Maximal families of Gorenstein algebras

Jan O. Kleppe

Abstract

The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring R. Let GradAlg^{H}(R) be the scheme parametrizing graded quotients of R with Hilbert function H. We prove there is a close relationship between the irreducible components of GradAlg^{H}(R) whose general member is a Gorenstein codimension (c + 1) quotient, and the irreducible components of GradAlg^{H′}(R) whose general member B is a codimension c Cohen-Macaulay algebra of Hilbert function H′ related to H. If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of B, this relationship actually determines a well-defined injective mapping from such “Cohen-Macaulay” components of GradAlg^{H′}(R) to “Gorenstein” components of GradAlg^{H}(R), in which generically smooth components correspond. Moreover the dimension of the “Gorenstein” components is computed in terms of the dimension of the corresponding “Cohen-Macaulay” component and a sum of two invariants of B. Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main Theorem. AMS Subject Classification. 14C05, 13D10, 13D03, 13D07, 13C40. Keywords. Parametrization, Gorenstein algebra, Artinian algebra, liaison, licci, Cohen-Macaulay, canonical module, normal module, Hilbert scheme.

1 Introduction

The main goal of this paper is to contribute to the classification of graded Gorenstein quotients of a polynomial ring R of dimension n + c over an algebraically closed field k. Let GradAlg^{H}(R) be the scheme parametrizing graded quotients A of R of depth A ≥ min(1, dim A) and with Hilbert function H. We let CM_{c}^{H}(R) (resp. Gor_{c}^{H}(R)) be the open subscheme parametrizing Cohen-Macaulay (resp. Gorenstein) quotients of codimension c in R. In the Artinian case Gor_{c}^{H}(R) and the scheme PGor(H) defined by certain catalecticant minors ([21]) are essentially the same (see Theorem 11). We call an irreducible component of GradAlg^{H}(R) which has non-empty intersection with CM_{c}^{H}(R) (resp. Gor_{c}^{H}(R)) a ”Cohen-Macaulay (resp. “Gorenstein) codimension c” component. The main Theorem of this paper establishes a strong connection between “Gorenstein codimension (c + 1)” components of GradAlg^{H}(R) and “Cohen-Macaulay codimension c” components of GradAlg^{H′}(R) for some H′. Indeed the connection between these components allows us to deduce the generic smoothness and the dimension of the “Gorenstein codimension (c + 1)” component from the corresponding property and number for the “Cohen-Macaulay codimension c” component, and vica versa. If the socle degree of an Artinian reduction of the Gorenstein quotient is large enough (about twice the Castelnuovo-Mumford regularity, reg(B), of B), we then prove that this connection actually determines a well-defined injective mapping between the set of “Cohen-Macaulay codimension c components of GradAlg^{H′}(R) and the set of “Gorenstein codimension (c + 1)” components of GradAlg^{H}(R) whose general members satisfy the Weak Lefschetz property (Theorem 24).

We all know the nice description of CM_{2}^{H}(R), given by the Hilbert-Burch theorem, and of Gor_{3}^{H}(R), proved by Buchsbaum and Eisenbud, in terms of the maximal minors (resp. the Pfaffians) of a suitable matrix. In this case the schemes CM_{2}^{H}(R) and Gor_{3}^{H}(R) are smooth and irreducible. There is, in our opinion, little hope of finding such a single nice description of Gor_{4}^{H}(R) in codimension 4 because this scheme has several irreducible components ([3] and [22]). We think a more promising strategy should consist of classifying the irreducible components of Gor_{4}^{H}(R) together with describing the members of each family concretely. In this context the main theorems show that the corresponding classification of CM_{3}^{H′}(R) is central for the classification of Gor_{4}^{H}(R), and they provide a method for answering Iarrobino and Srinivasan’s question ([22]) about whether the non-irreducibility of Gor_{4}^{H}(R) is a rare phenomenon, or not. In fact we give several classes of examples of reducible schemes Gor_{4}^{H}(R) whose general members are one dimensional Gorenstein quotients of R (Example 26 and Remark 27).

