Unobstructedness and dimension of families of Gorenstein algebras

Jan O. Kleppe

Abstract

The goal of this paper is to develop tools to study maximal families of Gorenstein quotients A of a polynomial ring R. We prove a very general Theorem on deformations of the homogeneous coordinate ring of a scheme Proj(A) which is defined as the degeneracy locus of a regular section of the dual of some sheaf M̃ of rank r supported on say an arithmetically Cohen-Macaulay subscheme Proj(B) of Proj(R). Under certain conditions (notably; M maximally Cohen-Macaulay and ∧^{r}M̃ ≃ K̃_{B}(t) a twist of the canonical sheaf), then A is Gorenstein, and under additional assumptions, we show the unobstructedness of A and we give an explicit formula the dimension of any maximal family of Gorenstein quotients of R with fixed Hilbert function obtained by a regular section as above. The theorem also applies to Artinian quotients A.

The case where M itself is a twist of the canonical module (r = 1) was studied in a previous paper, while this paper concentrates on other low rank cases, notably r = 2 and 3. In these cases regular sections of the first Koszul homology module and of normal sheaves to licci schemes (of say codimension 2) lead to Gorenstein quotients (of e.g. codimension 4) whose parameter spaces we examine. Our main applications are for Gorenstein quotients of codimension 4 of R since our assumptions are almost always satisfied in this case. Special attention are paid to arithmetically Gorenstein curves in ℙ^{5}. AMS Subject Classification. 14C05, 13D10, 13D03, 13C14, 13D02, 14F05. Keywords. Parametrization, Deformation, Hilbert scheme, Gorenstein scheme, Cohen-Macaulay scheme, Artinian algebra, sections of sheaves, special sheaves.

Contents

1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Families of Gorenstein quotients of low codimension on Cohen-Macaulay algebras 12

3 Rank two and three sheaves on codimension two quotients 19

4 Appendix: Deformations of quotients of zerosections 29

1 Introduction

In this paper we study deformations of arithmetically Gorenstein (AG) subschemes X = Proj(A) of a projective space ℙ^{N} = Proj(R). In particular we consider deformations of schemes X defined as the degeneracy locus of a regular section of some maximal Cohen-Macaulay sheaf M̃ supported on an arithmetically Cohen-Macaulay (ACM) subscheme Y = Proj(B) of ℙ^{N}. Our results allow us to study well three natural parameter spaces classifying AG subschemes of ℙ^{N} (resp. graded Gorenstein quotients of R) with Hilbert polynomial p (resp. Hilbert function H). The three schemes we have in mind are the open subscheme of Grothendieck’s Hilbert scheme Hilb^{p}(ℙ^{N}) consisting of AG subschemes of positive dimension, the open part of the postulation Hilbert scheme Hilb^{H}(ℙ^{c}) consisting of AG zeroschemes with Hilbert function H and Iarrobino-Kanev’s determinantal scheme PGor(H) classifying graded Artinian Gorenstein quotients of R via the Macaulay correspondence.

A common denominator of these parameter spaces is the k-scheme GradAlg^{H}(R) of [22] which parametrizes graded quotients B = R/I_{B} such that depth B ≥ min(1, dim B) and H_{B} = H, H_{B} the Hilbert function of B. Its open subscheme consisting of Gorenstein quotients B of codimension c in R, Gor_{c}^{H}(R), is naturally isomorphic to the three schemes in the mentioned order provided the degree of p is greater or equal 1, equal 0 and equal −1 (i.e. p(t) = 0) respectively. The isomorphism is even scheme-theoretic except possibly for PGor(H) where the isomorphism is at least infinitesimal and topological, making questions of dimension and smoothness of PGor(H) equivalent to the corresponding questions for GradAlg(H) := GradAlg^{H}(R), cf. [23] for details and [16] for a generalization of GradAlg(H).

If the codimension c ≤ 3, the smoothness, the irreducibility and the dimension of Gor_{c}^{H}(R) are known ([22], [10], [6], [24]). Hence we focus on Gorenstein quotients of codimension 4 of R which, nowadays, get quite a lot of attention ([9], [19], [23], [18], [3]). Moreover since some of our results require that the degree of p is greater or equal to 1, we mostly consider AG curves in ℙ^{5}, having also the case PGor(H) in mind since we have tried very much, and partially succeeded, to generalize our results to the Artinian case.

Let B → A be a morphism of graded quotients of R. In the background section we recall a basic result of how we can deduce the smoothness and the dimension of GradAlg^{H_{A}}(R) from the corresponding properties of GradAlg^{H_{B}}(R) (Theorem 15 and Corollary 18). The assumptions of Theorem 15 are satisfied for a complete intersection (c.i.) B and for some other classes of quotients with nice properties (licci is often enough) except for the vanishing “_{0}H^{2}(B, A, A) = 0” of the second algebra cohomology group. Theorem 15 and Corollary 18 are enough to treat the rank r = 1 case of M satisfactorily ([23]), while, for higher rank cases, they are insufficient. A main result, Theorem 22, of this paper replaces the assumption “_{0}H^{2}(B, A, A) = 0” with assumptions which are more natural to the application we have in mind; deforming schemes X defined as the degeneracy locus of a regular section. Note, however, that Theorem 15 and Corollary 18 apply also to quotients which not necessarily are given by such a regular section, and we take the opportunity to include a result which generalizes [18], Thm. 4.17 (Corollary 20).

To state the assumptions of Corollary 18 and Theorem 22 and its dimension formulas in a more computable form we consider low rank cases on a licci scheme Y = Proj(B). Let K_{B} be the canonical module of B. If M is a graded maximal Cohen-Macaulay B-module of rank r = 2 such that M̃|_{U} is locally free and ∧^{2}M̃|_{U} ≃ K̃_{B}(t)|_{U} in a large enough open set U = Y − Z of Y, then a regular section σ of M̃^{∗}(s)|_{U}, M^{∗} = Hom_{B}(M, B), defines a graded Gorenstein quotient A given by the exact sequence

0 → K_{B}(t − 2s) → M(−s) →^{σ} B → A → 0 ,  (1)

similar to what happens in the usual Hartshorne-Serre correspondence ([14], Thm. 4.1). Moreover M ≃ Hom_{B}(M, K_{B}(t)) (Theorem 8). Let S_{2}(M) be the second symmetric power of M. Using small letters for the k-dimension of the v-graded piece _{v}Ext_{B}^{i}(−, −) of Ext_{B}^{i}(−, −), we define γ(S_{2}M)_{v} and δ(Q)_{v} by

δ(Q)_{v} = _{v}hom_{B}(I_{B}/I_{B}^{2}, Q) − _{v}ext_{B}^{1}(I_{B}/I_{B}^{2}, Q) , and
γ(S_{2}M)_{v} = _{v}hom_{B}(S_{2}(M), K_{B}(t)) − _{v}ext_{B}^{1}(S_{2}(M), K_{B}(t)) .

Note that if depth_{I(Z)} B ≥ 3 (resp. depth_{I(Z)} B ≥ 4) and char(k) ≠ 2, one may show that γ(S_{2}M)_{0} = _{0}hom_{B}(M, M) − _{0}ext_{B}^{1}(M, M) − 1 (resp. _{0}Ext_{B}^{i}(S_{2}(M), K_{B}(t)) ≃ _{0}Ext_{B}^{i}(M, M) for i = 1, 2) since r = 2, cf. (12). A main result of Section 2 is the following Theorem (Theorem 25) which applies also to an Artinian A.

Theorem 1. Let B = R/I_{B} be a graded licci quotient of R, let M be a graded maximal Cohen-Macaulay B-module, and suppose M̃ is locally free of rank 2 in U := Proj(B) − Z, that dim B − dim B/I(Z) ≥ 2 and ∧^{2}M̃|_{U} ≃ K̃_{B}(t)|_{U}. Let A be defined by a regular section σ of M̃^{∗}(s) on U, i.e. given by (1), and suppose _{s}Ext_{B}^{1}(M, B) = 0.

A) If _{t}Ext_{B}^{2}(S_{2}(M), K_{B}) = 0 and _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, M) = 0 ,

then A is unobstructed as a graded R-algebra (indeed _{0}H^{2}(R, A, A) = 0), A is Gorenstein, and

dim_{(A)} GradAlg(H_{A}) = dim(N_{B})_{0} + dim(M^{∗})_{s} − 1 − γ(S_{2}M)_{0} + dim(K_{B})_{t−2s} + δ(K_{B})_{t−2s} − δ(M)_{−s}.

Moreover if char(k) = 0 and (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}), then the codimension of the stratum of quotients given by (1) around (A) is

−sext_{B}^{1}(I_{B}/I_{B}^{2}, M) − dim(im β)

where β is the homomorphism _{−2s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}(t)) → _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) induced by (1).

B) If (M, B) is unobstructed along any graded deformation of B and _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) = 0, then A is Gorenstein and H_{B}-generic. Moreover A is unobstructed as a graded R-algebra and the dimension formula for dim_{(A)} GradAlg^{H_{A}}(R) of part A) holds.

Here N_{B} = Hom_{B}(I_{B}/I_{B}^{2}, B), I_{A/B} := ker(B → A) and “(M, B) unobstructed along any deformation of B” means that for every deformation (M_{S}, B_{S}) of (M, B), S local and Artinian, there is a deformation of M_{S} to any deformation of B_{S} (cf. Definition 11). Moreover “(B → A) is general with respect to γ := _{0}hom_{R}(I_{B}, I_{A/B})” if and only if _{0}hom_{R}(I_{B′}, I_{A′/B′}) ≥ γ for every (B′ → A′) in an open neighbourhood of (B → A) in GradAlg(H_{B}, H_{A}) (see Remark 16(ii)). Furthermore “A is H_{B}-generic” essentially means that the codimension of the stratum mentioned in Theorem 1A) is zero, see the text before Theorem 15 and Theorem 25 for precise definitions.

Remark 2. If max n_{2,j} is the largest degree of a minimal relation of I_{B} and a is an integer such that M_{v} = 0 for v ≤ a, then

−sExt_{B}^{1}(I_{B}/I_{B}^{2}, M) = 0 provided s ≥ max n_{2,j} − a.

In this case dim(K_{B})_{t−2s} + δ(K_{B})_{t−2s} − δ(M)_{−s} = 0. This significantly simplifies the dimension formula of Theorem 1B) provided we use the theorem for such s.

On a Cohen-Macaulay (CM) quotient B = R/I_{B} of codimension 2 in R with minimal resolution

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

Theorem 1 applies to M = N_{B}, the normal module, and to M = H_{1}, the 1. Koszul homology of I_{B}. In Section 3 we study these applications in detail. To see that the assumptions of Theorem 1 are satisfied, let Proj(B) − Z ↪ ℙ^{N} be a local complete intersection (l.c.i.). Then we prove

Proposition 3. Let B = R/I_{B} be a codimension two CM quotient of R and suppose dim B − dim B/I(Z) ≥ 2. Then

(i) Ext_{B}^{2}(I_{B}/I_{B}^{2}, H_{1}) = Ext_{B}^{1}(H_{1}, B) = 0 and S_{2}(H_{1}) is a maximal CM B-module.

(ii) Ext_{B}^{i}(I_{B}/I_{B}^{2}, I_{B}/I_{B}^{2}) = 0 for 1 ≤ i ≤ dim B − dim B/I(Z).

Using Proposition 3(i) we get the following result, with a sufficiently weak assumption on depth_{I(Z)} B = dim B − dim B/I(Z) so that the result applies also to an Artinian A.

Corollary 4. Let B = R/I_{B} be a graded codimension two CM quotient of R, let U = Proj(B)−Z ↪ ℙ^{N} be an l.c.i. and suppose the number of minimal generators of I_{B} is μ = 4 and depth_{I(Z)} B ≥ 2. If A is defined by a regular section of H̃_{1}^{∗}(s) on U, then A is unobstructed as a graded R-algebra (indeed H^{2}(R, A, A) = 0), A is Gorenstein of codimension 4 in R, and

dim_{(A)} GradAlg(H_{A}) = dim(N_{B})_{0} + dim(H_{1}^{∗})_{s} − 1 − γ(S_{2}H_{1})_{0} + dim(K_{B})_{t−2s} + δ(K_{B})_{t−2s} − δ(H_{1})_{−s}

where t = dim R − ∑ n_{1,i}. Moreover if s > max n_{2,j} − min n_{2,j}, then A is H_{B}-generic, and

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim(H_{1}^{∗})_{s} − 1 − _{0}hom_{B}(S_{2}(H_{1}), K_{B}(t)).

Furthermore in the situation of Corollary 4 we show how to compute every term of the dimension formulas and we give several examples.

Applying Theorem 1 to the normal module N_{B} we need the vanishing of the Ext-groups of Proposition 3(ii) to prove that the assumptions of Theorem 1 hold. To do so we have to increase depth_{I(Z)} B. Hence the following result does not automatically apply to Gorenstein quotients A of dimension less or equal to 1 unless we in a given example are able to verify the assumptions of Theorem 1. In Section 3 we give examples of the result.

Corollary 5. Let B = R/I_{B} be a graded codimension two CM quotient of R, let U = Proj(B)−Z ↪ ℙ^{N} be an l.c.i. and suppose depth_{I(Z)} B ≥ 4. If A is given by a regular section of Ñ_{B}^{∗}(s) on U, and if s > 2 max n_{2,j} − min n_{1,i} and char(k) ≠ 2, then A is H_{B}-generic and unobstructed as a graded R-algebra. Moreover A is Gorenstein of codimension 4 in R, and letting X = Proj(A) and η(v) := dim(I_{B}/I_{B}^{2})_{v}, we have

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(X)} Hilb^{p}(ℙ^{N}) = η(s) + ∑_{j=1}^{μ−1} η(n_{2,j}) − ∑_{i=1}^{μ} η(n_{1,i}) .

Finally we prove a Theorem, similar to Theorem 1, in the rank r = 3 case which also applies to an Artinian A (Theorem 30). It admits the following Corollary,

Corollary 6. Let B = R/I and U = Proj(B) − Z be as in Corollary 4 and suppose that μ = 5, char(k) ≠ 2 and depth_{I(Z)} B ≥ 3. If A is defined by a regular section of H̃_{1}^{∗}(s) on U, then A is unobstructed as a graded R-algebra (indeed H^{2}(R, A, A) = 0), A is Gorenstein of codimension 5 in R, and dim_{(A)} GradAlg^{H_{A}}(R) =

dim(N_{B})_{0} + dim(H_{1}^{∗})_{s} + _{−s}hom_{B}(S_{2}(H_{1}), K_{B}(t)) − _{0}hom_{B}(H_{1}, H_{1}) − dim(K_{B})_{t−3s} − δ ,

where δ := δ(H_{1})_{−s} + δ(K_{B})_{t−3s} − δ(H_{2})_{−2s} and t = dim R − ∑ n_{1,i}. If in addition s > max n_{2,j} − min n_{2,j}, then A is H_{B}-generic, δ = 0 and dim(K_{B})_{t−3s} = 0.

We thank Chris Peterson and Johannes Kleppe for useful discussions. Indeed we are in this work inspired by the joint works [27] and [28] where we in various ways construct Gorenstein quotients of R. This paper uses deformation theory to vary every object and morphism of these constructions, to see how large the corresponding stratum of its parameter space will be. As we see by the results above, the stratum is often a component. In some cases, however, it is a proper stratum of a codimension which we make explicit. Moreover we thank Rosa M. Miró-Roig for interesting discussions on PGor(H). In particular Corollary 20 was included in the paper after a discussion with her.

1.1 Background

We largely keep the notations of [23], and we recommend the section of Preliminaries of that paper for an overview of cohomology groups and deformation theory. Let B be an n-dimensional graded quotient of a polynomial ring R in n + c variables (of degree 1) over an algebraically closed field k, and let M and N be finitely generated graded B-modules. Let depth_{J} M denote the length of a maximal M-sequence in a homogeneous ideal J and let depth M = depth_{𝔪} M where 𝔪 is the irrelevant maximal ideal. Let H_{J}^{i}(−) be the right derived functor of the functor, Γ_{J}(−), of sections with support in Spec(B/J). Then

depth_{J} M ≥ r if and only if (iff) H_{J}^{i}(M) = 0 for i < r.  (2)

(cf. [13]). Let Y = Proj(B) and let Z be closed in Y and U = Y − Z. Put H_{∗}^{0}(U, M̃) := ⊕_{v} H^{0}(U, M̃(v)). Then we have an exact sequence 0 → H_{I(Z)}^{0}(M) → M → H_{∗}^{0}(U, M̃) → H_{I(Z)}^{1}(M) → 0 and isomorphisms H_{I(Z)}^{i}(M) ≃ H_{∗}^{i−1}(U, M̃) for i ≥ 2. If depth_{I(Z)} N ≥ i + 1, then the graded group Ext_{B}^{i}(M, N) injects into the corresponding global Ext_{O_{U}}^{i} -group of sheaves. Indeed we have by [12], exp. VI, an exact sequence

0Ext_{B}^{i}(M, N) ↪ Ext_{O_{U}}^{i} (M̃|_{U}, Ñ|_{U}) → _{0}Hom_{B}(M, H_{I(Z)}^{i+1}(N)) → _{0}Ext_{B}^{i+1}(M, N) → Ext_{O_{U}}^{i+1}(M̃|_{U}, Ñ|_{U}) (3)

where the form of the middle term comes from a spectral sequence discussed in [12].

A Cohen-Macaulay (resp. maximal Cohen-Macaulay) B-module satisfies depth M = dim M (resp. depth M = dim B) by definition, or equivalently, H_{𝔪}^{i}(M) = 0 for i < dim M (resp. i < dim B). If B is Cohen-Macaulay, then the v-graded piece of H_{𝔪}^{i}(M) is via Gorenstein duality given by vH_{𝔪}^{i}(M) ≃ _{−v}Ext_{B}^{n−i}(M, K_{B})^{∨} where K_{B} = Ext_{R}^{c}(B, R(−n − c)) is the canonical module of B. In this case B = R/I_{B} has a minimal R-free resolution of the following form

0 → G_{c} → ... → G_{1} → R → B → 0 ,  G_{j} = ⊕_{i=1}^{r_{j}}R(−n_{j,i})  (4)

and the R-dual sequence 0 → R → G_{1}^{∗} → ... → G_{c}^{∗} → K_{B}(n + c) → 0 is exact. Moreover the Castelnuovo-Mumford regularity of I_{B} is reg(I_{B}) = max{n_{c,i}} − c + 1. A coherent non-trivial O_{Y} - module, M, is called a maximal Cohen-Macaulay sheaf on U if Ext_{O_{Y}}^{i} (M, ω_{Y})|_{U} = 0 for i > 0 where ω_{Y} = K̃_{B}.

Lemma 7. Let B be Cohen-Macaulay. Let r and t be integers. Let J ⊆ B be an ideal satisfying depth_{J} B ≥ r and let M be a finitely generated B-module satisfying depth_{𝔪} M ≥ dim B − t. Then depth_{J} M ≥ r − t.

