including ker τ X/Y and im δ in the diagram. From this enlarged diagram it follows that (2.11) and depth m B ≥ 2. The displayed formula also holds if τ X/Y is injective.
Moreover every deformation of a general (X) ∈ W (b; a) comes from deforming A.
For c > 2 this result is really [26], Thm. 5.1 and Cor. 5.3 except for the final statement. Since, however, the proof of Theorem 4.10 is to apply Proposition 4.6 inductively to the flag (2.7), starting with the codimension 2 case where we by Lemma 4.11 know that the final statement holds, the proof of [26] extends to get Theorem 4.10 for c > 2. If c = 2 we get the other conclusions (and even more) from [11] for n > 2 and from works of Gotzmann and others for n = 2 as explained in [24], Rem. 22 (i), or see [26], Rem. 4.6 for a direct approach to dim W (b; a) = λ 2 . (We also get all conclusions for c = 2 by combining Lemma 4.11 and Lemma 4.4.) Lemma 4.11. If X = Proj(A), A = R/I t (A), is a standard determinantal scheme of codimension 2 in P n and n ≥ 2, then every deformation of X comes from deforming A.
Proof. Let A = (f ij ) be a homogeneous t × (t + 1) matrix which represents the morphism ϕ * of (2.1) and let R T → A T be a graded deformation of R → A to a local ring T . To see that A T is of the form A T = R T /I t (A T ) for some matrix A T reducing to A via (−) ⊗ T k, we consider the canonical module K A = Ext 2 R (A, R)(−n − 1). Note that since c = 2 we have K A (n + 1 − ℓ 1 ) = M, where G * ϕ * −→ F * → M → 0 is a part of the Buchsbaum-Rim complex, cf. (2.3) and (2.6). Now we observe that K A T := Ext 2 R T (A T , R T )(−n − 1) is a (flat) graded deformation of K A to T because Ext i R (A, R) = 0 for i = 2 ([7], Proposition (A1)). It follows that K A T (n + 1 − ℓ 1 ) = coker(ϕ * T ) where ϕ * T corresponds to some matrix A T := (f ij,T ), as in (4.1).
Let A ′ := R T /I t (A T ). It suffices to show that A ′ and A T are isomorphic as R T quotients. Looking to the Eagon-Northcott complex associated to A ′ over R T and dualizing, i.e. applying Hom R T (−, R T ) to it, we get back the part of the Buchsbaum-Rim complex where ϕ * T appeared (up to twist). It follows that 1). Note that the Buchsbaum-Rim complex above is a free resolution of K A ′ (n + 1 − ℓ 1 ) over R T since it is T -flat and reduces to a R-free resolution via a (−) ⊗ T k. Applying . Similarly by dualizing twice an R Tfree resolution of A T we show that A T ≃ Ext 2 R T (K A T , R T )(−n − 1) and we are done. Remark 4.12. If we apply Proposition 4.6 successively to the flag (2.7) it is straightforward to generalize Thm. 5.1 of [26] to the zero dimensional case, i.e. we may replace the condition n ≥ 1 of [26], Thm. 5.1 by n ≥ 0 provided we in (i) of Thm. 5.1 (and correspondingly in (ii) of Thm. 5.1) replace the depth ≥ 3 condition by the condition (1) of Proposition 4.6.
There is a variation to Proposition 4.6 that we will use in the case n = c (dim X = 0) in which we mainly replace the injectivity assumption in (1) for the Ext 1 -groups with the injectivity assumption for the corresponding Ext 2 -groups. More precisely let

B) is injective, and (3) Y is unobstructed (this is weaker than (3) in Proposition 4.6).
Then A is unobstructed and the postulation Hilbert scheme Hilb H A (P n ) satisfies Proof. Let T → S be a small artinian surjection with kernel a, and let R S → A S be any graded deformation of R → A to S. By Remark 4.8, there is a graded deformation R S → B S of R → B and a morphism B S → A S . By assumption (3) and (1.1) there exists a deformation R T → B T of R S → B S to T . It is well known that the algebra cohomology group 0 H 2 (B, A, A) ⊗ k a contains the obstruction of deforming B S → A S further to B T and that there is an injection 0 H 2 (B, A, A) ֒→ 0 Ext 1 B (I X/Y , A) ( [16], exp. VI, [23], §1.1). The rightmost group vanishes by (1), and it follows that A is unobstructed.