More precisely let the graded algebra A be a codimension one quotient of B, well defined by a twist of the canonical module K_{B}, by which we mean that there is a graded exact sequence

0 → K_{B}(−s) → B → A → 0,  (1)

where B is assumed generically Gorenstein (see the beginning of Section 2). Let K_{B}^{∗} be the B-dual of the canonical module of B = R/I_{B}, let _{v}Ext_{B}^{i}(−, −), i ≥ 0, be the graded piece of Ext_{B}^{i}(−, −) of degree v and let _{v}ext_{B}^{i}(−, −) be its dimension as a k-vectorspace. Then we show (cf. Theorem 16)

Theorem 1. Let B = R/I_{B} be a graded Cohen-Macaulay quotient of a polynomial ring R, and let A be a graded codimension one quotient of B, well defined by a twist of the canonical module K_{B}.

A) If B is licci, then A is unobstructed as a graded R-algebra, A is Gorenstein and,

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

where δ(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, then the codimension of the H_{B}-stratum of A at (B → A) is _{−s} ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}).

B) If Proj(B) is locally licci and s >> 0, then A is H_{B}-generic and Gorenstein, 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 if and only if B is unobstructed as a graded R-algebra.

If we assume K_{B}(−s)_{v} = 0 for v less or equal the largest degree of a minimal generator of I_{B}, then the codimension of the H_{B}-stratum of A at (B → A), mentioned in Theorem 1A, is just the codimension at (A) of the stratum of GradAlg^{H_{A}}(R) of quotients R → A′ for which there exists some factorization B′ → A′ such that B′ has Hilbert function H_{B}. Near (A) this stratum turns out to be the stratum of quotients A′ given by (1), with A′, B′ replacing A, B in (1) (Lemma 7 and Proposition 13A). A is H_{B}-generic if there is an open subset U ∋ (A) of GradAlg^{H_{A}}(R) contained in this stratum.

Note that if B = R/I and R/J are geometrically linked by a complete intersection, then A = R/(I + J) is given by (1). In this case the unobstructedness of A is known if B is licci ([39]). It is standard to use sums of geometrically linked ideals to construct Gorenstein algebras (e.g. [8], [9], [10], [15]). By the corresponding more general construction of using geometric Gorenstein linkage, we also get Gorenstein algebras (e.g. [34], see also [35], Ex. 10.2). Since both constructions are given by (1) (e.g. [34], [30], p. 785), Theorem 1 applies in both cases.

In Theorem 1A, A is in fact shown to be unobstructed in the sense H^{2}(R, A, A) = 0, a formula which holds also when A is non-graded (Theorem 30). Moreover we prove a more general result that implies Theorem 1 (Proposition 13). To use Proposition 13 we need to verify the vanishing of certain Ext-groups. These vanishings always hold in the licci case. We also prove that the scheme ZGor(H) consisting of not necessarily graded Artinian Gorenstein quotients of R is smooth at a graded quotient (A) given by (1) provided B is licci, and we give a formula for its dimension (Proposition 29). In the final Section 3 we show how we can use linkage to compute the mentioned Ext-groups as well as to find the other dimension invariants of Theorem 1 (Proposition 33 and Corollary 37, see Example 2 below).