For a proof, see [28], Lem. 5. (What there looks like sheaf-Ext^{j} is actually Ext^{j} as B-modules). Now we recall a main result of [27] in which B is Cohen-Macaulay. Indeed a main idea of this

paper is to use deformation theory to see how general the construction of Gorenstein quotients, given by the Theorem below, is. Note that in this case depth_{J} B = dim B − dim B/J for any ideal J of B, cf. [7].

Theorem 8. Let R be a polynomial ring, let B = R/I_{B} be a codimension c graded CM quotient of R and let M be a finitely generated graded maximal CM B-module. Let Y = Proj(B), let Z be a closed scheme such that dim(B) − dim(B/I(Z)) ≥ max(r, 2) and let U = Y − Z. Let M_{i} = H_{∗}^{0}(U, ∧^{i}M̃) for i ≥ 0, and suppose that M̃|_{U} is locally free (of rank r) and that

∧^{r}M̃|_{U} ≃ K̃_{B}(t)|_{U} for some integer t .

Moreover suppose M_{i} is a maximal CM B-module for 2 ≤ i ≤ r/2. Then any regular section, σ ∈ H^{0}(U, M̃^{∗}(s)) (s an integer), defines a Gorenstein quotient R ↠ A of codimension r + c given by the exact sequence

0 → M_{r}(−rs) → M_{r−1}((1 − r)s) → ⋯ → M_{2}(−2s) → M(−s) →^{σ} B → A → 0.  (5)

Indeed all M_{i}, 0 ≤ i ≤ r, are maximal CM B-modules, and M_{r−i} ≃ Hom_{B}(M_{i}, K_{B})(t) for 0 ≤ i ≤ r.

Remark 9. By replacing M by M_{1} in (5) we see that the maximal CM assumption on M is really superfluous in the rank 2 Artinian case (r = dim(B) = 2). In [28] we succeeded generalizing Theorem 8 by weakening the maximal CM assumptions on M and M_{i} in several ways.

In [27] we applied Theorem 8 to different modules M on licci schemes (i.e. schemes whose homogeneous coordinate ring is in the linkage class of a complete intersection, cf. [30] for a survey), notably the normal module, N_{B}, and the 1. Koszul homology, H_{1} = H_{1}(I_{B}), built on a set of minimal generators of I_{B}. Note that the module H_{1} is essentially given by an exact sequence

0 → H_{2}(R, B, B) → H_{1} → G_{1} ⊗_{R} B → I_{B}/I_{B}^{2} → 0.  (6)

in which H_{2}(R, B, B) is the 2. algebra homology [31]. An ideal I_{B} of R is called syzygetic if H_{2}(R, R/I_{B}, R/I_{B}) = 0. Using (6) one shows that if R → B is generically a complete intersection and B is licci, then H_{2}(R, B, B) = 0 because H_{1} is a maximal CM module in the licci case [17]. If B is licci one also knows that N_{B} is a maximal CM B-module and that Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 ([5] and [26]). Note that if μ(I_{B}) is the number of minimal generators of I_{B}, we see from (6) that

rank H_{1} = μ(I_{B}) − c since the rank of N_{Y} = Ñ_{B} = { \atop I_{B}/I_{B}^{2}}^{∗} is the same as the codimension, c, of Y in Proj(R). We proved

Proposition 10. Let B = R/I_{B} be licci of codimension c ≥ 2, and let Y = Proj(B) ↪ ℙ^{n+c−1} = Proj(R) be a local complete intersection (l.c.i.) in some open U = Y − Z.

1. If dimB −dimB/I(Z) ≥ c, then H_{∗}^{0}(U, ∧^{i}N_{Y}) are maximal CM B-modules for every 1 ≤ i ≤ c, and any regular section of H^{0}(U, N_{Y}^{∗} (s)) defines a Gorenstein quotient R ↠ A of codimension 2c.

2. If dimB − dimB/I(Z) ≥ max(2, μ(I_{B}) − c), then H_{∗}^{0}(U, ∧^{i}H̃_{1}) are maximal CM B-modules for every 1 ≤ i ≤ μ(I_{B}) − c and any regular section of H^{0}(U, H̃_{1}^{∗}(s)) defines a Gorenstein quotient R ↠ A of codimension μ(I_{B}).

For the graded group H^{2}(R, B, B) we just remark that there is an exact sequence

0 → _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, N) → _{0}H^{2}(R, B, N) → _{0}Hom_{B}(H_{2}(R, B, B), N) →  (7)

induced from some well known spectral sequence, cf. [1], Prop. 16.1.

Let p ∈ ℚ[t] be a non-trivial polynomial and let ℙ = ℙ^{n+c−1}. Then the scheme GradAlg(H) = GradAlg^{H}(R) which we study in this paper is the stratum of Grothendieck’s Hilbert scheme Hilb^{p}(ℙ) (cf. [11]) consisting of points (Y ⊂ ℙ) with Hilbert function H_{Y} = H (i.e. its corresponding functor deforms the homogeneous coordinate ring, B, of Y flatly as a graded R-algebra), cf. [22]. Note that we define the Hilbert function of Y, or B, by H_{Y} (v) = H_{B}(v) := dim B_{v}. GradAlg^{H}(R) has a natural scheme structure whose tangent (resp. “obstruction”) space at (Y ⊂ ℙ) is _{0}Hom_{B}(I_{B}/I_{B}^{2}, B) ≃ 0Hom_{R}(I_{B}, B) (resp. _{0}H^{2}(R, B, B)) [21]. Since H(v) does not vanish for large v (i.e. B is non-Artinian), we may look upon GradAlg^{H}(R) as parametrizing graded R-quotients, R → B, satisfying depth_{𝔪} B ≥ 1 and with Hilbert function H_{B} = H. If B is Artinian, then GradAlg^{H}(R) still represents a functor parametrizing graded R-quotients with Hilbert function H_{B} = H. It contains an open subscheme of Gorenstein quotients which, at least topologically and infinitesimally, coincides with PGor(H), the corresponding scheme of forms with “catalecticant structure” (see [22] or [23] for details). B is called unobstructed as a graded R-algebra if GradAlg^{H}(R) is smooth at (R → B). Similarly a closed subscheme Y of ℙ is called unobstructed if Hilb^{p}(ℙ) is smooth at (Y ⊂ ℙ). By [21], Thm. 3.6 and Rem. 3.7,

GradAlg^{H}(R) ≃ Hilb^{p}(ℙ) at (Y ⊂ ℙ)  (8)

provided _{0}Hom_{R}(I_{B}, H_{𝔪}^{1}(B)) = 0 (e.g. provided depth_{𝔪} B ≥ 2, in which case (8) also follows from [8]). In particular if the degree of p is positive, then the open subschemes of GradAlg^{H}(R) (resp. of Hilb^{p}(ℙ)) of Gorenstein quotients (resp. of AG subschemes) are isomorphic as schemes.

Similarly we let GradAlg(H_{B}, H_{A}) be the representing object of the functor deforming flags (surjections) B → A of graded quotients of R of positive depth (if B and/or A are non-Artinian) and with Hilbert functions H_{B} and H_{A} of B and A respectively. Let p be the second projection

p : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R)

induced by sending (B′ → A′) onto (A′), and let q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R) be the first projection.

Definition 11. Let R → B be a graded quotient, let M be a graded B-module and let φ : M → B be a homogeneous B-linear map. Let (T, m_{T}) → (S, m_{S}) be a small Artin surjection (i.e. of local Artinian k-algebras with residue fields k whose kernel 𝔞 satisfies 𝔞 · m_{T} = 0). A graded deformation B_{S} of B to S is a graded S-flat quotient of R ⊗_{k} S satisfying B_{S} ⊗_{S} k ≃ B.

(i) (M, φ) is said to be unobstructed along any graded deformation of B if for every small Artin T ↠ S and for every graded deformation M_{S} → B_{S} (of S-flat B_{S}-modules) of M →^{φ} B to S, there exists, for every graded deformation B_{T} of B_{S} to T, a graded deformation M_{T} → B_{T} over M_{S} → B_{S} (i.e. a morphism of T-flat B_{T} -modules reducing to M_{S} → B_{S} via (−) ⊗_{T} S).

(ii) One correspondingly defines (M, B) to be unobstructed along any graded deformation of B by forgetting φ, i.e. by considering pairs (M_{S}, B_{S}) instead of M_{S} → B_{S}. The unobstructedness of (M, B) and (M(−s), B), s an integer, are obviously equivalent. Moreover a surjection of graded quotients of R, B → A, is unobstructed if every graded deformation B_{S} → A_{S} deforms further to T for every small Artin T ↠ S. Similarly a quotient A of R is unobstructed if every graded deformation A_{S} deforms further to T.

Note that we have defined unobstructedness of graded objects by considering graded deformations only. Moreover note that, in deforming quotients B and A of R to S the corresponding deformation of R is always the trivial one (i.e. R ⊗_{k} S) while the deformations B_{S} and A_{S} need not be trivial.

Remark 12. Let T → S be a small Artin surjection with kernel 𝔞. The obstruction of deforming a graded B_{S}-module M_{S} (resp. a graded morphism M_{S} → N_{S}) to T sits in _{0}Ext_{B}^{2}(M, M) ⊗ 𝔞 (resp. 0Ext_{B}^{1}(M, N)⊗𝔞) and _{0}Ext_{B}^{1}(M, M)⊗𝔞 (resp. _{0}Hom_{B}(M, N)⊗𝔞) corresponds to their set of graded deformations respectively. For a generically complete intersection, R ↠ B, its obstructions sit in 0Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ⊗ 𝔞. Note that, by definition, the vanishing of the obstruction is equivalent to the existence of corresponding desired deformation.

Since the whole deformation theory fits functorially together (by e.g. Laudal’s work on deformations of categories in [29]), we get in particular Proposition 13. Let M be a finitely generated graded B-module.

(i) If φ : M → B is a B-module homomorphism and _{0}Ext_{B}^{1}(M, B) = 0, then (M, φ) is unobstructed along any graded deformation of B if and only if (M, B) is unobstructed along any graded deformation of B.

(ii) If _{0}Ext_{B}^{2}(M, M) = 0, then (M, B) is unobstructed along any graded deformation of B.

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

Proof. (i) If T → S is a small Artin surjection (Definition 11), it follows from the exact sequence

0 → B ⊗_{k} 𝔞 → B_{T} → B_{S} → 0

that _{0}Hom_{B_{T}} (M_{T} , B_{T}) → _{0}Hom_{B_{S}}(M_{S}, B_{S}) is surjective and hence that _{0}Hom_{B_{S}}(M_{S}, B_{S}) → 0Hom_{B}(M, B) is surjective by induction. This implies that φ lifts to deformations of M and B, and we easily get (i). (Remark 12 may provide a quicker proof.)

Finally (ii) and (iii) follow from Remark 12. Here we leave a few details to the reader, remarking only that for (iii) the assumption _{0}Ext_{B}^{1}(M, M) = 0 implies that an isomorphism M_{S} → M′_{S} of deformations of M to B_{S} lifts further to an isomorphism of given deformations of M_{S} and M′_{S} to B_{T}, i.e. all deformations of M to B_{S} are isomorphic.

Example 14. Let B = R/I_{B} be a graded n-dimensional licci quotient of R of codimension c and suppose depth_{I(Z)} B ≥ 2 where Y − Z is an l.c.i. in Proj(R).

i) Then (K_{B}, B) is unobstructed along any graded deformation of B by Proposition 13(ii). Indeed Proposition 13(iii) also applies because K_{B_{T}} :=Ext_{B_{T}}^{c} (B_{T} , R_{T} (−n − c)) is a graded deformation of K_{B} to B_{T} by [20], Prop. A1.

ii) If c = 2, then (I_{B}/I_{B}^{2}, B) is unobstructed along any graded deformation of B by Proposition 13(ii) and Proposition 3. Moreover if depth_{I(Z)} B ≥ 4 we will see in Section 3 that (N_{B}, B) is unobstructed along any graded deformation of B because the assumptions of Proposition 13(iii) holds (here Proposition 13(ii) may not apply, cf. Remark 42).

We will need the following result on how deformations of R → A is related to deformations of R → B provided A ≃ B/I_{A/B} is a graded quotient of B ([23], Thm. 5 and Rem. 6). To state it, let U ⊂ GradAlg^{H_{A}}(R) be a sufficiently small open subset containing (A) and let p : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R) be the second projection, i.e. given by p((B′ → A′)) = (A′). Then the k-points of the subset p(p^{−1}(U)) of U correspond to quotients R → A′ with Hilbert function H_{A} for which there exist some factorization B′ → A′ such that B′ has Hilbert function H_{B}. We will call it a stratum of H_{B}-factorizations around (A), and dim U − dim p(p^{−1}(U)) the codimension of the H_{B}-stratum of A. At least if U is smooth, it is the ordinary codimension of p(p^{−1}(U)) in U. A is called H_{B}-generic if there is an open subset U_{A} of GradAlg^{H_{A}}(R) such that (A) ∈ U_{A} ⊂ p(p^{−1}(U)). The H_{B}-stratum of A at (B → A), (resp. its codimension), is defined to be p(U′) (resp. dim U − dim p(U′)) where U′ ⊂ p^{−1}(U) is an open subset in the union of the irreducible components of p^{−1}(U) which contain (B → A), and (B → A) ∈ U′. If (I_{A/B})_{v} = 0 for v ≤ the largest degree of the minimal generators of I_{B}, then p|_{U′} will be unramified and universally injective and the two concepts of codimension above coincide ([23], Lem. 7).

Theorem 15. Let R be a graded polynomial k-algebra, let B = R/I_{B} ↠ A ≃ B/I_{A/B} be a graded morphism of quotients of R and suppose _{0}H^{2}(B, A, A) = 0, depth_{𝔪} A ≥ min(1, dim A) and depth_{𝔪} B ≥ min(1, dim B).

A) If _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, A) = 0 and (I_{B})_{℘} is syzygetic for any graded ℘ of Ass(A), then A is unobstructed as a graded R-algebra (indeed _{0}H^{2}(R, A, A) = 0), and

dim_{(A)} GradAlg^{H_{A}}(R) = _{0}hom_{R}(I_{B}, B) + _{0}hom_{B}(I_{A/B}, A) − _{0}hom_{R}(I_{B}, I_{A/B})+
0ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, B).

Moreover let B be unobstructed as a graded R-algebra, let k be of characteristic zero and suppose (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}). Then the codimension of the H_{B}-stratum of A at (B → A) is

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

B) If _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = 0 and (I_{B})_{℘} is syzygetic for any graded prime ℘ of Ass(I_{A/B}),

then A is H_{B}-generic. Moreover A is unobstructed as a graded R-algebra if and only if B is unobstructed as a graded R-algebra. Indeed

0hom_{R}(I_{A}, A) − dim_{(A)} GradAlg^{H_{A}}(R) = _{0}hom_{R}(I_{B}, B) − dim_{(B)} GradAlg^{H_{B}}(R), and
dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + _{0}hom_{B}(I_{A/B}, A) − _{0}hom_{R}(I_{B}, I_{A/B}) .

Remark 16. (i) Theorem 15 follows directly from Thm. 5 and Rem. 6 of [23]. Moreover one may replace the assumption “ _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = 0 and (I_{B})_{℘} syzygetic for any graded prime ℘ of Ass(I_{A/B})” of Theorem 15B) by “ _{0}Ext_{R}^{1}(I_{B}, I_{A/B}) = 0” and conclude exactly as in Theorem 15B) because, what’s needed to prove part B) is _{0}H^{2}(R, B, I_{A/B}) = 0 (cf. the proof of Thm. 5 of [23]). Since we in general have an injection _{0}H^{2}(R, B, I_{A/B}) ↪ _{0}Ext_{R}^{1}(I_{B}, I_{A/B}), we get the claim.

(ii) To find the codimension of H_{B}-stratum of A at (B → A), the proof of [23], Thm. 5 uses generic smoothness. Indeed it computes the dimension of the image p(U′), see the text before Theorem 15 above, in terms of the dimension of GradAlg(H_{B}, H_{A}) at (B → A), and the dimension, 0hom_{R}(I_{B}, I_{A/B}), of a general fiber. Unfortunately the word “general”, i.e. the assumption “(B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B})” is missing in [23], Thm. 5 as well as in Theorem 15 of the published version, [Collect. Math. 58, 2 (2007), 199-238], of this paper. Indeed it is easily seen from the proof in [23] (cf. [23], Prop 4(i)) that we may suppose U′ above is a smooth irreducible scheme such that the restriction of the first projection q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R), q((B′ → A′)) = (B′), to U′ is a smooth morphism. Hence if we let “(B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B})” mean that, for a given (B → A), _{0}hom_{R}(I_{B}, I_{A/B}) obtains its least possible value in U′, or equivalently that _{0}hom_{R}(I_{B}, I_{A/B}) = _{0}hom_{R}(I_{B′}, I_{A′/B′}) for every (B′ → A′) in an open neighbourhood of (B → A) in U′, the proof is complete. Note that this assumption of generality is always satisfied if _{0}hom_{R}(I_{B}, I_{A/B}) = 0, in which case we do not need to suppose “char(k) = 0” either because then p|_{U′} is unramified. In particular in [23]; Theorem 1, Proposition 13, Theorem 16 (and hence in Theorem 23 of [Collect. Math. 58, 2 (2007), 199-238]), for the codimension statement in the A)-part of the results we need to assume that “(B) is general with respect to _{−s}hom_{R}(I_{B}, K_{B})”, i.e. that _{−s}hom_{R}(I_{B}, K_{B}) obtains its least possible value in the open subset q(U′) of GradAlg^{H_{B}}(R), since we in this application use [23], Thm. 5 with I_{A/B} := K_{B}(−s). Several results of the published version of this paper in Collect. Math. 58, 2 (2007), 199-238, as well as the first version of this paper on the arXiv, i.e. Thm. 1, Cor. 18, Cor. 20, Thm. 25, Thm. 30, Cor. 37 and Cor. 44, use Theorem 15 and lack the generality assumption on (B → A) above which is needed for the validity of the codimension statements of the strata. In this version all these results are corrected, and in the examples where the codimensions of the strata are stated and computed, one may check that _{0}hom_{R}(I_{B}, I_{A/B}) = 0, so no corrections are needed there.

The following lemma is a slight improvement of [23], Lem. 15 which we will need to transform Theorem 15 into a result on a Gorenstein quotient of a CM algebra.

Lemma 17. Let B be a graded CM quotient of R and let A ≃ B/I_{A/B} be Gorenstein such that the canonical module K_{A} ≃ A(j) and dim B − dim A = r. We have

(i) If r > 0 ( resp. r = 0 ), then _{0}H^{2}(B, A, A) = 0 if and only if _{−j} Ext_{B}^{r}(S_{2}(I_{A/B}), K_{B}) = 0 (resp. iff the natural map_{−j} Hom_{B}(I_{A/B}, K_{B}) →_{−j} Hom_{B}(S_{2}(I_{A/B}), K_{B}) is surjective).

(ii) _{0}hom_{B}(I_{A/B}, A) = _{−j} ext_{B}^{r}(B/I_{A/B}^{2}, K_{B}) − 1 for any r ≥ 0. Moreover if r ≥ 2, then 0hom_{B}(I_{A/B}, A) = _{−j} ext_{B}^{r−1}(S_{2}(I_{A/B}), K_{B}) − 1 .