Finally we get the dimension formula from (4.2). Indeed the arguments are almost exactly the same as in the proof of Proposition 4.6 with the variation that (4.3) now implies deformation B S → A S of B → A to S, there exists, for every graded deformation B T of B S to T , a graded deformation B T → A T reducing to B S → A S via (−) ⊗ T S. It is clear from the proof above that 0 Ext 1 B (I X/Y , A) = 0 implies (*) and moreover that we can generalize Proposition 4.13 by replacing the assumption 0 Ext 1 B (I X/Y , A) = 0 by (*). Now we come to the main results of this paper which are direct consequences of the above propositions. We start with determinantal curves whose result we will need in the zero dimensional case. Note that the result below is known ( [25], Cor. 10.15 and Rem. 10.9 for c = 3, [26], Rem. 5.4 and Cor. 5.7 for 4 ≤ c ≤ 5) except for the final statement of (i) and most statements on the codimension in (ii) and (iii). With notations as in Proposition 4.6 we have  Proof. In all cases we use Theorem 3.5 to get dim W (b; a) = λ c + K 3 + ... + K c .
(ii) The vanishing of the Ext 1 -group of (ii) implies that X is unobstructed (X is l.c.i. by Remark 2.2), and moreover that im δ ≃ 0 Ext 1 B (I X/Y , A), cf. the diagram (4.2) and continue the horizontal exact sequence as a long exact sequence of algebra cohomology. Since we have h 0 (N Y ) − dim W (b; a ′ ) = 0 by Theorem 4.10 and 0 hom(I X/Y , A) = dim M B (a t+c−2 ) 0 − 1 by (4.3) and Remark 4.7, we conclude by Remark 4.9 and the final dimension formula of Theorem 3.5.
(iii) As in (ii) we get 0 hom(I X , A) − dim W (b; a) = dim ker τ X/Y − im δ and hence the inequality of (iii). If the 0 Ext 1 B (I X/Y , A) vanishes, then im δ = 0, and since one knows that X is unobstructed if and only if we have equality in h 0 (N X ) ≥ dim (X) Hilb p (P n ), we conclude easily. (i) Since Y is licci ( [25], Def. 2.10) for c = 3 , we always have 0 Ext 1 B (I Y /I 2 Y , B) = 0 by [1] (or see [19] or [25], Prop. 6.17) and hence we get ker τ X/Y ≃ 0 Ext 1 B (I Y /I 2 Y , I X/Y ).
(ii) It is shown in [25], Cor. 10.11 (for n − c = 1) and Cor. 10.17 (for n − c = 0) that 0 Ext 1 B (I Y /I 2 Y , I X/Y ) = 0 provided a t+1 > a t + a t−1 − b 1 . Indeed the proofs of [25] (or [26], Cor. 5.10 (i)) show 0 Ext 1 R (I Y , I X/Y ) = 0 by mainly using the R-free minimal resolution of I Y and the degree of the minimal generators of I X/Y which we get from (2.2). Hence we can conclude by the injection 0 Ext 1 is, however, much more common than given by the above argument. Indeed examining many examples by Macaulay 2 ( [13]) in the range a 0 > b t = b 1 we almost always got 0 Ext 1 We can also analyze the cases m = 1 and 2 by using Proposition 4.15 (i). Note that we now delete the column of degree 3 polynomials to define B, i.e. we let Y be defined by the maximal minors of the 2 × 3 matrix B consisting of linear (resp. degree m) entries in the first and second (resp. third) column. If m = 1 (resp. m = 2) one verifies that Remark 4.18. We have checked the vanishing of 0 Ext 1 A (I Xm /I 2 Xm , A) for several m ≥ 1 in Example 4.17 (ii). It seems that this group is always non-zero for every m ≥ 1. This, we think, shows that the results presented here are quite strong because it is hard to show unobstructedness and to find dim (Xm) Hilb H (P n ) when even the "smallest known obstruction group, 0 Ext 1 A (I Xm /I 2 Xm , A)," does not vanish. Now we consider zero dimensional determinantal schemes (n − c = 0). Indeed Proposition 4.6 with depth I(Z) B = 2 and Proposition 4.13 are designed to take care of this case. We restrict our attention to a general X which through Remark 2.2 imply that all depth conditions of the propositions are satisfied. Then our result leads e.g. to the unobstructedness of A where X = Proj(A). In fact for special choices of X, A may be obstructed [36]. First we consider codimension c = 3 determinantal subschemes and schemes with c ≥ 4 which we may treat similarly.     We remark that we have checked a possible vanishing of 0 Ext 1 A (I Xm /I 2 Xm , A) for several m ≥ 3, and in the range 3 < m ≤ 6 this group is non-zero.