In Theorem 1B, we may make “s >> 0” quite explicit. For instance if Proj(B) is a zero dimensional scheme of length d, it may be replaced by “s ≥ 2 reg(I_{B})” (Remark 22; cf. (7)). In this case dim(K_{B}^{∗})_{s} = d and the h-vector of A contains a consecutive subsequence of the form (d, d, d). It is precisely in such ranges of s that Theorem 24 applies, telling that there is a welldefined injective application π from the set of irreducible components of GradAlg^{H_{B}}(R) whose general members satisfy the assumptions of Theorem 1B, to the set of irreducible “Gorenstein components” of GradAlg^{H_{A}}(R) whose general members satisfy the Weak Lefschetz property. This implies in particular that the h-vector of A is an SI-sequence ([15]). In this range of s one knows that any codimension one Gorenstein quotient is given by (1) ([4]). Theorem 24 is a consequence of a thoroughly studied correspondence (Proposition 23). If the codimension of B is c = 2, then Theorem 1B, the application π and the mentioned correspondence are well understood by Iarrobino-Kanev’s results on “annihilating schemes”, in which they construct the algebra B from a given A ([21], Ch. 5). Their approach applies also to c ≥ 3 and leads e.g. to reducible Gor_{c+1}^{H}(R) for c ≥ 4 ([21], Ch. 6) and to a dimension formula of Gor_{c+1}^{H}(R) provided the “first half” of H coincides with the Hilbert function of a complete intersection ([21], Thm. 4.17). The application π is surjective for c = 2, while Boij points out that such a surjectivity is not always true in higher codimensions ([3]). Also, results on the codimension of the Betti-strata can be related to the codimension statement in Theorem 1A (see [21], Sect. 5.3.5 and Remark 25(b)). Note that it is known that not every codimension 3 quotient is obtained by (1) ([36], [10]), i.e. that maximal families of quotients given by (1) may correspond to proper strata. This fits nicely with Theorem 1A in which the number −sext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B})) essentially measures the codimension of such strata.

In order to prove our results we study in full generality deformations of a quotient A of R which themselves are quotients of some deformation of B. First we recall and further develop some partially known results on the unobstructedness and the “family-dimension” of such a quotient A of R with fixed Hilbert function H_{A} (Theorem 5). Here we neither assume B to be Cohen-Macaulay, nor A to be Gorenstein. In this generality the deformation theory developed by Laudal is central ([31]). The Theorem 5 is important for the main application of this paper because it treats simultaneously the two cases of interest: when the general member of an irreducible family R → A factors via some deformation of B → A (i.e. A is H_{B}-generic), and when it does not.

To illustrate some of our results we consider the following example. Here the Hilbert function, H_{B}(v) := dim_{k} B_{v}, of B is written as a vector (H_{B}(0), H_{B}(1), H_{B}(2), ...). If A is Artinian, we call (H_{A}(0), H_{A}(1), ..., H_{A}(s)) the h-vector of A, where s, the socle degree of A, is by definition the smallest number such that H_{A}(v) = 0 for v > s.

Example 2. Let B = R/I_{B} be a one dimensional almost complete intersection obtained by linking a complete intersection (c.i.) B′ of three hypersurfaces of degree 2,2,2 to B via a c.i. D of three hypersurfaces of degree 3,3,3. A standard mapping cone construction yields

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

Hence the Castelnuovo-Mumford regularity reg(I_{B}) = 5 and it is easy to see that the Hilbert functions of B and B′ are (1, 4, 10, 16, 19, 19, 19, ...) and (1, 4, 7, 8, 8, 8, ...) respectively. Now B is obviously licci and Theorem 1 applies to any quotient A given by (1). Using (1) and duality one shows H_{A}(v) = H_{B}(v) + H_{B}(s − v) − 19. It follows that s is the socle degree of A and that the h-vector of A is the (s + 1)-tuple (1, 4, 7, 7, 4, 1), (1, 4, 10, 13, 10, 4, 1), (1, 4, 10, 16, 16, 10, 4, 1), (1, 4, 10, 16, 19, 16, 10, 4, 1) and (1, 4, 10, 16, 19..., 19, 16, 10, 4, 1) for s = 5, 6, 7, 8, and ≥ 9 respectively. It is known ([27], Proposition 1.7 or Proposition 33 of this paper) how to compute dim(N_{B})_{0}, the dimension of the tangent space of GradAlg^{H_{B}}(R) at (B), from dim(N_{B′})_{0}. Indeed

dim(N_{B})_{0} = dim(N_{B′})_{0} + 3H_{B}(3) − 3H_{B′}(3).

Since N_{B′} ≃ B′(2)^{⊕3} we get dim(N_{B})_{0} = 45. Moreover, due to Remark 22, we have δ(B)_{−s} = 0 for s ≥ 2 reg(I_{B}) = 10 and dim(K_{B}^{∗})_{s} = 19 for s ≥ 2 reg(I_{B}) − 3 = 7. Hence, for s ≥ 10 we get that A is unobstructed and H_{B}-generic and that dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(A)} PGor(H_{A}) = 63. Note that the conclusions so far only use Theorem 1B because we may replace the assumption s >> 0 of Theorem 1B by s ≥ 2 reg(I_{B}).