(iii) Let dim A = 0. If the surjection S_{2}(I_{A/B})_{j} → (I_{A/B}^{2})_{j} is an isomorphism, then_{0}H^{2}(B, A, A) = 0 and _{0}hom_{B}(I_{A/B}, A) = dim B_{j} − dim S_{2}(I_{A/B})_{j} − 1. In particular if S_{2}(I_{A/B})_{j} = 0, then 0H^{2}(B, A, A) = 0 and _{0}hom_{B}(I_{A/B}, A) = dim B_{j} − 1.

Proof. (i) By [23], Lemma 15 (whose proof is quite close to the proof below), we get the vanishing of 0H^{2}(B, A, A) from_{0}Ext_{B}^{r}(S_{2}(I_{A/B}), K_{B}(−j)) = 0, and conversely if r > 0. By paying a little extra attention to the case r = 0 we get (i).

(ii) We have by A ≃ K_{A}(−j) and Gorenstein duality (applied to both A and B) that

0Hom_{A}(I_{A/B}/I_{A/B}^{2}, A) ≃ _{j}H_{𝔪}^{dim A}(I_{A/B}/I_{A/B}^{2})^{∨} ≃ _{−j}Ext_{B}^{r}(I_{A/B}/I_{A/B}^{2}, K_{B}).

Moreover since Ext_{B}^{r}(A, K_{B}) ≃ Hom_{A}(A, K_{A}) ≃ A(j) and Ext_{B}^{r−i}(D, K_{B}) = 0 for i > 0 provided D is an A-module, we get _{0}hom_{B}(I_{A/B}, A) =_{−j} ext_{B}^{r}(B/I_{A/B}^{2}, K_{B}) − 1 by the long exact sequence of Hom(−, K_{B}) applied to 0 → I_{A/B}/I_{A/B}^{2} → B/I_{A/B}^{2} → A → 0. Moreover applying Hom(−, K_{B}) onto 0 → I_{A/B}^{2} → B → B/I_{A/B}^{2} → 0, we get _{−j} Ext_{B}^{r}(B/I_{A/B}^{2}, K_{B}) ≃ _{−j} Ext_{B}^{r−1}(I_{A/B}^{2}, K_{B}) provided r > 1. Since the kernel of the surjection S_{2}(I_{A/B}) → I_{A/B}^{2} is an A-module (namely H_{2}(B, A, A), cf. [31], Section 2.1), we get Ext_{B}^{r−1}(I_{A/B}^{2}, K_{B}) ≃ Ext_{B}^{r−1}(S_{2}(I_{A/B}), K_{B}) and we have (ii).

(iii) If dim A = 0 and the surjection S_{2}(I_{A/B})_{j} → (I_{A/B}^{2})_{j} is an isomorphism, we get dim B = r and _{j}H_{2}(B, A, A) = 0 and hence

−j Ext_{B}^{r}(S_{2}(I_{A/B}), K_{B})^{∨} ≃_{j}H_{𝔪}^{0}(S_{2}(I_{A/B})) ≃_{j}H_{𝔪}^{0}(I_{A/B}^{2})

by Gorenstein duality. If r > 0 (resp. r = 0), the last group vanishes (resp. injects into_{j} H_{𝔪}^{0}(I_{A/B})). Hence we get_{0}H^{2}(B, A, A) = 0 by (i) and since_{−j} Ext_{B}^{r}(B/I_{A/B}^{2}, K_{B})^{∨} ≃_{j}H_{𝔪}^{0}(B/I_{A/B}^{2}) = (B/I_{A/B}^{2})_{j}, we conclude by (ii) and by assumption.

Corollary 18. Let B = R/I_{B} be Cohen-Macaulay and let A ≃ B/I_{A/B} be a graded Gorenstein quotient such that K_{A} ≃ A(j) and dim B − dim A = r. If

(i) _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 and (I_{B})_{℘} is syzygetic for any graded ℘ of Ass(A) ∪ Ass(B),

(ii) _{−j}Ext_{B}^{r}(S_{2}(I_{A/B}), K_{B}) = 0, and

(iii) _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, I_{A/B}) = 0,

then A is unobstructed as a graded R-algebra (indeed _{0}H^{2}(R, A, A) = 0). Moreover, if r ≥ 2, then

dim_{(A)} GradAlg^{H_{A}}(R) = _{0}hom_{R}(I_{B}, B) +_{−j} ext_{B}^{r−1}(S_{2}(I_{A/B}), K_{B}) − 1 − δ(I_{A/B})_{0}

where δ(I_{A/B})_{0} := _{0}hom_{R}(I_{B}, I_{A/B}) − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}). Furthermore if char(k) = 0 and (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}), then the codimension of the H_{B}-stratum of A at (B → A) is _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}).

Proof. By (i) we have that B is unobstructed. Moreover (i), (iii) and the long exact sequence of Hom_{B}(I_{B}/I_{B}^{2}, −) applied to 0 → I_{A/B} → B → A → 0 show the first line of assumptions of Theorem 15A). Hence we conclude by Lemma 17.

Remark 19. We get _{v}H^{2}(R, A, A) = 0 by twisting the three Ext_{B}^{i}-vanishing assumptions of Corollary 18 by v because Theorem 15A) admits such a generalization (cf. [23], Rem. 6(a)).

Corollary 20. Let B = R/I_{B} be Cohen-Macaulay and let A ≃ B/I_{A/B} be a graded Artinian Gorenstein quotient such that K_{A} ≃ A(j) and dim B − dim A = r. If

(i) _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 and (I_{B})_{℘} is syzygetic for any graded ℘ of Ass(A) ∪ Ass(B),

(ii) dim S_{2}(I_{A/B})_{j} = dim(I_{A/B}^{2})_{j} , and

(iii) _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, I_{A/B}) = 0,

then A is unobstructed as a graded R-algebra (indeed _{0}H^{2}(R, A, A) = 0), and

dim_{(A)} PGor(H_{A}) = _{0}hom_{R}(I_{B}, B) + dim B_{j} − dim S_{2}(I_{A/B})_{j} − 1 − δ(I_{A/B})_{0}

where δ(I_{A/B})_{0} := _{0}hom_{R}(I_{B}, I_{A/B}) − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}). Furthermore if char(k) = 0 and (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}), then the codimension of the H_{B}-stratum of A at (B → A) is _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}).

The proof is similar to the preceding Corollary. Note that if B is a c.i. and (I_{A/B})_{v} = 0 for v ≤ j/2, then all assumptions of Corollary 20 are satisfied. Hence [18], Thm. 4.17 is generalized by Corollary 20 and we get in addition that PGor(H_{A}) is generically smooth along the component of [18], Thm. 4.17.

Remark 21. If the Gorenstein algebra A is given by (5), then condition (iii) of Corollary 18 and 20 is satisfied provided

0Ext_{B}^{i+1}(I_{B}/I_{B}^{2}, M_{i}(−is)) = 0 for 1 ≤ i ≤ r  (9)

letting M_{1} = M. Indeed splitting the exact sequence (5) into short exact sequences, we get

0 → Z_{i} → M_{i}(−is) → Z_{i−1} → 0

where Z_{i} = ker(M_{i}(−is) → M_{i−1}((1 − i)s)) for i ≥ 1 and Z_{0} = I_{A/B}. Successively applying _{0}Ext_{B}^{i+1}(I_{B}/I_{B}^{2}, −) onto this sequences and using (9) we get _{0}Ext_{B}^{i+1}(I_{B}/I_{B}^{2}, Z_{i−1}) = 0 from 0Ext_{B}^{i+1}(I_{B}/I_{B}^{2}, M_{i}(−is)) = 0 and _{0}Ext_{B}^{i+2}(I_{B}/I_{B}^{2}, Z_{i}) = 0, i.e. we get (iii) because Z_{r−1} = M_{r}(−rs).

It is desirable to weaken the general assumption_{0}H^{2}(B, A, A) = 0 of Theorem 15. In the appendix we prove a nice result, Theorem 47, on deformations of the degeneracy locus of a regular section of a maximal CM sheaf, which leads to a variation of Theorem 15B) (Theorem 22 below) in which the mentioned assumption is replaced by assumptions closer to set-up of Theorem 8. The main idea of the proof of Theorem 47 and hence of Theorem 22 is to use the complete intersection property (in large enough open subset) of a regular section to control all deformations of the degeneracy locus. Since the proofs are probably of most interest only for specialists in deformation theories, we delay them to the appendix. For the case r = 1 and depth_{I(Z)} B = 1, which requires special attention, we refer to [23].

Theorem 22. Let R be a finitely generated polynomial k-algebra, let B = R/I_{B} be a graded Cohen-Macaulay quotient of R of codimension c, and let M be a finitely generated graded maximal Cohen-Macaulay B-module. Let Y = Proj(B), let Z be a closed scheme such that dim B − dim B/I(Z) ≥ max(r, 2) and let U = Y − Z. Let A be defined by a regular section σ of M̃^{∗}(s) on U, i.e. given by (5). Let M_{i} = H_{∗}^{0}(U, ∧^{i}M̃) for i ≥ 0, and suppose M̃|_{U} is locally free (of rank r > 0) and

∧^{r}M̃|_{U} ≃ K̃_{B}(t)|_{U} (for some integer t).

Moreover suppose M_{i} are maximal Cohen-Macaulay B-modules for 2 ≤ i ≤ r/2. If

(i) _{0}Ext_{B}^{i}(M, M_{i}(−(i − 1)s)) = 0 for 2 ≤ i ≤ r − 1, and

(ii) (M(−s), σ) is unobstructed along any graded deformation of B, and

(iii) either _{0}Ext_{B}^{i}(I_{B}/I_{B}^{2}, M_{i}(−is)) = 0 for 1 ≤ i ≤ r and (I_{B})_{℘} is syzygetic for any ℘ of Ass(B), or _{0}Ext_{R}^{i}(I_{B}, M_{i}(−is)) = 0 for 1 ≤ i ≤ r,

then _{0}hom_{R}(I_{B}, B) − dim_{(B)} GradAlg^{H_{B}}(R) = _{0}hom_{R}(I_{A}, A) − dim_{(A)} GradAlg^{H_{A}}(R), A is H_{B}-generic and Gorenstein, and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + _{0}hom_{B}(I_{A/B}, A) − _{0}hom_{R}(I_{B}, I_{A/B}).

Moreover A is unobstructed as a graded R-algebra if and only if B is unobstructed as a graded R-algebra.

Fortunately, by results of the next sections we will see that many of the Ext-groups of Corollary 18 and Theorem 22 vanish, and that the dimension formula turns out to be computable, provided we apply it to a licci quotient B of R of small enough codimension.

2 Families of Gorenstein quotients of low codimension on Cohen-Macaulay algebras

An important issue of this paper is to study families of graded Gorenstein quotients A obtained by taking regular sections of the dual of a maximal Cohen-Macaulay sheaf of rank r, i.e. quotients given by Theorem 8. In the following we will see how Corollary 18 and Theorem 22 enable us to treat the low rank cases r ≤ 3 on a licci scheme satisfactorially. The different values of r, 1 ≤ r ≤ 3, require special attention and lead to the three theorems of this section.

The case r = 1 was considered in [23]. There we proved the following result in which N_{B} := Hom_{B}(I_{B}/I_{B}^{2}, B) and K_{B}^{∗} := Hom_{B}(K_{B}, B) ≃ Hom_{B}(S_{2}(K_{B}), K_{B}). The conclusions below about when “the codimension of the stratum is zero” overlap results proved by others ([4], Thm. 3.2, cf. [15], Thm. 3.5). Note that we say “the stratum of quotients given by (10) around or at (A)” for “the H_{B}-stratum of A at (B → A)” provided any (B′ → A′) in a small enough open neighbourhood of (B → A) in GradAlg(H_{B}, H_{A}) is given by (10).

Theorem 23. Let B = R/I_{B} be a generically Gorenstein, graded Cohen-Macaulay quotient of a polynomial ring R, and let A be a graded codimension one quotient of B, given by an exact sequence

0 → K_{B}(−s) → B → A → 0 , s an integer .  (10)

A) If B is licci, then A is unobstructed as a graded R-algebra (indeed H^{2}(R, A, A) = 0), A is Gorenstein and,

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + dim(K_{B}^{∗})_{s} − 1 − δ(K_{B})_{−s}

where δ(K_{B})_{−s} = _{−s} hom_{B}(I_{B}/I_{B}^{2}, K_{B}) − _{−s} ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}). Moreover if char(k) = 0 and (B) is general with respect to _{−s}hom_{B}(I_{B}/I_{B}^{2}, K_{B}), then the codimension of the stratum of quotients given by (10) around (A) is _{−s} ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}).

B) If s >> 0 and Proj(B) is locally licci, then A is Gorenstein and the codimension of the stratum of quotients given by (10) around (A) is zero (so A is H_{B}-generic), and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + dim(K_{B}^{∗})_{s} − 1.

Moreover A is unobstructed as a graded R-algebra iff B is unobstructed as a graded R-algebra.

Example 24. (Arithmetically Gorenstein curves Proj(A) in ℙ^{5}, obtained by (10).) Let R be a polynomial ring in 6 variables, and let Y = Proj(B) be a generically Gorenstein ACM-surface with resolution

0 → R(−7)^{3} → R(−6)^{3} ⊕ R(−5)^{3} → R(−3)^{4} → I_{B} → 0

and Hilbert function H_{B}(v) = 19\binom{v+1}{2} − 26v + 16, v ≥ 2. We may obtain B by taking a c.i. B′ of type (2, 2, 2) and linking it to B via some c.i. of type (3, 3, 3). Since the linkage result, Prop. 33 of [23], connects all invariants of B, appearing in Theorem 23, to the corresponding invariants of B′ and since N_{B′} ≃ B′(2)^{⊕3}, Hom_{B′}(I_{B′}/I_{B′}^{2}, K_{B′}) ≃ K_{B′}(2)^{⊕3} and K_{B′} ≃ B′, we get by [23] Prop. 33 that dim(N_{B})_{0} = dim(N_{B′})_{0} + 3H_{B}(3) − 3H_{B′}(3) = 96 , and that

dim(K_{B}^{∗})_{v} = dim B_{v−3} + 3 dim(I_{B/D})_{v} − δ(K_{B′})_{v−6} = 67 + (19v^{2} − 99v)/2

because δ(K_{B′})_{v} = _{v} hom_{B′}(I_{B′}/I_{B′}^{2}, K_{B′}). Similarly, δ(K_{B})_{v−6} = dim B′_{v−3} + 3 dim(I_{B′/D})_{v} − dim(K_{B′}^{∗})_{v}. Hence we get δ(K_{B})_{v} = 0 for v ≤ −7 and (δ(K_{B})_{−5}, δ(K_{B})_{−6}) = (−6, −1). Let A be defined by (10) for s ≥ 5, so X = Proj(A) is an AG curve in ℙ^{5} with Hilbert polynomial p_{X}(v) = (19s − 33)v − s(19s − 33)/2. Then A is unobstructed by Theorem 23 and (cf. (8)),

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(X)} Hilb^{p_{X}} (ℙ^{5}) = 162 + (19s^{2} − 99s)/2 − δ(K_{B})_{−s} ,

where _{−s} ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = −δ(K_{B})_{−s} represents the codimension of quotients given by (10) at (A).

In the following we concentrate on the case r > 1, notably r = 2. If r = 2, then Theorem 8 provides us with an exact sequence

0 → K_{B}(t − 2s) → M(−s) →^{σ} B → A → 0 ,  (11)

and we have M ≃ Hom_{B}(M, K_{B})(t). We define γ(S_{2}M)_{v} by

γ(S_{2}M)_{v} = _{v}hom_{B}(S_{2}(M), K_{B}(t)) − _{v}ext_{B}^{1}(S_{2}(M), K_{B}(t)) .

Note that if char(k) ≠ 2, then we get an exact sequence 0 → Hom_{B}(S_{2}(M), K_{B}(t)) → Hom_{B}(M, M) → B → 0 from the split exact sequence 0 → ∧^{2}M → M ⊗ M → S_{2}(M) → 0 and (3). The split exact sequence and (3) also lead to isomorphisms

Ext_{B}^{i}(M, M) ≃ Ext_{B}^{i}(S_{2}(M), K_{B}(t)) for i = 1 (resp. i = 2)  (12)

provided depth_{I(Z)} B ≥ 3 (resp. depth_{I(Z)} B ≥ 4). Indeed if depth_{I(Z)} B ≥ 3 say, we have

Ext_{B}^{1}(M, M) ≃ Ext_{O_{U}}^{1} (M̃|_{U}, M̃|_{U}) ≃ Ext_{O_{U}}^{1} (M̃ ⊗ M̃^{∗} ⊗ K̃_{B}|_{U}, K̃_{B}|_{U}) ≃ Ext_{B}^{1}(S_{2}(M), K_{B}(t)) where we have used { \atop (M^{∗} ⊗ K_{B})}|_{U} ≃ { \atop M(−t)}|_{U} which follows from M ≃ Hom_{B}(M, K_{B})(t). Hence if depth_{I(Z)} B ≥ 3 and char(k) ≠ 2, then γ(S_{2}M)_{v} is given by

γ(S_{2}M)_{v} = _{v}hom_{B}(M, M) − _{v}ext_{B}^{1}(M, M) − dim B_{v}.

If Q is a finitely generated B-module, we define δ(Q)_{v} as previously by

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

If A is a quotient of B given by Theorem 8 and hence defined by (11), then a stratum of quotients given by (11) around (A) is of the form p(W), p : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R) the second projection, where W is a maximal closed subset of a small enough open affine neighbourhood of (B → A) in GradAlg(H_{B}, H_{A}), consisting of quotients (B′ → A′) for which there exists an extension ξ′ : 0 → K_{B′}(t − 2s) → M′(−s) → I_{A′/B′} → 0 which is induced by a regular section of some M̃′|_{U′} as in Theorem 8 (M̃′|_{U′} locally free and Proj(B′) − U′ of codimension at least two). Moreover if (B_{W} → A_{W}) is the pullback to W of the universal element of GradAlg(H_{B}, H_{A}) and I_{A_{W} /B_{W}} = ker(B_{W} → A_{W}), then there is an element ξ_{W} ∈ Ext_{B_{W}}^{1} (I_{A_{W} /B_{W}} , K_{B_{W}} (t − 2s)) whose obvious pullbacks are ξ′ above and the extension given by (11). Note that if W is open in GradAlg(H_{B}, H_{A}), then the stratum defined above coincides with the H_{B}-stratum of A at (B → A).

Theorem 25. Let B = R/I_{B} be a graded licci quotient of R, let M be a graded maximal Cohen-Macaulay B-module, and suppose M̃ is locally free of rank 2 in U := Proj(B) − Z, that dim B − dim B/I(Z) ≥ 2 and ∧^{2}M̃|_{U} ≃ K̃_{B}(t)|_{U}. Let A be defined by a regular section σ of M̃^{∗}(s) on U, i.e. given by (11) and suppose _{s}Ext_{B}^{1}(M, B) = 0.