Finally to be complete we consider the cases m = 1 and m = 2 in which case we will delete the column of degree 3 polynomials to define B and hence Y . If m = 1 we get by Macaulay 2 dim ker ρ 1 = 0 ext 1  We let A = [B, v] be a general 2×4 matrix whose columns consist of general polynomials of the same degree, 2, 2, 4 and m, m ≥ 4 respectively. The vanishing of all 2 × 2 minors of A defines a reduced scheme X =: X m of 20m + 16 points in P 3 . This time Macaulay 2 computations show 0 ext 1 B (I Xm/Y , I Xm/Y ) = 0 and 0 Ext 1 B (I Y /I 2 Y , I Xm/Y ) = 0 (resp. = 1) for every m > 4 (resp. m = 4). It follows from Theorem 4.19 (i) that W (b; a) is a generically smooth irreducible component of Hilb H (P 3 ) of dimension λ 3 + K 3 = 20m + 49 for m > 4. For m = 4 we have verified that 0 ext 1 A (I Xm /I 2 Xm , A) = 3 and in this particular case we have not been able to verify whether A is unobstructed or not. But for every m > 4, A is unobstructed by Theorem 4.19 (i)! Moreover we have checked a possible vanishing of 0 Ext 1 A (I Xm /I 2 Xm , A) for many m ≥ 3, and combined with some theoretical arguments (which we don't take here) we can conclude that this group is always non-zero for every m ≥ 4. Again, we think, this shows that the results presented here are quite strong because it is really hard to show unobstructedness when even the "smallest known obstruction group, 0 Ext 1 A (I Xm /I 2 Xm , A)," does not vanish.
In the final case 4 ≤ c ≤ 6 and ker τ Y /V = 0 where a general X = Proj(A) ⊂ Y = Proj(B) ⊂ V = Proj(C) is given by deleting columns as above we can not apply Proposition 4.6 to X ⊂ Y because there is no reason to expect condition (3) of Proposition 4.6 to be true (that condition is closely related to ker τ Y /V = 0). But we can still use Proposition 4.13 since condition (3) of Proposition 4.13 is weakened to "Y unobstructed". The natural condition for "Y unobstructed" which also give a formula for h 0 (N Y )−dim W (b; a ′ ) is 0 Ext 1 B (I Y /I 2 Y , B) = 0, cf. the proof of Proposition 4.15 (ii). We get By the same argument we have h 0 (N Y ) − dim W (b; a ′ ) ≤ dim ker τ Y /V and moreover, if 0 Ext 1 C (I Y /V , B) = 0, then equality holds. Hence we get the inequality of (ii), and furthermore, if the two Ext 1 -groups of (ii) vanish then the inequality of (ii) is an equality if and only if dim (X) Hilb H (P c ) = 0 hom(I X , A) and we are done. Let A = [B, v] be a general 2 × 5 matrix with linear (resp. quadratic) entries in the first, second and third (resp. fourth) column and let both entries of the column v be of degree m ≥ 2. Keeping the notations of Proposition 4.24, we get that the vanishing of all 2 × 2 minors defines a reduced scheme X of 7m + 2 points in P 4 . One verifies that dim ker ρ 1 = 0 ext 1 B (I X/Y , I X/Y ) = 3, 0 Ext 1 B (I X/Y , A) = 0 and that (3.3) holds by Remark 3.6. Note that we have dim ker τ Y /V = 1 and 0 Ext 1 C (I Y /V , B) = 0 from Suppose m = 2. Since dim ker τ X/Y = 0 ext 1 B (I Y /I 2 Y , I X/Y ) = 4 and 0 Ext 1 A (I X /I 2 X , A) = 0, it follows from Proposition 4.24 (ii) that A is unobstructed and that dim W (b; a) = λ 4 + K 3 + K 4 = 44. Hence W (b; a) is contained in a unique generically smooth irreducible component of Hilb H (P 4 ) and, codim Hilb H (P 4 ) W (b; a) = dim ker ρ 1 + dim ker τ X/Y + dim ker τ Y /V = 3 + 4 + 1 = 8.
In this case we see that all three kernels of Proposition 4.24 (ii) contribute to the codimension of W (b; a) in Hilb H (P 4 )!