However, to deal with s < 10 we need Theorem 1A and that B is licci (or that the explicitly mentioned Ext-groups of Proposition 13A vanish). We also have to compute the numbers dim(K_{B}^{∗})_{s} and δ(B)_{−s}. To do so, we have a nice connection between these invariants using liaison. Indeed by Proposition 33

dim(K_{B}^{∗})_{v} = dim B_{v−5} + 3 dim(I_{B/D})_{v−2} − δ(B′)_{v−10}

and symmetrically, δ(B)_{v−10} = dim B′_{v−5} + 3 dim(I_{B′/D})_{v−2} − dim(K_{B′}^{∗})_{v} for any v. Since B′ is a c.i., we easily compute δ(B′)_{v} and (K_{B′}^{∗})_{v}. Indeed K_{B′} ≃ B′(2) and δ(B′)_{v} = v hom_{B′}(I_{B′}/I_{B′}^{2}, K_{B′}) = dim(K_{B′})_{v+2}^{⊕3}. We get that dim(K_{B}^{∗})_{v} = H_{B}(v − 5) + 3H_{B′}(v − 5) − 3H_{B′}(v − 6), i.e. dim(K_{B}^{∗})_{s} = 4, 13, 19, 19, ... for s = 5, 6, 7, 8... respectively. Correspondingly, δ(B)_{−s} = 20, 2, −4, −1, 0, 0... for s = 5, 6, 7, 8, 9, 10, ... respectively. Hence dim_{(A)} PGor(H_{A}) =

dim_{(A)} GradAlg^{H_{A}}(R) = 28, 55, 67, 64, 63, 63, ... for s = 5, 6, 7, 8, 9, 10, ... respectively.

If s ≥ 7 we see from the resolution of I_{B} that _{−s} hom_{B}(I_{B}/I_{B}^{2}, K_{B}) = 0, and hence that −s ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = −δ(B)_{−s} = 4, 1, 0 for s = 7, 8, 9 respectively. By Theorem 1A the numbers 4,1,0 represent the codimensions of the strata of quotients R → A′, H_{A′} = H_{A}, given by (1). It follows that these strata form a 63 dimensional irreducible family at (A) for all s ≥ 7, while the dimension of the corresponding irreducible component is 63+_{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}). That the conclusion of Theorem 1A holds for s = 7, 8, shows the existence of a quotient A with Hilbert function H_{A} which is not given by (1) for some B with Hilbert function H_{B}! For a complete such analysis on any codimension 3 almost c.i. B, see Example 40.

In this work the author was inspired by joint work with Chris Peterson where we constructed Gorenstein quotients of R in various ways. We showed for instance that if M is a maximal Cohen-Macaulay B-module of rank r and ∧^{r}M̃ ≃ K̃_{B}(s) as sheaves (in a large enough open set U of Proj(B)), then any regular section of M̃^{∗}(s) defines a Gorenstein quotient R ↠ A of codimension r + c provided H_{∗}^{0}(U, ∧^{i}M̃) for 2 ≤ i ≤ r/2 are maximally Cohen-Macaulay (cf. [30] for details). The idea of this paper is to look at the construction and to use deformation theory to vary every object and morphism in the construction as much as possible, in order to see how large the corresponding stratum in GradAlg^{H}(R) will be. This paper looks at the r = 1 case because so far it is only when r = 1 that the author is also able to show his results for Artinian Gorenstein rings. In a forthcoming paper the author generalizes the results of this paper considerably by considering deformations of regular sections of a maximal Cohen-Macaulay module M of any rank, but, unfortunately, there we must partially restrict to positively dimensional Gorenstein algebras. We thank Olav A. Laudal, Roy Skjelnes and Johannes Kleppe for clarifying discussions. The author heartily thanks the referee for many valuable comments and questions.