A) If _{t}Ext_{B}^{2}(S_{2}(M), K_{B}) = 0 and _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, M) = 0 ,

then A is unobstructed as a graded R-algebra (indeed _{0}H^{2}(R, A, A) = 0), A is Gorenstein, and

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim(M^{∗})_{s} − 1 − γ(S_{2}M)_{0} + dim(K_{B})_{t−2s} + δ(K_{B})_{t−2s} − δ(M)_{−s}.

Moreover if char(k) = 0 and (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}), then the codimension of the H_{B}-stratum of A at (B → A) is

0ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, M) − dim(im β)

where β is the homomorphism _{−2s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}(t)) → _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) induced by (11). This codimension also equals the codimension of the stratum of quotients given by (11) around (A).

B) If (M, B) is unobstructed along any graded deformation of B and _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) = 0, then A is Gorenstein and the stratum of quotients given by (11) around (A) is open in GradAlg^{H_{A}}(R) (so A is H_{B}-generic). Moreover A is unobstructed as a graded R-algebra and the dimension formula for dim_{(A)} GradAlg^{H_{A}}(R) of part A) holds.

Remark 26. i) Looking to the proof below we can weaken the assumption “B is a licci quotient” of Theorem 25B) to “B is a generically syzygetic unobstructed CM quotient satisfying_{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}(t− 2s)) = 0”, and conclude as in Theorem 25B).

ii) The most natural vanishing condition for _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) in Theorem 25B) seems to be s ≥ max n_{2,j} −a where max n_{2,j} is the largest degree of a minimal relation of I_{B} (cf. (4)) and a is an integer which satisfies M_{v} = 0 for v ≤ a. In this case we get _{−s}Hom_{R}(I_{B}, M) = _{−s}Ext_{R}^{1}(I_{B}, M) = 0 and hence δ(M)_{−s} = 0. Since K_{B}(t)_{v} = 0 for v ≤ a − s by (11) we get _{−2s}Ext_{R}^{i}(I_{B}, K_{B}(t)) = 0 for i ≤ 1 and δ(K_{B})_{t−2s} = 0, as well as dim(K_{B})_{t−2s} = 0. It follows that

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim(M^{∗})_{s} − 1 − γ(S_{2}M)_{0}.

iii) Arguing as in ii) one shows _{−v}Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}) = 0 for i ≤ 1 and hence δ(K_{B})_{−v} = 0 provided v > max n_{c,i} + max n_{2,i} − n − c, e.g. provided v > 2 reg(I_{B}) − n.

Remark 27. If dim B − dim B/I(Z) ≥ 4 we can replace GradAlg^{H_{A}}(R) by Hilb^{p}(ℙ) in all conclusions, obviously modified ( A by Proj(A) etc.), of Theorem 25 by (8), cf. Remark 49.

Proof. A) We need to verify the assumptions of Corollary 18. Since B is licci and generically Gorenstein (K̃_{B}(t)|_{U} is locally free), one knows that Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 ([5]) and Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}) = 0 for i ≥ 2 ([31], Thm.4.2.6), and that I_{B} is syzygetic (see [23], proof of Thm. 16). It follows from Lemma 28 below and Remark 21 that all assumptions of Corollary 18 hold. To see that the dimension formulas of GradAlg^{H_{A}}(R) in Theorem 25 and Corollary 18 coincide, we use (11) and we get that δ(I_{A/B})_{0} = δ(M)_{−s} − δ(K_{B})_{t−2s} as well as

0ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, M) − dim(im β).

Then we conclude by Lemma 28. For the final codimension statement, we refer to Lemma 29.

B) A generically Gorenstein licci quotient is generically syzygetic (cf. [23], proof of Thm. 16) and satisfies _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}(t − 2s)) = 0. Hence we get Theorem 25B) from Proposition 13(i) and Theorem 22, since the dimension formulas of Theorem 25 coincide with that in Theorem 22 by Lemma 28 and Lemma 29 (see part A) of the proof).

Lemma 28. Let B be Cohen-Macaulay, let M and Z be as in (the three first lines of) Theorem 25 and let A ≃ B/I_{A/B} be given by (11) (so r = dim B − dim A = 2 and K_{A} ≃ A(j) where j = 2s − t). If _{s}Ext_{B}^{1}(M, B) = 0, then

0hom_{B}(I_{A/B}, A) = dim(M^{∗})_{s} − 1 − γ(S_{2}M)_{0} + dim(K_{B})_{t−2s}.

If in addition _{t}Ext_{B}^{2}(S_{2}(M), K_{B}) = 0, then _{−j}Ext_{B}^{2}(S_{2}(I_{A/B}), K_{B}) = 0, i.e. _{0}H^{2}(B, A, A) = 0.

Proof. To show the dimension formula we remark that _{0}hom_{B}(I_{A/B}, A) =_{−j}ext_{B}^{1}(S_{2}(I_{A/B}), K_{B})−1 by Lemma 17. Moreover note that there exists an exact sequence

M ⊗ K_{B}(t − 3s) → S_{2}(M(−s)) → S_{2}(I_{A/B}) → 0  (13)

whose leftmost map becomes injective if we sheafify and restrict to U. (Indeed ∧^{2}K̃_{B}|_{U} = 0, or if we prefer, the morphism U ∩ Proj(A) ↪ Proj(B) is an l.c.i.). Let MK := ker(S_{2}(M(−s)) → S_{2}(I_{A/B})) and ∧ := ker(M ⊗ K_{B}(t − 3s) → MK). Since ∧ is supported at Z, we get Hom_{B}(∧, K_{B}) = 0. It follows that Hom_{B}(MK, K_{B}) ≃ Hom_{B}(M ⊗ K_{B}(t − 3s), K_{B}) and that

t−2sExt_{B}^{1}(MK, K_{B}) ⊆_{t−2s}Ext_{B}^{1}(M ⊗ K_{B}(t − 3s), K_{B}) ≃ _{0}Ext_{B}^{1}(M(−s), B) = 0.

Hence (13) induces a long exact sequence (i.e. exact in degree t − 2s)

... → Hom_{B}(M(t − 3s), B) → Ext_{B}^{1}(S_{2}(I_{A/B}), K_{B}) → Ext_{B}^{1}(S_{2}(M(−s)), K_{B}) → 0. (14)

Since it is easy to see that the socle degree j of A is j = 2s − t from (11), we get the conclusion by counting dimensions in (14) in degree t − 2s and by observing that Hom_{B}(S_{2}(I_{A/B}), K_{B}) ≃ Hom_{B}(I_{A/B}^{2}, K_{B}) ≃ Hom_{B}(I_{A/B}, K_{B}) ≃ K_{B}. Indeed these isomorphisms follow from the fact that ker(S_{2}(I_{A/B}) → I_{A/B}^{2}) and coker(I_{A/B}^{2} ↪ I_{A/B}) are A-modules (see the proof of Lemma 17).

By Lemma 17, we get _{0}H^{2}(B, A, A) ≃ _{−j}Ext_{B}^{2}(S_{2}(I_{A/B}), K_{B}). Now continuing the exact sequence (14) we get an injection _{−j}Ext_{B}^{2}(S_{2}(I_{A/B}), K_{B}) ↪ _{t−2s}Ext_{B}^{2}(S_{2}(M(−s)), K_{B}) and we conclude easily.

Lemma 29. Let B be Cohen-Macaulay, let M and Z be as in (the three first lines of) Theorem 25 and let A ≃ B/I_{A/B} be given by (11) (so r = dim B − dim A = 2 and K_{A} ≃ A(2s − t)). Then the H_{B}-stratum of A at (B → A) and the stratum of quotients given by (11) around (A) coincide. Moreover if _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = 0, then the mentioned strata are open in GradAlg^{H_{A}}(R).

Proof. To see that the H_{B}-stratum and the stratum of quotients given by (11) coincide, it suffices to show that any (B′ → A′) in a small enough open neighbourhood of t := (B → A) in G := GradAlg(H_{B}, H_{A}) is given by (11). Let (S, m_{S}) be the local ring of G at t and let S_{i} = S/m_{S}^{i}. Since G is a scheme of finite type which represents the corresponding functor of graded flat quotients, there exists a universal quotient whose pullback to Spec(S) is denoted by B_{S} → A_{S} (with kernel I_{A_{S}/B_{S}}, flat over S). Since K_{B_{S}} := Ext_{R_{S}}^{c}(B_{S}, R_{S}(−n−c)) where R_{S} := R⊗_{k} S, is S-flat (cf. [20], Prop.A1), it suffices to show that the natural map _{t}Ext_{B_{S}}^{1}(I_{A_{S}/B_{S}}, K_{B_{S}}(−2s)) → _{t}Ext_{B}^{1}(I_{A/B}, K_{B}(−2s)) is surjective because then, there is an extension of I_{A_{S}/B_{S}} by K_{B_{S}}(t − 2s) over S, which reduces to the given extension of I_{A/B} by K_{B}(t−2s), and which extends to extensions in an open neighbourhood of t in G (the assumptions of Theorem 8 needed to get (11), i.e. M̃′|_{U′} is locally free and Proj(B′) − U′ is of codimension at least two is quite easy to get. Note that the maximal Cohen-Macaulayness of M′ may be deduced from the extension). To show the surjectivity, let F_{S} be a graded B_{S}- free module which surjects into I_{A_{S}/B_{S}}, and let Q_{S} := ker(F_{S} → I_{A_{S}/B_{S}}). Let Q_{S_{i}} := Q_{S} ⊗_{S} S_{i} and Q := Q_{S} ⊗_{S} k. A simple diagram chasing shows that it suffices to prove the surjectivity of tHom(Q_{S}, K_{B_{S}}(−2s)) → _{t}Hom(Q, K_{B}(−2s)). As in the proof of [23], Prop. 13, it suffices to show that

η_{i} : _{t}Hom(Q_{S_{i}}, K_{B_{S_{i}}} (−2s)) → _{t}Hom(Q_{S_{i−1}}, K_{B_{S_{i−1}}} (−2s))

is surjective. Since Ext_{B}^{1}(Q, K_{B}) ≃ Ext_{B}^{2}(I_{A/B}, K_{B}) = 0, this follows by applying _{t−2s}Hom(Q_{S_{i}}, −) onto 0 → B ⊗_{k} 𝔞 → B_{S_{i}} → B_{S_{i−1}} → 0 where 𝔞 := m_{S}^{i−1}/m_{S}^{i}, i.e. we get the surjectivity of η_{i} from tExt_{B}^{1}(Q, K_{B}(−2s)) = 0 and we get what we want. Finally note that by Prop. 4(ii) and (5) of [23] the assumption _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = 0 implies that the second projection GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R) maps small enough open sets of (B → A) to open sets of (A) and we are done.

Since we need some vanishing results on Ext-groups to use Theorem 25 effectively, we will delay specific examples to the next section. Instead we consider the rank 3 case which will allow us to treat certain Gorenstein families of codimension 5 or more. Here there are even more Ext-groups involved. Fortunately they all vanish for “good” modules on licci schemes. Now if r = 3, then Theorem 8 provides us with an exact sequence

0 → K_{B}(t − 3s) → M^{∨}(t − 2s) → M(−s) → B → A → 0 ,  (15)

where M^{∨} = Hom_{B}(M, K_{B}). Let

γ(S_{2}M)_{v} = _{v}hom_{B}(S_{2}(M), K_{B}(t)) − _{v}ext_{B}^{1}(S_{2}(M), K_{B}(t)) + _{v}ext_{B}^{2}(S_{2}(M), K_{B}(t))

(one more term than in the rank 2 case !), let γ(M, M)_{v} = _{v}hom_{B}(M, M) − _{v}ext_{B}^{1}(M, M) while let δ(Q)_{v} = _{v}hom_{B}(I_{B}/I_{B}^{2}, Q) − _{v}ext_{B}^{1}(I_{B}/I_{B}^{2}, Q) be as previously.

Theorem 30. Let B = R/I_{B} be a graded licci quotient of R, let M be a graded maximal Cohen-Macaulay B-module, and suppose M̃ is locally free of rank 3 in U := Proj(B) − Z, that dim B − dim B/I(Z) ≥ 3 and ∧^{3}M̃|_{U} ≃ K̃_{B}(t)|_{U}. Let A be defined by a regular section σ of M̃^{∗}(s) on U, i.e. given by (15) and suppose _{s}Ext_{B}^{1}(M, B) = 0.

A) If _{0}Ext_{B}^{2}(M^{∨} ⊗ M, K_{B}) = _{−s}Ext_{B}^{3}(S_{2}(M), K_{B}(t)) = 0 and _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, M) = −2sExt_{B}^{3}(I_{B}/I_{B}^{2}, M^{∨}(t)) = 0 , then A is unobstructed as a graded R-algebra (indeed _{0}H^{2}(R, A, A) = 0), A is Gorenstein, and

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim(M^{∗})_{s} + γ(S_{2}M)_{−s} − γ(M, M)_{0} − dim(K_{B})_{t−3s} − δ ,

where

δ := δ(M)_{−s} + δ(K_{B})_{t−3s} − δ(M^{∨})_{t−2s} −_{−2s} ext_{B}^{2}(I_{B}/I_{B}^{2}, M^{∨}(t)).

Moreover if char(k) = 0 and (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}), then the codimension of the H_{B}-stratum of A at (B → A) is

0ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, M) + _{−2s} ext_{B}^{2}(I_{B}/I_{B}^{2}, M^{∨}(t)) − dim(im β)

where β is the homomorphism _{−2s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M^{∨}(t)) → _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) induced by (15).

B) If (M, B) is unobstructed along any graded deformation of B and _{−s}Ext_{B}^{2}(M, M^{∨}(t)) = −sExt_{B}^{2}(S_{2}(M), K_{B}(t)) = 0 and _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) = _{−2s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, M^{∨}(t)) = 0, then A is Gorenstein and H_{B}-generic. Moreover A is unobstructed as a graded R-algebra and the dimension formula for dim_{(A)} GradAlg^{H_{A}}(R) of part A) holds (this formula simplifies a little due to the assumed vanishing of the Ext-groups).

Remark 31. i) Using (3) one may see that _{−s}Ext_{B}^{2}(M, M^{∨}(t)) ≃ _{−s}Ext_{B}^{2}(S_{2}(M), K_{B}(t)) provided depth_{I(Z)} B ≥ 4 and char(k) ≠ 2, cf. the assumptions of Theorem 30B).

ii) Moreover one may replace “_{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) = _{−2s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, M^{∨}(t)) = 0” in Theorem 30B) by

−sExt_{R}^{1}(I_{B}, M) = _{−2s}Ext_{R}^{2}(I_{B}, M^{∨}(t)) = 0

and still conclude as in Theorem 30B). This follows from Theorem 22. This variation is particularly useful if the codimension of B in R is 2, in which case Ext_{R}^{2}(I_{B}, M^{∨}(t)) vanishes. A natural vanishing condition for _{−s}Ext_{R}^{1}(I_{B}, M) is again s ≥ max n_{2,j} −a where max n_{2,j} is the largest degree of a minimal relation of I_{B} and a is an integer which satisfies M_{v} = 0 for v ≤ a. In this case we get _{−s}Ext_{R}^{i}(I_{B}, M) = 0 for i = 0, 1 and hence δ(M)_{−s} = 0.

Lemma 32. Let B be Cohen-Macaulay, let M and Z be as in (the three first lines of) Theorem 30 and let A ≃ B/I_{A/B} be given by (15) (so r = dim B − dim A = 3 and K_{A} ≃ A(j) where j = 3s − t) and suppose _{s}Ext_{B}^{1}(M, B) = 0.

A) If _{0}Ext_{B}^{2}(M^{∨} ⊗ M, K_{B}) = 0, then

0hom_{B}(I_{A/B}, A) = dim(M^{∗})_{s} + γ(S_{2}M)_{−s} − γ(M, M)_{0} − dim(K_{B})_{t−3s}

If in addition _{−s}Ext_{B}^{3}(S_{2}(M), K_{B}(t)) = 0, then _{−j}Ext_{B}^{3}(S_{2}(I_{A/B}), K_{B}) ≃ _{0}H^{2}(B, A, A) = 0.

B) If _{−s}Ext_{B}^{2}(S_{2}(M), K_{B}(t)) = 0 then _{0}hom_{B}(I_{A/B}, A) is given as in part A).

Proof. A) Thanks to Lemma 17 we have _{0}hom_{B}(I_{A/B}, A) =_{−j}ext_{B}^{2}(S_{2}(I_{A/B}), K_{B}) − 1. To compute −jext_{B}^{2}(S_{2}(I_{A/B}), K_{B}), we look to the exact sequence

M^{∨} ⊗ M(t − 3s) → S_{2}(M(−s)) → S_{2}(I_{A/B}) → 0  (16)

where one knows that the kernel of { \atop M^{∨} ⊗ M}(t−3s)|_{U} → { \atop S_{2}(M)}(−2s)|_{U} is (∧^{2}M̃^{∨}(2t−4s)⊕K̃_{B}(t− 3s))|_{U} (U ∩ Proj(A) ↪ Proj(B) is an l.c.i.). Let M^{∨}M := ker(S_{2}(M(−s)) → S_{2}(I_{A/B})). Since ∧^{2}M̃^{∨}|_{U} ≃ M̃ ⊗ K̃_{B}(−t)|_{U}, we have an exact sequence

0 → (K̃_{B}(t − 3s) ⊕ (M̃ ⊗ K̃_{B})(t − 4s))|_{U} → { \atop M^{∨} ⊗ M}(t − 3s)|_{U} → { \atop M^{∨}M}|_{U} → 0

to which we apply Hom(−, K̃_{B}|_{U}). Since Ext_{B}^{i}(−, N) ≃ Ext_{O_{U}}^{i} (˜−|_{U}, Ñ|_{U}) for i ≤ 1 for every maximal CM module N by (3), we get a long exact sequence starting as

0 → Hom_{B}(M^{∨}M, K_{B}(−j)) → Hom_{B}(M^{∨} ⊗ M(t − 3s), K_{B}(−j)) → Hom_{B}(K_{B}, K_{B}) ⊕ H → ...

where H := Hom_{B}(M ⊗ K_{B}(t − 4s), K_{B}(−j)) ≃ Hom_{B}(M(−s), B) and stopping at _{0}Ext_{B}^{1}(M ⊗ K_{B}(t − 4s), K_{B}(−j)) ≃ _{s}Ext_{B}^{1}(M, B) = 0. Noting that

0Ext_{B}^{i}(M^{∨} ⊗ M, K_{B}) ≃ Ext_{O_{U}}^{i} (M̃^{∗} ⊗ K̃_{B} ⊗ M̃|_{U}, K̃_{B}|_{U}) ≃ _{0}Ext_{B}^{i}(M, M) for i ≤ 1,

we get

∑_{i=0}^{1}(−1)^{i}_{−j}ext_{B}^{i}(M^{∨}M, K_{B}) = ∑_{i=0}^{1}(−1)^{i}_{0}ext_{B}^{i}(M, M) − 1 −_{s}hom_{B}(M, B) .