For the convenience of the reader we include a section of preliminaries where we give some space to the theory of deformations. Indeed it is not always easy to find a nice reference for the results we use because treatments in the literature are often either too general and concern general cohomology groups of algebras, or too special, assuming certain groups are zero or very manageable.

1.1 Preliminaries

Let B be an n-dimensional graded quotient of a polynomial k-algebra R (k is algebraically closed) in n + c variables (of degree 1), and let M and N be finitely generated graded Bmodules. 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). Recall that depth_{J} M = inf{depth M_{℘} | ℘ ⊇ J, ℘ graded} and that depth_{J} M ≥ r iff H_{J}^{i}(M) = 0 for i < r, cf. [12] or [16].

If Y = Proj(B) and Z is closed in Y and U = Y − Z, we let 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  (2)

and isomorphisms H_{I(Z)}^{i}(M) ≃ H_{∗}^{i−1}(U, M̃) for i ≥ 2. In the case depth_{I(Z)} N ≥ 2 the graded Ext_{B}^{1}(M, N) injects into the corresponding global Ext_{O_{U}}^{1} -group of sheaves. Indeed we have in this case an exact sequence in degree zero ([12], exp. VI)

0Ext_{B}^{1}(M, N) ↪ Ext_{O_{U}}^{1} (M̃|_{U}, Ñ|_{U}) → _{0}Hom_{B}(M, H_{I(Z)}^{2}(N)) → _{0}Ext_{B}^{2}(M, N) → Ext_{O_{U}}^{2} (M̃|_{U}, Ñ|_{U})

(3)

where the form of the middle term comes from a spectral sequence also treated 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 and K_{B} = Ext_{R}^{c}(B, R(−n − c)) is the canonical module of B, we know by Gorenstein duality that v-graded piece of H_{𝔪}^{i}(M) satisfies

vH_{𝔪}^{i}(M) ≃ _{−v}Ext_{B}^{n−i}(M, K_{B})^{∨} .

Recall that two graded quotients, R/J and R/J′, are said to be linked by a complete intersection if there exists a homogeneous complete intersection ideal L such that J = L : J′ and J′ = L : J (with L ⊆ J ∩ J′). The relationship of being linked generates the equivalence relation, “linkage”. B = R/I_{B} is said to be licci (and hence Cohen-Macaulay) if it is in the linkage class of a complete intersection (cf. [34] for a survey). We define licci for a quotient of a regular local ring correspondingly. Moreover Proj(B) is said to be locally licci if each of its local rings are licci.

Even though we in this paper try to avoid the algebraic (co)homology groups H_{2}(R, B, B) and H^{2}(R, B, B) and use instead the corresponding Tor- or Ext-groups, we will occasionally need them. We recall that the former group is given by an exact sequence

0 → H_{2}(R, B, B) → H_{1} → G_{1} ⊗_{R} B → I_{B}/I_{B}^{2} → 0.  (4)

in which G_{1} is R-free, G_{1} ↠ I_{B} is surjective and minimal, and H_{1} = H_{1}(I_{B}) is the degree-one Koszul homology of I_{B} [38]. An ideal I_{B} of R is called syzygetic if H_{2}(R, R/I_{B}, R/I_{B}) = 0. If I_{B} is syzygetic, then (I_{B})_{℘} is syzygetic for any prime ideal ℘ of R. I_{B} is called generically syzygetic if (I_{B})_{℘} is syzygetic for any graded prime ℘ of Ass(B). We define genericity of other properties similarly. Using (4) 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 Cohen-Macaulay module in the licci case [20]. Thus, generically complete intersection licci ideals are syzygetic. For the graded group H^{2}(R, B, B) we 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) →  (5)

This comes from the spectral sequence Ext_{B}^{p}(H_{q}(R, B, B), N), which converges to H^{p+q}(R, B, N), and I_{B}/I_{B}^{2} ≃ H_{1}(R, B, B), cf. [1], Prop. 16.1 or [31].