Note that the long exact sequence above and the assumption _{0}Ext_{B}^{2}(M^{∨} ⊗ M, K_{B}) = 0 also show 0Ext_{B}^{2}(M^{∨}M, K_{B}) = 0. Now combining with the long exact sequence which we get by applying 0Hom(−, K_{B}(−j)) = 0 to 0 → M^{∨}M → S_{2}(M)(−2s) → S_{2}(I_{A/B}) → 0;

... → _{0}Ext_{B}^{1}(M^{∨}M, K_{B}(−j)) → _{0}Ext_{B}^{2}(S_{2}(I_{A/B}), K_{B}(−j)) → _{0}Ext_{B}^{2}(S_{2}(M)(−2s), K_{B}(−j)) → 0 (17)

which implies

∑_{i=0}^{2}(−1)^{i}_{−j}ext_{B}^{i}(S_{2}(I_{A/B}), K_{B}) = γ(S_{2}M)_{−s} − ∑_{i=0}^{1}(−1)^{i}_{−j}ext_{B}^{i}(M^{∨}M, K_{B}) ,

we get the dimension formula because

Hom_{B}(S_{2}(I_{A/B}), K_{B}) ≃ Hom_{B}(I_{A/B}^{2}, K_{B}) ≃ Hom_{B}(I_{A/B}, K_{B}) ≃ K_{B}

and Ext_{B}^{1}(S_{2}(I_{A/B}), K_{B}) ≃ Ext_{B}^{1}(I_{A/B}, K_{B}) = 0 (ker(S_{2}(I_{A/B}) → I_{A/B}^{2}) and coker(I_{A/B}^{2} ↪ I_{A/B}) are A-modules). Part B) is proven in exactly the same way (the sequence (17) stops by “one module earlier”). Moreover continuing (17), we get the “if in addition” statement of part A) from the assumption _{−s}Ext_{B}^{3}(S_{2}(M), K_{B}(t)) = 0.

Proof of Theorem 30. A) follows from Corollary 18, Remark 21 and Lemma 32 because B is licci and generically Gorenstein (cf. the proof of Theorem 25 for details). Note that we need to split (15) into two short exact sequences, and apply _{0}Hom_{B}(I_{B}/I_{B}^{2}, −) to them, to see the formula of δ in Theorem 30 (δ obviously equals δ(I_{A/B})_{0} by definition of the latter). The same splitting, together with _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, M) = 0 and _{−3s}Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}(t)) = 0 for i > 1, shows that

0ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, M) + _{−2s}ext_{B}^{2}(I_{B}/I_{B}^{2}, M^{∨}(t)) − dim(im β),

and we get the codimension statement.

B) We get Theorem 30B) from Proposition 13(i), Theorem 22 and Lemma 32 (see part A) of the proof for the dimension formula).

Remark 33. If we in Theorem 25A) and Theorem 30A) replace the vanishing of all _{v}Ext_{B}^{i}-groups by the vanishing of the corresponding Ext_{B}^{i}-groups (skipping the index v), we get that A is strongly unobstructed in the sense H^{2}(R, A, A) = 0 (cf. Remark 19).

3 Rank two and three sheaves on codimension two quotients

In this section we will see that Theorem 25 applies to the normal module, M = N_{B} and Theorem 25 and 30 to the 1. Koszul homology module, M = H_{1}. Of course Theorem 25 also applies to modules of the form M = S_{2}(K_{B})⊕K_{B}^{∗}(t), where t is an integer. For such decomposable modules Theorem 25 does not really lead to new results since they may be treated by applying Theorem 23.

Before giving our applications we need two propositions to handle the vanishing of the Ext-groups involved. In what follows, B = R/I is an n-dimensional codimension two CM quotient of R,

0 → G_{2} := ⊕_{j=1}^{μ−1}R(−n_{2,j}) → G_{1} := ⊕_{i=1}^{μ}R(−n_{1,i}) → I → 0  (18)

is a minimal resolution and Y = Proj(B) is an l.c.i. in an open set U = Y −Z where depth_{I(Z)} B ≥ 1. Taking R-duals, we get a minimal resolution

0 → R → ⊕R(n_{1,i}) → ⊕R(n_{2,j}) → K_{B}(n + 2) → 0  (19)

to whom we apply Hom(−, B) to see the exactness to the left in the exact sequence

0 → K_{B}(n + 2)^{∗} → ⊕B(−n_{2,j}) → ⊕B(−n_{1,i}) → I/I^{2} → 0  (20)

Note that (20) splits into two short exact sequences “via ⊕B(−n_{2,j}) ↠ H_{1} ↪ ⊕B(−n_{1,i})”, one of which is (6) with H_{2}(R, B, B) = 0.

Proposition 34. Let l be a natural number, let B = R/I be a codimension two CM quotient of R, and suppose depth_{I(Z)} B ≥ l. Then

Ext_{B}^{i}(I/I^{2}, I/I^{2}) = 0 for 1 ≤ i ≤ l.

Proof. Since pd_{R}I = 1, we have Ext_{R}^{i}(I, I/I^{2}) = 0 and Tor_{i}^{R}(I, B) = 0 for i ≥ 2. Moreover, using the spectral sequence Ext_{B}^{p}(Tor_{q}^{R}(I, B), I/I^{2}) which converges to Ext_{R}^{p+q}(I, I/I^{2}), we get isomorphisms Ext_{B}^{i−2}(Tor_{1}^{R}(I, B), I/I^{2}) ≃ Ext_{B}^{i}(I/I^{2}, I/I^{2}) for i > 2 and an exact sequence

0 → Ext_{B}^{1}(I/I^{2}, I/I^{2}) → Ext_{R}^{1}(I, I/I^{2}) → Hom_{B}(Tor_{1}^{R}(I, B), I/I^{2}) → Ext_{B}^{2}(I/I^{2}, I/I^{2}) → 0.

Note that Ext_{R}^{1}(I, −) is right exact by pd_{R}I = 1. In particular Ext_{R}^{1}(I, R) ≃ Ext_{R}^{1}(I, A) and it follows that N_{B} ≃ Ext_{R}^{1}(I, I) ≃ Ext_{R}^{1}(I, R) ⊗ I ≃ K_{B}(n + 2) ⊗ I/I^{2}. Similarly we get

Ext_{R}^{1}(I, I/I^{2}) ≃ Ext_{R}^{1}(I, R) ⊗ I/I^{2} ≃ N_{B} .

Hence there is an injection H_{I(Z)}^{0}(Ext_{B}^{1}(I/I^{2}, I/I^{2})) ↪ H_{I(Z)}^{0}(N_{B}). Since Ĩ/I^{2}|_{U} is locally free, we get H_{I(Z)}^{0}(Ext_{B}^{1}(I/I^{2}, I/I^{2})) ≃ Ext_{B}^{1}(I/I^{2}, I/I^{2}). Moreover one knows that N_{B} is a maximal CM module (cf. the text after (6)), and we conclude that Ext_{B}^{1}(I/I^{2}, I/I^{2}) = 0.

Let p ≤ l be a natural number and suppose we have proved Ext_{B}^{i}(I/I^{2}, I/I^{2}) = 0 for 1 ≤ i < p. By the spectral sequence (resp. the two displayed formulas) above, it suffices to show Ext_{B}^{p−2}(Tor_{1}^{R}(I, B), I/I^{2}) = 0 for p > 2 (resp. Ext_{B}^{p−2}(Tor_{1}^{R}(I, B), I/I^{2}) ↪ N_{B} a graded injection for p = 2). We have H_{I(Z)}^{i}(I/I^{2}) = 0 for i ≤ p − 2 by (6). Hence we get an injective graded map

Ext_{B}^{p−2}(Tor_{1}^{R}(I, B), I/I^{2}) ↪ Ext_{O_{U}}^{p−2}({ \atop Tor_{1}^{R}(I, B)}|_{U}, { \atop I/I^{2}}|_{U}) ≃ Ext_{O_{U}}^{p−2}({ \atop Tor_{1}^{R}(I, K_{B}(n + 2))}|_{U}, Ñ_{B}|_{U}), (21)

(cf. (3)), noting that { \atop Tor_{1}^{R}(I, K_{B})} ≃ { \atop Tor_{1}^{R}(I, B) ⊗ K_{B}} and { \atop I/I^{2} ⊗ K_{B}} ≃ Ñ_{B} are isomorphic on U. Since H_{I(Z)}^{i}(N_{B}) = 0 for i ≤ p − 1, the rightmost Ext-group in (21) is, by (3) mainly, further isomorphic to Ext_{B}^{p−2}(Tor_{1}^{R}(I, K_{B}(n + 2)), N_{B}). We claim that Tor_{1}^{R}(I, K_{B}(n + 2)) ≃ R/I. Indeed we have by (19) that Tor_{1}^{R}(I, K_{B}(n + 2)) is the homology group in the middle of the complex

0 → I → ⊕I(n_{1,i}) → ⊕I(n_{2,j}) → 0.

Applying Hom_{R}(−, I) to (18), we see that ker[⊕I(n_{1,i}) → ⊕I(n_{2,j})] ≃ Hom(I, I) ≃ R and putting things together we get the claim. Now using the claim we get Ext_{B}^{p−2}(Tor_{1}^{R}(I, B), I/I^{2}) ≃ Ext_{B}^{p−2}(R/I, N_{B}) which vanishes for p > 2 and equals N_{B} for p = 2, and we are done.

Remark 35. Applying Hom_{R}(−, I/I^{2}) to (18), noting that Hom_{R}(I, I/I^{2}) ≃ Hom_{B}(I/I^{2}, I/I^{2}) and Ext_{R}^{1}(I, I/I^{2}) ≃ N_{B} (see the proof above) we get the following exact sequence

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

This sequence of graded B-modules can be used to find _{v}hom_{B}(I/I^{2}, I/I^{2}) because dim(I^{2})_{v} and dim(N_{B})_{v} can be computed from

0 → ∧^{2}(⊕R(−n_{2,j})) → (⊕R(−n_{1,i})) ⊗ (⊕R(−n_{2,j})) → S_{2}(⊕R(−n_{1,i})) → I^{2} → 0 ,  (22)
0 → G_{1}^{∗} ⊗_{R} G_{2} → ((G_{1}^{∗} ⊗_{R} G_{1}) ⊕ (G_{2}^{∗} ⊗_{R} G_{2}))/R → G_{2}^{∗} ⊗_{R} G_{1} → N_{B} → 0  (23)

Indeed the latter sequence is deduced from the exact sequence 0 → R → ⊕I(n_{1,i}) → ⊕I(n_{2,j}) → N_{B} → 0 which we get by applying Hom_{R}(−, I) to (18), (cf. [23], (26)). Note that if s > max n_{2,j} − 2 min n_{1,i}, then I^{2}(n_{i,j})_{−s} = 0 for all i, j and we get _{−s}Hom_{B}(I/I^{2}, I/I^{2}) ≃ R_{−s}.

Now we consider the 1.Koszul homology module H_{1}. In this case all assumptions of Theorem 25A) are satisfied, due to

Proposition 36. Let B = R/I be a codimension two CM quotient of R, and suppose depth_{I(Z)} B ≥

2. Then S_{2}(H_{1}) is a maximal CM B-module. Moreover

Ext_{B}^{1}(H_{1}, H_{1}) ≃ Ext_{B}^{2}(I/I^{2}, H_{1}) = 0 and Ext_{B}^{1}(H_{1}, B) = 0.

Proof. In the sequence (6), H_{2}(R, B, B) = 0. Applying Hom(I/I^{2}, −) to (6) we get isomorphisms Ext_{B}^{1}(I/I^{2}, I/I^{2}) ≃ Ext_{B}^{2}(I/I^{2}, H_{1}) because Ext_{B}^{i}(I/I^{2}, B) = 0 for 1 ≤ i ≤ 2 ([27], (7)). Hence Ext_{B}^{2}(I/I^{2}, H_{1}) = 0 by Proposition 34. Moreover applying Hom(−, B) (resp. Hom(−, H_{1})) to (6) we get Ext_{B}^{1}(H_{1}, B) ≃ Ext_{B}^{2}(I/I^{2}, B) = 0 (resp. Ext_{B}^{1}(H_{1}, H_{1}) ≃ Ext_{B}^{2}(I/I^{2}, H_{1})). Finally to see that S_{2}(H_{1}) is a maximal CM module, we consider the short exact sequence deduced from (20) “to the left”. It induces an exact sequence

0 → K_{B}(n + 2)^{∗} ⊗ (⊕B(−n_{2,j})) →^{ψ} S_{2}(⊕B(−n_{2,j})) → S_{2}(H_{1}) → 0  (24)

because ker ψ, which is supported at Z and contained in a maximal CM module, must vanish. It follows from (24) that S_{2}(H_{1}) has codepth at most one. Moreover dualizing (24) we get

0 → Hom_{B}(S_{2}(H_{1}), K_{B}) → K_{B} ⊗ S_{2}(⊕B(n_{2,j})) → ⊕S_{2}(K_{B})(n_{2,j} + n + 2)  (25)

because Hom_{B}(S_{2}(⊕B(−n_{2,j})), K_{B}) ≃ K_{B} ⊗ S_{2}(⊕B(n_{2,j})) and

Hom_{B}(K_{B}(n + 2)^{∗} ⊗ (⊕B(−n_{2,j})), K_{B}) ≃ ⊕S_{2}(K_{B})(n_{2,j} + n + 2).

Note that the cokernel of K_{B} ⊗ S_{2}(⊕B(n_{2,j})) → ⊕S_{2}(K_{B})(n_{2,j} + n + 2) is Ext_{B}^{1}(S_{2}(H_{1}), K_{B}). One may, however, easily see that this map is surjective because it is, via symmetrization and tensorization, obtained from the surjective map ⊕B(n_{2,j}) → K_{B}(n+2). Hence Ext_{B}^{1}(S_{2}(H_{1}), K_{B}) = 0, and by Gorenstein duality and the fact that the

codepth of S_{2}(H_{1}) is at most one, we get that S_{2}(H_{1}) is maximally Cohen-Macaulay.

If the number of minimal generators of I is μ(I) = 4, then the rank of H_{1} is r = 2 by (6) and Theorem 25A) applies to B-module M = H_{1}. Since all assumptions of Theorem 8 are satisfied ([27] and Proposition 10) we have an exact sequence

0 → K_{B}(t − 2s) → H_{1}(−s) → I_{A/B} → 0  (26)

with t = n + 2 − ∑ n_{1,i}. Hence

Corollary 37. Let B = R/I be a graded codimension two CM quotient of R, let U = Proj(B) − Z ↪ ℙ^{n+1} be an l.c.i. and suppose μ(I) = 4 and depth_{I(Z)} B ≥ 2. If A is defined by a regular section of H̃_{1}^{∗}(s) on U, i.e. given by (26), then A is unobstructed as a graded R-algebra (indeed 0H^{2}(R, A, A) = 0), A is Gorenstein of codimension 4 in R, and

dim_{(A)} GradAlg(H_{A}) = dim(N_{B})_{0} + dim(H_{1}^{∗})_{s} − 1 − γ(S_{2}H_{1})_{0} + dim(K_{B})_{t−2s} + δ(K_{B})_{t−2s} − δ(H_{1})_{−s}.

Moreover if char(k) = 0 and (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}), then the stratum of quotients given by (26) around (A) is irreducible and its codimension is _{−s}ext_{B}^{1}(I/I^{2}, H_{1}) − dim(im β) where β is the homomorphism _{−2s}Ext_{B}^{1}(I/I^{2}, K_{B}(t)) → _{−s}Ext_{B}^{1}(I/I^{2}, H_{1}) induced by (26). Furthermore if (B′) ∈ GradAlg^{H_{B}}(R) satisfies the same assumptions as B above and defines A′ as B defined A, then the closures in GradAlg^{H_{A}}(R) of the stratum of quotients given by (26) around (A) and the corresponding stratum around (A′) coincide. If in addition s > max n_{2,j} − min n_{2,j}, then the stratum of quotients given by (26) around (A) is open (so A is H_{B}-generic), and

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim(H_{1}^{∗})_{s} − 1 − _{0}hom_{B}(S_{2}(H_{1}), K_{B}(t)) .

Note that dim(H_{1}^{∗})_{s} is easily computed from the exact sequence

0 → N_{B} → ⊕_{i=1}^{μ}B(n_{1,i}) → H_{1}^{∗} → 0  (27)

(the B-dual sequence of (6) is short exact due to Ext_{B}^{1}(I/I^{2}, B) = 0). Moreover γ(S_{2}H_{1})_{0} = 0hom_{B}(S_{2}(H_{1}), K_{B}(t)) is computed from (25) and the minimal resolution ([7], p. 595),

0 → ∧^{2}(⊕R(n_{1,i})) → (⊕R(n_{1,i})) ⊗ (⊕R(n_{2,j})) → S_{2}(⊕R(n_{2,j})) → S_{2}(K_{B})(2n + 4) → 0 (28)

Proof. Since all assumptions of Theorem 25A) are taken care of by Proposition 36, we get Corollary 37 from Theorem 25A) except the irreducibility of the stratum, its uniqueness (i.e. that the strata above coincide, up to closure) and the final statement of Corollary 37. The irreducibility is trivial because the stratum is the image of an irreducible set. Indeed GradAlg(H_{B}, H_{A}) is smooth at (B → A) because B is unobstructed and _{0}H^{2}(B, A, A) = 0 by Lemma 28. The proof of the uniqueness is essentially the same as for [23], Prop. 23(i) and Thm. 24 (see the two first lines of the proof of [23], Thm. 24) because the open subscheme of GradAlg^{H_{B}}(R) consisting of CM quotients is irreducible. Moreover, since ⊕B(−n_{2,i}) ↠ H_{1} is surjective (cf. (20) ) we have (H_{1})_{v} = 0 for v < min n_{2,j}. By Remark 26(ii) and the assumption s > max n_{2,j} − min n_{2,j}, it follows that −sExt_{B}^{1}(I/I^{2}, H_{1}) = 0. Hence A is H_{B}-generic and the formula for dim_{(A)} GradAlg^{H_{A}}(R) simplifies as in Corollary 37 according to Remark 26(ii).

We illustrate Corollary 37 by a example in which we compute all numbers in the dimension formula of dim_{(A)} GradAlg^{H_{A}}(R) as well as the number leading to the codimension of the H_{B}- stratum. Even though the example is relatively simple, our methods of computation may be used quite generally (to treat sections of H_{1}^{∗}(s) on a codimension 2 CM quotient of R generated by μ(I) = 4 generators).

Example 38. It is well known how to find a minimal resolution of H_{1}. Indeed by [2],

0 → ∧^{2}G_{2} → ∧^{2}G_{1} → G_{2} → H_{1} → 0  (29)

where the free modules G_{i} belong to (18). Applying the mapping cone construction to (26) we get the following resolution of the Gorenstein algebra A = R/I_{A} of Corollary 37,

0 → R(b) → G_{1}^{∗}(b) ⊕ ∧^{2}G_{2}(−s) → G_{2}^{∗}(b) ⊕ ∧^{2}G_{1}(−s) ⊕ G_{2} → G_{2}(−s) ⊕ G_{1} → I_{A} → 0 (30)

where b := − ∑ n_{1,i} − 2s and μ(I) = 4. As a special case we suppose the resolution (18) is linear (i.e. all n_{1,i} = 3). We get the following resolution (obviously minimal for s > 0) of A,

\begin{matrix}0 → R(−12 − 2s) → R(−9 − 2s)^{4} ⊕ R(−8 − s)^{8} \\ → R(−8 − 2s)^{3} ⊕ R(−6 − s)^{6} ⊕ R(−4)^{3} → R(−4 − s)^{3} ⊕ R(−3)^{4} → R → A → 0 . \end{matrix}  (31)

Let R be a polynomial ring in four variables. Then Y = Proj(B) is a curve of degree d = 6 and with Hilbert polynomial p_{Y} (v) = 6v − 2. Moreover A is Artinian of socle degree 2s + 8 and with h-vector (1, 4, 10, .., 6s + 16, 6s + 19, 6s + 16, .., 10, 4, 1). We suppose s ≥ −1 to avoid discussing very special cases. If Y is an l.c.i., then A is unobstructed by Corollary 37 of a dimension which we now calculate. Indeed dim(N_{B})_{0} = h^{0}(Ñ_{B}) = 4d = 24 while, by (27),

dim(H_{1}^{∗})_{s} = 4 · p_{Y} (s + 3) − h^{0}(Ñ_{B}(s)) = 24(s + 3) − 8 − 4d − 2ds = 12s + 40

for s ≥ −1 (one may see that (N_{B})_{s} = 12 = 4d + 2ds also for s = −1 by (23)). Moreover

0hom_{B}(S_{2}(H_{1}), K_{B}(−8)) = 6 dim(K_{B})_{0} − 3 dim(S_{2}(K_{B})_{0}) = 0

by (25), (19) and (28). Hence by Corollary 37, if s > 0, then dim_{(A)} PGor(H_{A}) =

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim(H_{1}^{∗})_{s} − 1 − _{0}hom_{B}(S_{2}(H_{1}), K_{B}(−8)) = 12s + 63

and A is H_{B}-generic, i.e. the closure of the family given by (26) forms a (12s + 63)-dimensional generically smooth, irreducible component of GradAlg^{H_{A}}(R). It is interesting to observe that we can also use Lemma 17(iii) to compute_{0}hom_{B}(I_{A/B}, A) and hence dim(H_{1}^{∗})_{s}−1−_{0}hom_{B}(S_{2}(H_{1}), K_{B}(−8)) (cf. Lemma 28). Indeed since we get dim S_{2}(I_{A/B})_{2s+8} = 6 from (31), it follows that

0hom_{B}(I_{A/B}, A) = dim B_{2s+8} − dim S_{2}(I_{A/B})_{2s+8} = 12s + 40

which again leads to dim_{(A)} GradAlg^{H_{A}}(R) = 12s + 63.

Finally suppose s ≤ 0. We still have δ(K_{B})_{−8−2s} = 0 and _{−2s}Ext_{B}^{1}(I/I^{2}, K_{B}(−8)) = 0 for s > −2 by Remark 26(iii) while (K_{B})_{−8−2s} = 0 by (19). It remains to compute δ(H_{1})_{−s} and −sext_{B}^{1}(I/I^{2}, H_{1}). To do so we apply Hom(I/I^{2}, −) to (6), and we get

0 → Hom(I/I^{2}, H_{1}) → Hom(I/I^{2}, ⊕B(−n_{1,i})) → Hom(I/I^{2}, I/I^{2}) → Ext_{B}^{1}(I/I^{2}, H_{1}) → 0  (32)

Note that (N_{B})_{v} = 0 for v < −1 by (23). Hence

−sext_{B}^{1}(I/I^{2}, H_{1}) =_{−s}hom(I/I^{2}, I/I^{2}) = dim R_{−s}

and δ(H_{1})_{−s} = −_{−s}ext_{B}^{1}(I/I^{2}, H_{1}) for −1 ≤ s ≤ 0 by Remark 35. By Corollary 37,

dim_{(A)} GradAlg^{H_{A}}(R) = 12s + 63 +_{−s}ext_{B}^{1}(I/I^{2}, H_{1}) for − 1 ≤ s ≤ 0,

and the codimension, _{−s}ext_{B}^{1}(I/I^{2}, H_{1}), of the H_{B}-stratum of A is 1, 4 for s = 0, −1 respectively.

Remark 39. In view of Remark 35, we can always use (32) to find δ(H_{1})_{−s}. Moreover δ(K_{B})_{t−2s} is always computable from the exact sequence we get by applying Hom(−, K_{B}(t)) to (6), cf. [23], Remark 14(d). Since dim(K_{B})_{t−2s} is given by (19), we see that all members of the dimension formula for GradAlg^{H_{A}}(R) in Corollary 37 are easily computed, even by hand. On the other side, the formula for the codimension is not always straightforward. However, in the range max n_{2,j} − 2 min n_{1,i} < s ≤ max n_{2,j} − min n_{2,j} , one may show

−sext_{B}^{1}(I/I^{2}, H_{1}) = −δ(H_{1})_{−s} = dim R_{−s}.

Indeed by (32) it suffices to show N_{B}(−n_{1,i})_{−s} = 0 for every i. Since ⊕R(n_{2,j} − n_{1,i′}) → N_{B} is surjective and ⊕R(n_{2,j} −n_{1,i′} −n_{1,i})_{−s} = 0 we conclude by Remark 35. Finally note that in the range max n_{2,j} − 2 min n_{1,i} < s ≤ max n_{2,j} − min n_{2,j} , _{−s}ext_{B}^{1}(I/I^{2}, H_{1}) is the codimension of the H_{B}- stratum of Corollary 37 because we have _{−2s}Ext_{B}^{1}(I/I^{2}, K_{B}(t)) = 0. Indeed by Remark 26(iii) we get the last mentioned vanishing provided 2s−t > 2 max n_{2,j} −n−2, i.e. provided s > max n_{2,j} −∑ n_{1,i}/2 (by t = n + 2 − ∑_{i=1}^{i=4} n_{1,i}) which holds since ∑_{i=1}^{i=4} n_{1,i}/2 ≥ 2 min n_{1,i}.

Example 40. (Arithmetically Gorenstein curves Proj(A) in ℙ^{5}, obtained by (26).) Here we reconsider the preceding example, in dimension two higher. Let R be a polynomial ring in 6 variables, and let Y = Proj(B) be a threefold satisfying depth_{I(Z)} B ≥ 2 whose resolution (18) is linear with μ = 4. Let A be defined by a regular section of H̃_{1}^{∗}(s)|_{Y −Z}, so X = Proj(A) is an AG curve in ℙ^{5}. The Hilbert polynomial of Y is given by “integrating” 6v−2 two times, or more precisely p_{Y} (v) = 6\binom{v+2}{3}− 2\binom{v+2}{2}+3\binom{v+1}{1}. The h-vector of A is of course still (1, 4, 10, .., 6s+16, 6s+19, 6s+16, .., 10, 4, 1) and the Hilbert polynomial of X is p_{X}(v) = (6s^{2} + 44s + 81)(v − s − 3). This time we suppose s ≥ −2, only avoiding the degenerate case. A is unobstructed by Corollary 37. To find the dimension of GradAlg^{H_{A}}(R), let η(v) := dim(I/I^{2})_{v} and note η(v) = dim(I)_{v} for v < 6. Then dim(N_{B})_{0} = 3η(4)−4η(3) = 48 by Remark 35. Moreover, by (27), dim(H_{1}^{∗})_{s} = 4·dim B_{(3+s)} − dim(N_{B})_{s} where dim(N_{B})_{s} = 3 dim I_{(4+s)} − 4 dim I_{(3+s)} + dim R_{s} by Remark 35. Hence

dim(H_{1}^{∗})_{s} = 4 · dim R_{(3+s)} − 3 dim I_{(4+s)} − dim R_{s} = 4\binom{s + 8}{5} − 12\binom{s + 6}{5} + 8\binom{s + 5}{5} = 2(s + 4)^{2}(s + 5).

Moreover _{0}hom_{B}(S_{2}(H_{1}), K_{B}(−6)) = 6 dim(K_{B})_{2} − 3 dim S_{2}(K_{B})_{4} = 0 by (25), (19) and (28). Hence by Corollary 37 and (8), if s > 0, then

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(X)} Hilb^{p_{X}} (ℙ^{5}) = 2(s + 4)^{2}(s + 5) + 47.

and the quotients given by (26) generate a generically smooth, irreducible component of GradAlg^{H_{A}}(R). Now δ(K_{B})_{−6−2s} = 0 and _{−2s}Ext_{B}^{1}(I/I^{2}, K_{B}(−6)) = 0 for s > −2 by Remark 26(iii) while (K_{B})_{−6−2s} = 0 for s ≥ −2 by (19). To compute δ(H_{1})_{−s} and _{−s}ext_{B}^{1}(I/I^{2}, H_{1}) we use (32) and we get _{−s}ext_{B}^{1}(I/I^{2}, H_{1}) = dim R_{−s} and δ(H_{1})_{−s} = −_{−s}ext_{B}^{1}(I/I^{2}, H_{1}) for −1 ≤ s ≤ 0 by Remark 39. By Corollary 37,

dim_{(A)} GradAlg(H_{A}) = 2(s + 4)^{2}(s + 5) + 47 + _{−s}ext_{B}^{1}(I/I^{2}, H_{1}) for − 1 ≤ s ≤ 0,

and the codimension, _{−s}ext_{B}^{1}(I/I^{2}, H_{1}), of the H_{B}-stratum of A is 1, 6 for s = 0, −1 respectively. Furthermore, for s = −2, we compute δ(H_{1})_{2} and δ(K_{B})_{−2} exactly as described in Remark 39 above and we get its values to be −3 and 3 respectively. Hence dim_{(A)} GradAlg^{H_{A}}(R) = 2(s + 4)^{2}(s + 5) + 47 = 71 in this case.

Finally we pay some extra attention to the case s = 0 of AG curves of degree d = 81 and genus g = 244. In this case the stratum given by (26) forms a 207 dimensional irreducible family contained in an irreducible component of GradAlg^{H_{A}}(R) or of Hilb^{p_{X}} (ℙ^{5}) of dimension 208. It is interesting to observe that we have exactly the same degree and genus as for the AG curve with s = 6 of Example 24 where a 207 dimensional stratum in a 208 dimensional component was constructed by means of Theorem 23! Their resolutions, however, differ (see (31) for the AG curve of the first family where R(−4)^{3} appears as a repeated factor, and use linkage to see that R(−5)^{3} appears as a repeated factor in the resolution of the second family). We would like to pose the following two questions. Are these two strata subschemes of codimension one of the same irreducible component of GradAlg^{H_{A}}(R)? Is the minimal resolution of the generic Gorenstein algebra of this component obtained from the resolution of (31) by deleting all repeated factors?

Now we apply Theorem 25 to M = N_{B}. By Theorem 8 and Proposition 10 we have the exact sequence

0 → K_{B}(t − 2s) → N_{B}(−s) → I_{A/B} → 0  (33)

where t = n + 2. Since we have not been able to verify the assumption _{t}Ext_{B}^{2}(S_{2}(N_{B}), K_{B}) = 0 of Theorem 25A), even by increasing depth_{I(Z)} B to its largest possible value, we must use the B)-part of the Theorem. Here we get advantage of developing the concept “unobstructed along any graded deformation of B”, and we must suppose s “large enough”. Indeed we get

Corollary 41. Let B = R/I be a graded codimension two CM quotient of R, let U = Proj(B)−Z ↪ ℙ^{n+1} be an l.c.i. and suppose depth_{I(Z)} B ≥ 4. If A is given by a regular section of Ñ_{B}^{∗}(s) on U and if s > 2 max n_{2,j} − min n_{1,i} and char(k) ≠ 2, then A is unobstructed as a graded R-algebra, and the stratum of quotients given by (33) around (A) is open and irreducible (so A is H_{B}-generic). Moreover A is Gorenstein of codimension 4 in R, and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(X)} Hilb^{p}(ℙ^{n+1}) = dim(N_{B})_{0} + dim(I/I^{2})_{s} − _{0}hom(I/I^{2}, I/I^{2}) .

Letting η(v) := dim(I/I^{2})_{v}, we also have

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(X)} Hilb^{p}(ℙ^{n+1}) = η(s) + ∑_{j=1}^{μ−1} η(n_{2,j}) − ∑_{i=1}^{μ} η(n_{1,i}) .

Furthermore if (B′) ∈ GradAlg^{H_{B}}(R) satisfies the same assumptions as B above and defines A′ as B defined A, then the closures in GradAlg^{H_{A}}(R) of the stratum of quotients given by (33) around (A) and the corresponding stratum around (A′) coincide, i.e. they form the same irreducible component of GradAlg^{H_{A}}(R).

Proof. Firstly to see that (N_{B}, B) is unobstructed along any graded deformation of B, it suffices by Proposition 13(iii) to show that _{0}Ext_{B}^{1}(N_{B}, N_{B}) = 0 since N_{B_{T}} := Hom(I_{T} /I_{T}^{2} , B_{T}) (B_{T} := R_{T} /I_{T} a graded deformation of B to an Artinian T) is a graded deformation of N_{B} to B_{T} by [20], Prop. A1 and _{0}Ext_{B}^{1}(I/I^{2}, B) = 0. Since depth_{I(Z)} B ≥ 4 implies depth_{I(Z)} N_{B} ≥ 4 and depth_{I(Z)} I/I^{2} ≥ 3 we get by (3) and Proposition 34,

0Ext_{B}^{1}(N_{B}, N_{B}) ≃ H_{∗}^{1}(U, Hom_{O_{U}} (Ĩ/I^{2}^{∗}|_{U}, Ĩ/I^{2}^{∗}|_{U})) ≃ Ext_{B}^{1}(I/I^{2}, I/I^{2}) = 0

and we get what we want. Similarly Ext_{B}^{1}(N_{B}, B) ≃ H_{∗}^{1}(U, Hom_{O_{U}} (Ĩ/I^{2}^{∗}, B̃)) ≃ H_{∗}^{1}(U, Ĩ/I^{2}) = 0.

Secondly to show the remaining assumption of Theorem 25B), we use Remark 26(ii). Since ⊕R(n_{2,j} − n_{1,i}) → N_{B} is surjective by Remark 35, it follows that (N_{B})_{v} = 0 for v < min n_{1,i} − max n_{2,j}, and we conclude by Remark 26(ii). Note that also the dimension formula above follows from Remark 26 and (12) (see the text before (12) for _{0}hom(I/I^{2}, I/I^{2}) = _{0}hom(N_{B}, N_{B}) = 0hom(S_{2}(N_{B}), K_{B}(t)) + 1). Finally the proof of the irreducibility is trivial because the stratum is the image of an irreducible set. Indeed GradAlg(H_{B}, H_{A}) is smooth at (B → A) by the first conclusion of Theorem 47, cf. proofs of Theorem 25B) Theorem 22. The proof of the uniqueness (i.e. that the strata above coincide, up to closure) is essentially the same as for [23], Prop. 23(i) and Thm. 24 (see the two first lines of the proof of [23], Thm. 24) because the open subscheme of GradAlg^{H_{B}}(R) consisting of CM quotients is irreducible. For the connection with the Hilbert scheme, see (8).

Remark 42. Let depth_{I(Z)} B ≥ 4 and char(k) ≠ 2. Using (3) (see the proof above), we get

0Ext_{B}^{2}(N_{B}, N_{B}) ≃ Ext_{O_{U}}^{2} (Ĩ/I^{2}|_{U}, Ĩ/I^{2}|_{U}) ≃ _{0}Hom_{B}(I/I^{2}, H_{I(Z)}^{3}(I/I^{2})) ,

and H_{I(Z)}^{3}(I/I^{2}) ≃ H_{I(Z)}^{4}(I^{2}). Sometimes we can use this connection to prove the vanishing of 0Ext_{B}^{2}(N_{B}, N_{B}) in which case Theorem 25A) applies to M = N_{B} provided we can show_{−s}Ext_{B}^{2}(I/I^{2}, N_{B}) = 0. To show this vanishing, we play on N_{B} ≃ I/I^{2} ⊗ K_{B}(n + 2) and we get Ext_{B}^{2}(I/I^{2}, N_{B}) ≃ Ext_{B}^{2}(Hom(I/I^{2}, I/I^{2}), K_{B}(n + 2)) = 0 from the fact that the codepth of Hom(I/I^{2}, I/I^{2}) is at most one by Remark 35. The benefit of using Theorem 25A) (instead of Theorem 25B) which requires −sExt_{B}^{1}(I/I^{2}, N_{B}) = 0) is that we don’t need to assume s > 2 max n_{2,j} − min n_{1,i}. Hence if we in the case s ≤ 2 max n_{2,j} −min n_{1,i} suppose _{0}Hom_{B}(I/I^{2}, H_{I(Z)}^{4}(I^{2})) = 0, we get that A is an unobstructed graded Gorenstein quotient of codimension 4 in R, and

dim_{(A)} GradAlg^{H_{A}}(R) = ϵ + δ + dim(K_{B})_{t−2s}

where δ := δ(K_{B})_{t−2s} − δ(N_{B})_{−s} and ϵ := dim(N_{B})_{0} + dim(I/I^{2})_{s} − _{0}hom(I/I^{2}, I/I^{2}), i.e. ϵ is equal to the expression of η′s in Corollary 41. Continuing this argument (for this final statement we omit the details of the proof) we may even show that we can skip s > 2 max n_{2,j} − min n_{1,i} in Corollary 41 and at least get inequalities

ϵ + δ + dim(K_{B})_{t−2s} − _{0}ext_{B}^{2}(N_{B}, N_{B}) ≤ dim_{(A)} GradAlg^{H_{A}}(R) ≤ ϵ + δ + dim(K_{B})_{t−2s} .

Moreover, if the inequality to the right is an equality, then A is unobstructed.

Example 43. (Arithmetically Gorenstein curves Proj(A) in ℙ^{5}, obtained by (33).) Let R be a polynomial k-algebra in 6 variables, let B = R/I be a codimension two quotient with minimal resolution

0 → R(−3)^{2} → R(−2)^{3} → R → B → 0 .  (34)

and suppose Y = Proj(B) is an l.c.i in ℙ^{5}. Let A be given by a regular section of Ĩ/I^{2}(s). The Hilbert polynomial/function of Y is p_{Y} (v) = H_{B}(v) = 3\binom{v+2}{3}+\binom{v+2}{2} for v ≥ 0 by (34) (e.g. by “integrating” 3v + 1 two times). Thanks to (23) and (19) and the mapping cone construction applied to (33) we get the following resolution (obviously minimal for s > 4) of the Gorenstein algebra A = R/I_{A} of Corollary 41,

\begin{matrix}0 → R(−2s) → R(2 − 2s)^{3} ⊕ R(−1 − s)^{6} \\ → R(3 − 2s)^{2} ⊕ R(−s)^{12} ⊕ R(−3)^{2} → R(1 − s)^{6} ⊕ R(−2)^{3} → R → A → 0 . \end{matrix}  (35)

Let h^{i}(O_{X}(v)) = dim H^{i}(X, O_{X}(v)). Since K_{A} ≃ A(2s − 6), the Hilbert polynomial of X is of the form p_{X}(v) = h^{0}(O_{X}(v)) − h^{1}(O_{X}(v)) = d(v − s + 3) because h^{1}(O_{X}(s − 3)) = h^{0}(O_{X}(s − 3)). Moreover looking to the resolution of I_{A} we see that

p_{X}(s − 2) = h^{0}(O_{X}(s − 2)) − h^{0}(O_{X}(s − 4)) = h^{0}(O_{Y} (s − 2)) − h^{0}(O_{Y} (s − 4)) = 3s^{2} − 10s + 9.

So X = Proj(A) is an AG curve of degree d = 3s^{2} − 10s + 9 and arithmetic genus g = 1 + d(s − 3) in ℙ^{5}. If s > 2 max n_{2,j} − min n_{1,i} = 4, then Corollary 41 applies. Letting η(v) := dim(I/I^{2})_{v}, we get ∑ η(n_{2,j}) − ∑ η(n_{1,i}) = 2 dim I_{3} − 3 dim I_{2} = 23. Calculating η(s) and using (22) which implies

0 → R(−6) → R(−5)^{6} → R(−4)^{6} → I^{2} → 0  (36)

we get η(s) = (s + 1)(s − 1)^{2}. Hence, if s ≥ 5, then A is unobstructed and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(X)} Hilb^{p_{X}} (ℙ^{5}) = η(s) + 2η(3) − 3η(2) = (s + 1)(s − 1)^{2} + 23 .

Finally we discuss the cases 3 ≤ s ≤ 4. By (36), we get an injection H_{𝔪}^{4}(I^{2}) → H_{𝔪}^{6}(R(−6)). By Remark 42, _{0}Ext_{B}^{2}(N_{B}, N_{B}) = 0 and it follows that A is unobstructed and dim_{(A)} GradAlg^{H_{A}}(R) = (s + 1)(s − 1)^{2} + 23 + δ where δ := dim(K_{B})_{6−2s} + δ(K_{B})_{6−2s} − δ(N_{B})_{−s}. We have (K_{B})_{6−2s} = 0 by (19). Moreover applying Hom(−, K_{B}(6)) to (6), we get −δ(K_{B})_{6−2s} = _{−2s}hom(H_{1}, K_{B}(6)) for s ≥ 3. Since H_{1} has rank one by (6) we get H_{1} ≃ Hom(H_{0}, K_{B}(6 − ∑ n_{1,i})) ≃ K_{B} by the last conclusion of Theorem 8. Hence δ(K_{B})_{6−2s} = − dim B_{(6−2s)}, i.e. δ(K_{B})_{6−2s} = −1, 0 for s = 3, 4 respectively. It remains to compute δ(N_{B})_{−s} for which we use H_{1} ≃ K_{B} and the following exact sequence

0 → Hom(I/I^{2}, N_{B}) → Hom(I/I^{2}, ⊕B(n_{1,i})) → Hom(I/I^{2}, K_{B}^{∗}) → Ext_{B}^{1}(I/I^{2}, N_{B}) → 0  (37)

which we get by applying Hom(I/I^{2}, −) to (27). Since Hom(I/I^{2}, K_{B}^{∗}) ≃ Hom(I/I^{2}⊗K_{B}(6), B(6)) ≃ I/I^{2}(6), we get δ(N_{B})_{−s} = 2, −3 for s = 3, 4 respectively. By Remark 42, dim_{(A)} GradAlg^{H_{A}}(R) = 36, 71 for s = 3, 4 respectively. Note that this result confirms the well known formula

6d + 2(1 − g) ≤ dim_{(A)} GradAlg^{H_{A}}(R) ≤ 6d + 2(1 − g) + h^{1}(N_{X})

because for s = 3 (resp. s = 4) the curve has degree d = 6 and genus g = 0 (resp. d = 17 and genus g = 18, and one may verify h^{1}(N_{X}) = 3 by other methods).

We will finish the section by looking to the rank r = 3 case, i.e. we apply Theorem 30A) to the B-module M = H_{1} provided the number of minimal generators of I is μ(I) = 5. Note that (15) translates to t = n + 2 − ∑ n_{1,i}, H_{2} = H_{1}^{∨}(t) ≃ Hom(H_{1}, K_{B}(t)) and

0 → K_{B}(t − 3s) → H_{2}(−2s) → H_{1}(−s) → I_{A/B} → 0 .  (38)

Corollary 44. Let B = R/I be a graded codimension two CM quotient of R, let U = Proj(B)−Z ↪ ℙ^{n+1} be an l.c.i. and suppose μ(I) = 5, char(k) ≠ 2 and depth_{I(Z)} B ≥ 3. If A is defined by a regular section of H̃_{1}^{∗}(s) on U, i.e. given by (38), then A is unobstructed as a graded R-algebra (indeed _{0}H^{2}(R, A, A) = 0), A is Gorenstein of codimension 5 in R, and dim_{(A)} GradAlg^{H_{A}}(R) =

dim(N_{B})_{0} + dim(H_{1}^{∗})_{s} +_{−s}hom_{B}(S_{2}(H_{1}), K_{B}(t)) −_{0}hom_{B}(H_{1}, H_{1}) − dim(K_{B})_{t−3s} − δ ,

where δ := δ(H_{1})_{−s} + δ(K_{B})_{t−3s} − δ(H_{2})_{−2s}. Moreover if char(k) = 0 and (B → A) is general with respect to _{0}hom_{R}(I_{B}, I_{A/B}), then the codimension of the H_{B}-stratum of A at (B → A) is

−sext_{B}^{1}(I_{B}/I_{B}^{2}, H_{1}) − dim(im β)

where β is the homomorphism _{−2s}Ext_{B}^{1}(I/I^{2}, H_{2}) → _{−s}Ext_{B}^{1}(I/I^{2}, H_{1}) induced by (38). If in addition s > max n_{2,j} − min n_{2,j} , then A is H_{B}-generic, δ = 0 and dim(K_{B})_{t−3s} = 0.

Proof. This is a corollary to Theorem 30A). Indeed, thanks to Proposition 36 we only need to show 0Ext_{B}^{2}(H_{2}(−t) ⊗ H_{1}, K_{B}) = 0 and _{−2s}Ext_{B}^{i}(I/I^{2}, H_{2}) = 0 for 2 ≤ i ≤ 3 and the “if in addition”- statement. Using (6) we get _{−2s}Ext_{B}^{i}(I/I^{2}, H_{2}) ≃ _{0}Ext_{B}^{i−1}(H_{1}, H_{2}(−2s)). Since depth_{I(Z)} B ≥ 3, we have

0Ext_{B}^{1}(H_{1}, H_{2}(−2s)) ≃ Ext_{O_{U}}^{1} (H̃_{1}|_{U}, { \atop H_{1}^{∗} ⊗ K_{B}}(t − 2s)|_{U}) ≃ Ext_{O_{U}}^{1} ({ \atop H_{1} ⊗ H_{1}}|_{U}, K̃_{B}(t − 2s)|_{U})

by (3) which vanishes because 0 → H̃_{2}|_{U} ≃ ∧^{2}H̃_{1}|_{U} → { \atop H_{1} ⊗ H_{1}}|_{U} → S_{2}(H_{1})|_{U} → 0 is exact and H_{2} and S_{2}(H_{1}) are maximal CM modules (this argument also shows _{0}Ext_{B}^{2}(H_{1}, H_{2}(−2s)) = 0 provided depth_{I(Z)} B ≥ 4).

To show _{0}Ext_{B}^{2}(H_{1}, H_{2}(−2s)) = 0 under the assumption depth_{I(Z)} B ≥ 3, we use (20). Hence it suffices to show the _{v}Ext_{B}^{1}(K_{B}^{∗}, H_{2}) = 0 for every v. By (3) this Ext-group is isomorphic to

Ext_{O_{U}}^{1} (K̃_{B}^{∗}|_{U}, { \atop H_{1}^{∗} ⊗ K_{B}}(t + v)|_{U}) ≃ Ext_{O_{U}}^{1} ({ \atop K_{B}^{∗} ⊗ H_{1}}|_{U}, K̃_{B}(t + v)|_{U}) ≃ _{v}Ext_{B}^{1}(Hom_{B}(K_{B}, H_{1}), K_{B}(t))

which vanishes since Hom_{B}(K_{B}, H_{1}) is a maximal CM module. Indeed by using (20), we showed in [25] the exactness of

0 → Hom_{B}(K_{B}(n + 2), H_{1}) → ∧^{2}(⊕B(−n_{2,j})) → H_{2} → 0  (39)

(for that sequence it suffices to have depth_{I(Z)} B ≥ 2), from which we get that Hom_{B}(K_{B}, H_{1}) is a maximal CM B-module.

To prove _{t}Ext_{B}^{2}(H_{2} ⊗ H_{1}, K_{B}) = 0 , we use again (20) to get 0 → { \atop K_{B}(n + 2)}^{∗} ⊗ H̃_{2}|_{U} → (⊕B̃(−n_{2,j})) ⊗ H̃_{2}|_{U} → H̃_{2} ⊗ H̃_{1}|_{U} → 0 . Applying H_{∗}^{0}(U, −) to it, we get the exact sequence

0 → Hom_{B}(K_{B}(n + 2), H_{2}) → (⊕B(−n_{2,j})) ⊗ H_{2} → H_{2} ⊗ H_{1}/τ → 0

where τ := H_{I(Z)}^{0}(H_{2} ⊗ H_{1}). Invoking also (3) it follows that _{v−n−2}Ext_{B}^{2}(H_{2} ⊗ H_{1}/τ, K_{B}) ≃

vExt_{B}^{1}(Hom_{B}(K_{B}, H_{2}), K_{B}) ≃ Ext_{O_{U}}^{1} (K̃_{B}^{∗} ⊗ { \atop H_{1}^{∗} ⊗ K_{B}}(t)|_{U}, K̃_{B}(v)|_{U}) ≃ _{v}Ext_{B}^{1}(H_{1}^{∗}(t), K_{B}), and

that the latter vanishes since H_{1}^{∗} is a maximal CM B-module. Indeed due to (27), H_{1}^{∗} has codepth at most one, and applying Hom(−, K_{B}) to (27), we will see Ext_{B}^{1}(H_{1}^{∗}, K_{B}) = 0 as well because Hom(N_{B}, K_{B}) ≃ N_{B}(−n − 2) ≃ I/I^{2} ⊗ K_{B}. Then we conclude by the exact sequence

Ext_{B}^{2}(H_{2} ⊗ H_{1}/τ, K_{B}) → Ext_{B}^{2}(H_{2} ⊗ H_{1}, K_{B}) → Ext_{B}^{2}(τ, K_{B})

because Ext_{B}^{2}(τ, K_{B}) ↪ Ext_{O_{U}}^{2} (τ̃|_{U}, K̃_{B}|_{U}) is injective by (3) and τ̃|_{U} = 0.

Finally if s > max n_{2,j} − min n_{2,j} we get _{−s}Hom_{R}(I, H_{1}) = _{−s}Ext_{R}^{1}(I, H_{1}) = 0 and hence δ(H_{1})_{−s} = 0 by Remark 31(ii) and the “left” short exact sequence deduced from (20). Using (39) instead of (20) we similarly get δ(H_{2})_{−2s} = 0 as well as δ(K_{B})_{t−3s} = 0 and we conclude easily.

Remark 45. Due to Remark 33 the quotients A of Corollary 37 and Corollary 44 are strongly unobstructed in the sense H^{2}(R, A, A) = 0. This follows from the proofs of the corollaries because the proofs of the vanishing of the _{v}Ext_{B}^{i}-groups involved are easily extended to the vanishing of the corresponding Ext_{B}^{i}-groups. As a consequence of this, look to the scheme ZGor(H) parametrizing not necessarily graded Gorenstein quotients R → A with Hilbert function H ([18], p.126). Since vH^{2}(R, A, A) = 0 for v ≥ 0, we can use [22], Thm. 1.10 to prove the smoothness of ZGor(H) at (A), and one may also find the dimension of the scheme ZGor(H) at (A) by a formula analogous to the dimension formulas of Corollary 37 and Corollary 44, cf. [23], Prop. 29.

Example 46. Let R be a polynomial ring in five variables, and let B = R/I be a CM-quotient with μ(I) = 5 and with linear resolution (i.e. all n_{1,i} of (18) are n_{1,i} = 4). Then Y = Proj(B) is a surface with Hilbert polynomial p_{Y} (v) = 5v^{2} − 5v + 5. By [2],

0 → ∧^{3}G_{2} → ∧^{3}G_{1} → ∧^{2}G_{2} → H_{2} → 0  (40)

is exact, and the mapping cone construction applied to (38) lead to the following resolution of the Gorenstein algebra A = R/I_{A} of Corollary 44A),

\begin{matrix}0 → R(−20 − 3s) → R(−16 − 3s)^{5} ⊕ R(−15 − 2s)^{8} → R(−15 − 3s)^{4} ⊕ R(−12 − 2s)^{10}⊕ \\ R(−10 − s)^{6} → R(−10 − 2s)^{6} ⊕ R(−8 − s)^{6} ⊕ R(−5)^{4} → R(−5 − s)^{4} ⊕ R(−4)^{5} → I_{A} → 0. \end{matrix} (41)

By (38) or (41) A is Artinian of socle degree 3s + 15 and with symmetric h-vector (1, ..., 1) given by H_{A}(v) = p_{Y} (v) for 1 ≤ v ≤ s + 4, H_{A}(v) = p_{Y} (v) − 4 for v = s + 5 and

H_{A}(v) = p_{Y} (v) − 15(v − s − 4)^{2} + 35(v − s − 4) − 30 for s + 5 < v < 2(s + 5).

So if s = −3, −2, ..., then the h-vector of A is (1, 5, 11, 15, 11, 5, 1), (1, 5, 15, 31, 45, 45, 31, 15, 5, 1), ... respectively. Suppose s ≥ −3 and Y an l.c.i.. Then A is unobstructed by Corollary 44. By Remark 35, dim(N_{B})_{s} = 4 dim I_{s+5} − 5 dim I_{s+4} + dim R_{s}, so dim(N_{B})_{0} = 60. Moreover

dim(H_{1}^{∗})_{s} = 5 dim B_{s+4} − dim(N_{B})_{s} = 5\binom{s + 8}{4} − 4 dim I_{s+5} − \binom{s + 4}{4} = 15s^{2} + 125s + 265

by (27). Applying Hom(−, H_{1}) to (6), and we get

0 → Hom(I/I^{2}, H_{1}) → Hom(⊕B(−n_{1,i}), H_{1}) → Hom(H_{1}, H_{1}) → Ext_{B}^{1}(I/I^{2}, H_{1}) → 0  (42)

Hence _{0}hom(H_{1}, H_{1}) = 1 by Remark 39 and _{0}hom_{B}(S_{2}(H_{1}), K_{B}(−15 − s)) = 0 by (25) and (19). Finally dim(K_{B})_{−15−3s} = 0 and the sum dim(H_{1}^{∗})_{s} +_{−s}hom_{B}(S_{2}(H_{1}), K_{B}(t)) − _{0}hom_{B}(H_{1}, H_{1}) − dim(K_{B})_{t−3s} of Corollary 44 is equal to 15s^{2} + 125s + 264. We get

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(A)} PGor(H_{A}) = 15s^{2} + 125s + 324 − δ.

Moreover if s > 0, then δ = 0 and A is H_{B}-generic. Looking to s ≤ 0, we note that if 2s > max n_{2,j} − 2 min n_{2,j}, then _{−2s}Hom_{R}(I, H_{2}) = _{−2s}Ext_{R}^{1}(I, H_{2}) = 0 and hence δ(H_{2})_{−2s} = 0 by Remark 31 and (39). Combining with Remark 39, we get the codimension of the H_{B}-stratum of Corollary 44 to be −δ = −δ(H_{1})_{−s} =_{−s} ext_{B}^{1}(I/I^{2}, H_{1}) = dim R_{−s} for −2 ≤ s ≤ 0. In particular

dim_{(A)} GradAlg^{H_{A}}(R) = 15s^{2} + 125s + 324 + \binom{−s + 4}{4} for − 2 ≤ s ≤ 0.

In the final case s = −3 (where we skip a few details), one may see δ(H_{2})_{−2s} = 5 dim H_{2}(4)_{6} − 6hom_{B}(S_{2}(H_{1}), K_{B}(−15)) = 0 (from the sequence we get by applying Hom(−, H_{2}) to (6)), and

δ(H_{1})_{−s} = 5 dim N_{B}(−4)_{3} − _{3}hom_{B}(I/I^{2}, I/I^{2}) = 5 · 20 − (4 dim I_{8}^{2} + dim R_{3}) = 5

by Remark 39, and we get dim_{(A)} GradAlg^{H_{A}}(R) = 15s^{2} + 125s + 324 − 5 = 79.

4 Appendix: Deformations of quotients of zerosections

Let X = Proj(A) be a subscheme of Y = Proj(B) ⊂ ℙ = Proj(R) defined as the degeneracy locus of a regular section of some sheaf M supported on Y. In this appendix we prove a quite general result, Theorem 47, concerning the unobstructedness and the “family-dimension” of a quotient A obtained from B as the homogeneous coordinate ring of a zerosection as above, in which we neither assume B to be Cohen-Macaulay, nor A to be Gorenstein. This leads to a main result of this paper (Theorem 22). We have included a version of Theorem 47 for the Hilbert scheme (Corollary 48). This result essentially generalizes Thm. 9.4 of [26], which treats the case where M is locally free of rank r = 1, to higher ranks. Unfortunately our methods often lead to assumptions on depth_{I(Z)} B which imply that A is non-Artinian. We have, however, succeeded in proving Theorem 22 also for an Artinian Gorenstein algebra A. Below B = R/I_{B} → A = R/I_{A} is a graded surjection with kernel I_{A/B} and M^{∗} = Hom_{B}(M, B).

Theorem 47. Let r ≥ 1 and s be integers. Let B be a graded quotient of a finitely generated polynomial k-algebra R. Let M be a finitely generated graded B-module, let Y := Proj(B) and U = Y − Z be an open subset of Proj(B) such that depth_{I(Z)} B ≥ 2 and such that M̃|_{U} is locally free of rank r. Let M_{i} = H_{∗}^{0}(U, ∧^{i}M̃) for 0 ≤ i ≤ r, and let σ ∈ H^{0}(U, M̃^{∗}(s)) be a regular section on U. Let X = Proj(A) be the zero locus of σ defined by A := coker(H_{∗}^{0}(U, M̃(−s)) →^{σ} B). Let K_{1} = ker σ(s) and suppose

(i) H_{∗}^{i}(U, M̃_{i+1}) = 0 for 1 ≤ i ≤ r − 1

(ii) _{0}Ext_{B}^{2}(M_{1}, K_{1}) = 0, and

(iii) (M_{1}(−s), σ) is unobstructed along any graded deformation of B.

Then the first projection GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R) is smooth at (B → A) and,

dim_{(B→A)} GradAlg(H_{B}, H_{A}) = dim_{(B)} GradAlg^{H_{B}}(R) + _{0}hom_{B}(I_{A/B}, A).

Moreover, depth_{𝔪} A ≥ 1. If, in addition,

(iv) either _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) = 0 and (I_{B})_{℘} is syzygetic for any graded prime ℘ of Ass(I_{A/B}), or _{0}Ext_{R}^{1}(I_{B}, I_{A/B}) = 0,

then _{0} hom_{R}(I_{B}, B) − dim_{(B)} GradAlg^{H_{B}}(R) = _{0} hom_{R}(I_{A}, A) − dim_{(A)} GradAlg^{H_{A}}(R), A is H_{B}-generic, and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + _{0} hom_{B}(I_{A/B}, A) − _{0} hom_{R}(I_{B}, I_{A/B}).

Moreover A is unobstructed as a graded R-algebra if and only if B is unobstructed as a graded R-algebra.

Using the arguments appearing in (46) below, we will see that the condition (i) of Theorem 47 implies H_{I(Z)}^{0}(A) = 0. In particular A is the homogeneous coordinate ring of X. If we in addition suppose

(v) H_{∗}^{i}(U, M̃_{i}) = 0 for 1 ≤ i ≤ r ,

we can mainly argue as in (46) (or as in the proof of [28], Lemma 12) to see that H_{I(Z)}^{1}(A) = 0, i.e. that depth_{I(Z)} A ≥ 2. Hence GradAlg^{H}(R) ≃ Hilb^{p}(ℙ) at (X ⊂ ℙ) by (8).

Let X ⊂ Y be closed subschemes of ℙ of Hilbert polynomials p_{X} and p_{Y} respectively, let D(p_{X}, p_{Y}) be the Hilbert-flag scheme parametrizing all such “pairs” of closed subschemes and let p_{1} : D(p_{X}, p_{Y}) → Hilb^{p_{X}} (ℙ) be the projection induced by p_{1}((X′ ⊂ Y′)) = (X′). X is called p_{Y} -generic if there is an open subset U_{X} of Hilb^{p_{X}} (ℙ) such that (X) ∈ U_{X} ⊂ p_{1}(D(p_{X}, p_{Y})).

Corollary 48. In addition to the notations and assumptions of Theorem 47 (i)-(iv), suppose (v). Then X is p_{Y} -generic, and

dim_{(X)} Hilb^{p_{X}} (ℙ) = dim_{(Y)} Hilb^{p_{Y}} (ℙ) + hom_{O_{Y}} (I_{X/Y} , O_{X}) − hom_{O_{ℙ}}(I_{Y} , I_{X/Y}).

Moreover X is unobstructed if and only if Y is unobstructed.

Remark 49. Since depth_{I(Z)} B ≥ 2, the modules M_{i} of Theorem 47 satisfy M_{i} = (∧^{i}M)^{∗∗} (cf. [28], Remark 8). In particular M̃_{i} = (∧^{i}M̃)^{∗∗} and clearly M̃_{i}|_{U} = (∧^{i}M̃)|_{U}. Moreover note that Theorem 47(i) holds if

depth_{I(Z)} M_{i} ≥ i + 1 for 2 ≤ i ≤ r.

This holds in particular if each M_{i} is a maximal CM B-module and depth_{I(Z)} B ≥ r +1. In this case it follows from [28], Prop. 6 that A is equidimensional and satisfies Serre’s condition S_{1} provided B is Cohen-Macaulay. Indeed since a regular section by definition leads to an exact Koszul resolution of Ã on U, we get that U ∩ X is equidimensional and without embedded components, and [28], Prop. 6 applies. In the same way (v) above holds if each M_{i} is maximally Cohen-Macaulay and depth_{I(Z)} B ≥ r + 2, in which case A satisfies Serre’s condition S_{2} if B is Cohen-Macaulay.

Proof. Once we have proved the smoothness of the first projection

q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R) at (B → A),

we shall see that we get all the conclusions of the Theorem rather quickly. To prove the smoothness of q, let (T, m_{T}) → (S, m_{S}) be a small Artin surjection with kernel 𝔞. Let B_{S} → A_{S} be a (flat and graded) deformation of B → A to S and let B_{T} be a deformation of B_{S} to T. It suffices to find a deformation A_{T} of A_{S} to T and a map B_{T} → A_{T} over B_{S} → A_{S}. Let I_{A_{S}/B_{S}} = ker(B_{S} → A_{S}).

Firstly we claim that there exists a graded deformation ϕ_{S} : M_{1S}(−s) → I_{A_{S}/B_{S}} (of S-flat B_{S}-modules) of ϕ : M_{1}(−s) → I_{A/B} to S where ϕ composed with I_{A/B} ↪ B is the map σ in the definition of A. Indeed we can by induction suppose there is a graded deformation ϕ_{S_{1}} : M_{1S_{1}}(−s) → I_{A_{S_{1}}/B_{S_{1}}} of ϕ to S_{1} where S → S_{1} is small Artin surjection (with kernel 𝔞_{1}). Composing ϕ_{S_{1}} with I_{A_{S_{1}}/B_{S_{1}}} → B_{S_{1}} we deduce by the assumption of (iii) the existence of a deformation M′_{1S} of M_{1S_{1}} to S.

By Remark 12 the existence of a homogeneous map ϕ′_{S} : M′_{1S}(−s) → I_{A_{S}/B_{S}} such that ϕ′_{S} ⊗_{S} id_{S_{1}} = ϕ_{S_{1}} is equivalent to the vanishing of a well defined element (obstruction) o_{0}(M′_{1S}, I_{A_{S}/B_{S}}) ∈ 0Ext_{B}^{1}(M_{1}, I_{A/B}(s)) ⊗_{k} 𝔞_{1}. Since

0Ext_{B}^{1}(M_{1}, M_{1}) ⊗_{k} 𝔞_{1} →^{ψ} _{0}Ext_{B}^{1}(M_{1}, I_{A/B}(s)) ⊗_{k} 𝔞_{1} → _{0}Ext_{B}^{2}(M_{1}, K_{1}) ⊗_{k} 𝔞_{1}

is exact, there is by (ii) an element λ ∈ _{0}Ext_{B}^{1}(M_{1}, M_{1}) ⊗ 𝔞_{1} such that ψ(λ) = o_{0}(M′_{1S}, I_{A_{S}/B_{S}}). On the other hand, by deformation theory, one knows more generally that M′_{1S} − λ defines a graded deformation M_{1S} of M_{1S_{1}} such that ψ(λ) = o_{0}(M′_{1S}, I_{A_{S}/B_{S}}) − o_{0}(M_{1S}, I_{A_{S}/B_{S}}) (analogous to [23], last part of Remark 3). Hence o_{0}(M_{1S}, I_{A_{S}/B_{S}}) = 0 for some deformation M_{1S} and the claim is proved.

The composition of ϕ_{S} : M_{1S}(−s) → I_{A_{S}/B_{S}} with I_{A_{S}/B_{S}} → B_{S} yields a homogeneous map σ_{S} such that σ_{S} ⊗_{S} id_{k} = σ where id_{k} : k → k is the identity. By (iii) there is a deformation σ_{T} : M_{1T} (−s) → B_{T} over σ_{S}.

Secondly we claim that A_{T} := coker σ_{T} is a graded deformation of A_{S} to T, i.e. that the surjection I_{A_{T} /B_{T}} ⊗_{T} k ↠ I_{A/B} is an isomorphism (where I_{A_{T} /B_{T}} := im σ_{T}). Indeed since A_{T} ⊗_{T} S ≃ A_{S}, it follows that A_{T} is a deformation of A_{S} if it is T-flat, i.e. if Tor_{1}^{T} (A_{T} , k) = 0 or equivalently, if I_{A_{T} /B_{T}} ⊗_{T} k ≃ I_{A/B}. The rough idea for proving this isomorphism of ideals is just to see that the following part of the Koszul complex

∧^{2}M_{1}(−2s) ⟶ M_{1}(−s) ⟶ B ⟶ A  (43)

(induced by the section σ), which obviously commutes with the corresponding complex, ∧^{2}M_{1T} (−2s) → M_{1T} (−s) → B_{T} → A_{T} over T (because σ_{T} commutes with σ), is exact (or “exact enough”, cf. below for a precise formulation).

More precisely since σ is a regular section on U, the Koszul complex induced by σ is exact on U ([28], Thm. 7(4)). Applying H_{∗}^{0}(U, −) to this Koszul complex (which is really what we did in the proof of Theorem 8 in [28]), we get in particular the following part of a complex

M_{2}(−2s) ⟶ M_{1}(−s) ⟶ B ⟶ A ⟶ 0  (44)

where M_{1}(−s) ⟶ B ⟶ A ⟶ 0 is exact. Using correspondingly σ_{T} we get a complex M_{2T} (−2s) → M_{1T} (−s) → B_{T} → A_{T} over T which commutes with (44), where M_{2T} := H_{∗}^{0}(U, ∧^{2}M̃_{1T}) (slightly abusing the notation of U by letting U be the set in Proj(B_{T}) which corresponds to U ⊆ Proj(B)). Moreover note that M_{1T} ⟶^{≃} H_{∗}^{0}(U, M̃_{1T}) follows from H_{∗}^{0}(U, M̃_{1}) ≃ M_{1} and the fact that M_{1T} is a deformation of M_{1}. Indeed since we by induction may suppose H_{∗}^{0}(U, M̃_{1S}) ≃ M_{1S}, we conclude easily by applying H_{∗}^{0}(U, −) to the exact sequence

0 → M̃_{1}|_{U} ⊗_{k} 𝔞 → M̃_{1T} |_{U} → M̃_{1S}|_{U} → 0,  (45)

and by comparing with 0 → M_{1} ⊗_{k} 𝔞 → M_{1T} → M_{1S} → 0. Let Z_{i} = ker(M_{i}(−is) → M_{i−1}((1 − i)s)) for i ≥ 1 and let Z_{1T} := ker(M_{1T} (−s) → I_{A_{T} /B_{T}}). Applying (−) ⊗_{T} k to

Z_{1T} ⟶ M_{1T} (−s) ⟶ I_{A_{T} /B_{T}} ⟶ 0,

we see that the exactness of Z_{1} → M_{1}(−s) → I_{A/B} → 0 and the isomorphism M_{1T} ⊗_{T} k ≃ M_{1} imply the claim provided we can prove that Z_{1T} → Z_{1} is surjective.

Now we will show that H_{∗}^{1}(U, Z̃_{1}) = 0 implies the surjectivity of Z_{1T} → Z_{1}. To prove this we remark that the T -flatness of Z̃_{1T} |_{U} (which is true because the claim is true locally in U) yields an exact sequence (45) in which we have replaced every M̃_{1} by Z̃_{1}. Applying H_{∗}^{0}(U, −) to such an exact sequence, we get the exact sequence

Z_{1T} → Z_{1S} → H_{∗}^{1}(U, Z̃_{1}) ⊗_{k} 𝔞.

because we have H_{∗}^{0}(U, Z̃_{1T}) ≃ Z_{1T} from H_{∗}^{0}(U, M̃_{1T}) ≃ M_{1T}. If H_{∗}^{1}(U, Z̃_{1}) = 0, we get the surjectivity of Z_{1T} → Z_{1S} and hence the surjectivity of Z_{1T} → Z_{1} by induction.

Hence it suffices to prove H_{∗}^{1}(U, Z̃_{1}) = 0. Taking cohomology of the sequence 0 → Z_{i} → M_{i}(−is) → Z_{i−1} → 0, the assumption (i) leads to injections

H_{∗}^{1}(U, Z̃_{1}) ↪ H_{∗}^{2}(U, Z̃_{2}) ↪ ... ↪ H_{∗}^{r−1}(U, Z̃_{r−1})  (46)

and we get the vanishing H_{∗}^{1}(U, Z̃_{1}) = 0 because Z̃_{r−1}|_{U} ≃ M̃_{r}(−rs)|_{U} and H_{∗}^{r−1}(U, M̃_{r}) = 0 and the second claim is proved.

Combining the two claims we get the smoothness of the projection q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R) at (B → A).

From (46) we have H_{I(Z)}^{2}(Z_{1}) ≃ H_{∗}^{1}(U, Z̃_{1}) = 0 and using the exact sequence

0 → Z_{1} → M_{1}(−s) → B → A → 0  (47)

we get H_{I(Z)}^{1}(I_{A/B}) = 0 and H_{I(Z)}^{0}(A) = 0. It follows that depth_{𝔪} A ≥ 1.

Once we have the smoothness of q and the assumption (iv), all remaining conclusions of Theorem 47 follow by exactly the same proof as in the proof of Thm. 5B) in [23]. Indeed (iv) implies that the second projection p : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R) is smooth at (B → A) (cf. [23], Prop.4(ii) and Remark 16(i) to conclude that the assumption _{0}Ext_{R}^{1}(I_{B}, I_{A/B}) = 0 also implies the smoothness of p). Since the tangent space T_{B→A} of GradAlg(H_{B}, H_{A}) at (B → A) is given by the cartesian square in the following diagram of exact sequences

0Hom_{R}(I_{B}, I_{A/B})
↓
T_{B→A}  →^{T_{q}}  _{0}Hom_{R}(I_{B}, B)
↓  □  ↓
0 → _{0}Hom_{R}(I_{A/B}, A) → _{0}Hom_{R}(I_{A}, A) → _{0}Hom_{R}(I_{B}, A)

(48)

([22], (10)) where the top vertical map is injective and the tangent map T_{q} is surjective by the smoothness of q, we get

dim_{(B→A)} GradAlg(H_{B}, H_{A}) = dim T_{B→A} = dim_{(B)} GradAlg^{H_{B}}(R) + _{0}hom_{R}(I_{A/B}, A)

as well as the other conclusions (by e.g. using that the tangent map T_{p} : T_{B→A} → _{0}Hom_{R}(I_{A}, A) of p is surjective by (iv) and (48)).

Remark 50. If we, instead of (i) H_{∗}^{i}(U, M̃_{i+1}) = 0 for 1 ≤ i ≤ r − 1, assume

(i′) H_{∗}^{1}(U, M̃_{2}) = 0 and H_{∗}^{i}(U, M̃_{i+2}) = 0 for 1 ≤ i ≤ r − 2,

and keep the other assumptions of Theorem 47, we still get all conclusions of Theorem 47. Indeed, looking to the proof of (in particular the second claim of) Theorem 47, it suffices to prove that M_{2T} → M_{2} is surjective and that (44) is exact. Now applying H_{∗}^{0}(U, ∧^{2}−) to the exact sequence (45), remarking that M̃_{2}|_{U} ≃ ∧^{2}M̃_{1}|_{U}, we get the exact sequence M_{2T} → M_{2S} → H_{∗}^{1}(U, M̃_{2}) ⊗_{k} 𝔞. Since we have H_{∗}^{1}(U, M̃_{2}) = 0 by the assumption (i′), we get the surjectivity of M_{2T} → M_{2S} and hence the surjectivity of M_{2T} → M_{2} by induction. Finally to show that (44) is exact, it suffices to show the surjectivity of H_{∗}^{0}(U, M̃_{2}(−2s)) → H_{∗}^{0}(U, Z̃_{1}) or the vanishing of H_{∗}^{1}(U, Z̃_{2}). Since the conditions of (i′) lead to inclusions as in (46) provided we have replaced Z_{i} by Z_{i+1}, we get precisely H_{∗}^{1}(U, Z̃_{2}) = 0 and we are done. Finally note that the assumption H_{∗}^{1}(U, M̃_{2}) = 0 in (i′) is superfluous if we in proving the second claim above can show that there is a deformation M′_{2T} of M_{2} which locally on U is M_{2T} for any T. Indeed the argument in the proof of Theorem 47 where we showed H_{∗}^{0}(U, M̃_{1T}) ≃ M_{1T} as consequence of H_{∗}^{0}(U, M̃_{1}) ≃ M_{1} and the fact that M_{1T} is a deformation of M_{1}, apply to get M′_{2T} ≃ M_{2T} and hence the surjectivity of M_{2T} → M_{2}.

If we apply Theorem 47 under the assumptions of Theorem 8, we get Theorem 22 stated in the background section. Here we include a proof of Theorem 22.

Proof. It suffices to verify (i), (ii) and (iv) of Theorem 47. Firstly we show (ii). Note that (5) is given by applying H_{∗}^{0}(U, −) onto the Koszul resolution induced by the regular section σ. Now splitting the exact sequence (5) into short exact sequences, we get 0 → Z_{i} → M_{i}(−is) → Z_{i−1} → 0 where Z_{i} = ker(M_{i}(−is) → M_{i−1}((1 − i)s)) for i ≥ 2 and Z_{1}(s) = K_{1}. Hence _{0}Ext_{B}^{2}(M, K_{1}) = 0 provided _{0}Ext_{B}^{2}(M, M_{2}(−s)) = 0 and _{0}Ext_{B}^{3}(M, Z_{2}(s)) = 0, while _{0}Ext_{B}^{3}(M, Z_{2}(s)) = 0 provided 0Ext_{B}^{3}(M, M_{3}(−2s)) = 0 and _{0}Ext_{B}^{4}(M, Z_{3}(s)) = 0 etc. By the assumption (i) of Theorem 22, it suffices to show _{0}Ext_{B}^{r}(M, Z_{r−1}(s)) = 0. Since Z_{r−1} = M_{r}(−rs) = K_{B}(t − rs), cf. Theorem 8, we get this vanishing by Gorenstein duality. Hence we conclude by the assumption (i) of Theorem 22. Moreover (iv) of Theorem 47 is proven by the same argument, using (iii) of Theorem 22 (cf. Remark 21), and remarking that Ass(I_{A/B}) ⊂ Ass(B). Note that the argument shows that 0Ext_{R}^{i}(I_{B}, M_{i}(−is)) = 0 for 1 ≤ i ≤ r implies _{0}Ext_{R}^{1}(I_{B}, I_{A/B}) = 0 and we conclude as required.

It remains to show (i) of Theorem 47 or (i′) of Remark 50. If depth_{I(Z)} B = dim B−dim B/I(Z) ≥ r + 1, we have

H_{∗}^{i}(U, M̃_{i+1}) ≃ H_{I(Z)}^{i+1}(M_{i+1}) = 0 for 1 ≤ i ≤ r − 1

since all M_{i} are maximal CM B-modules by Theorem 8. If depth_{I(Z)} B = r and r ≥ 3 we similarly verify (i′) of Remark 50. In the final case depth_{I(Z)} B = r = 2 we can use the two final sentences in Remark 50 to conclude. Indeed in this case M_{2} = K_{B}(t) and since Ext_{R}^{j}(B, R(−n−c)) = 0 for j ≠ c and K_{B} = Ext_{R}^{c}(B, R(−n − c)) we may take M′_{2T} = K_{B_{T}} := Ext_{R_{T}}^{c} (B_{T} , R_{T} (−n − c)) and prove that K_{B_{T}} is a deformation of K_{B} to B_{T} by e.g. [20], Prop. (A1). This concludes the proof.

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OSLO UNIVERSITY COLLEGE, FACULTY OF ENGINEERING, PB. 4, ST. OLAVS PLASS, N-0130 OSLO, NORWAY. E-mail address: JanOddvar.Kleppe@iu.hio.no