A Cohen-Macaulay quotient B = R/I_{B} of codimension c = dim R − dim B in R has a minimal R-free resolution of the following form (cf. [7])

0 → G_{c} → ... → G_{1} → R → B → 0  ,  G_{j} = ⊕_{i=1}^{r_{j}}R(−n_{j,i})  (6)

In this case the Castelnuovo-Mumford regularity of I_{B} is given by (cf. [34], p. 8)

reg(I_{B}) = max_{j,i} {n_{j,i} − j + 1} = max_{i} {n_{c,i}} − c + 1 .  (7)

Let G_{i}^{∗} be the R-dual of G_{i}. Since K_{B} = Ext_{R}^{c}(B, R(−n − c)), the R-dual complex of (6) yields a minimal R-free resolution of the twisted canonical module K_{B}(n + c). In particular if c = 2, it induces a complex

0 → N_{B} → G_{1}^{∗} ⊗_{R} B → G_{2}^{∗} ⊗_{R} B → K_{B}(n + c) → 0  (8)

which is exact by e.g. [30], p. 795.

Let Hilb^{p}(ℙ) be Grothendieck’s Hilbert scheme parametrizing closed subschemes Y of ℙ = Proj R with Hilbert polynomial p ∈ ℚ[t], cf. [11]. A k-point of Hilb^{p}(ℙ) which corresponds to the subscheme Y is usually denoted by (Y ⊂ ℙ) or just (Y). A closed subscheme Y of ℙ is called unobstructed if Hilb^{p}(ℙ) is smooth at (Y ⊂ ℙ).

Let GradAlg^{H} := GradAlg^{H}(R) be the stratum of Hilb^{p}(ℙ) given by deforming Y ⊂ ℙ with constant Hilbert function H_{Y} = H (i.e. its functor deforms both Y ⊂ ℙ and its homogeneous coordinate ring, B, flatly), cf. [27]. GradAlg^{H}(R) allows a natural scheme structure whose tangent (resp. “obstruction”) space at (Y ⊂ ℙ) is _{0}Hom_{B}(I_{B}/I_{B}^{2}, B) ≃ _{0}Hom_{R}(I_{B}, B) (resp. _{0}H^{2}(R, B, B)), i.e. it is given by deforming B as a graded R-algebra [24]. In the case 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, GradAlg^{H}(R) still represents a functor parametrizing graded R-quotients with Hilbert function H_{B} = H (see Proposition 9), and its open subscheme of Gorenstein quotients coincides, at least topologically and infinitesimally, with PGor(H) (the corresponding scheme of forms with “catalecticant structure”, see Theorem 11 below and the material before it). B is called unobstructed as a graded R-algebra iff GradAlg^{H_{B}}(R) is smooth at (R → B). This definition of unobstructedness coincides with the one given in Remark 3 below by [31] or [37], p. 151. By [24], Remark 3.7,

GradAlg^{H}(R) ≃ Hilb^{p}(ℙ) at (Y ⊂ ℙ)  (9)

provided _{0}Hom_{R}(I_{B}, H_{𝔪}^{1}(B)) = 0. This happens for quotients B of depth_{𝔪} B ≥ 2.

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 (for non-Artinian quotients) and with Hilbert functions H_{B} and H_{A} of B and A respectively. The tangent space _{0}A_{B→A}^{1} of GradAlg(H_{B}, H_{A}) at (B → A) and the tangent maps T_{p} and T_{q} of the second projection p : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R) (essentially given by p((B′ → A′)) = (A′)) and the first projection q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R) are given by the cartesian square in the following diagram of exact sequences

0Hom_{R}(I_{B}, I_{A/B})
↓
0A_{B→A}^{1}  →^{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)

(10)

Here B → A ≃ R/I_{A} is a surjection of graded R-algebras with kernel I_{A/B}. Since the lower sequence in the diagram above may be continued by _{0}H^{2}(B, A, A) to make a long exact sequence of algebra cohomology groups, and since _{0}H^{2}(B, A, A) injects in general into 0Ext_{B}^{1}(I_{A/B}, A) ([14], exp. VI), it follows that _{0}Hom_{R}(I_{A}, A) → _{0}Hom_{R}(I_{B}, A) is surjective provided _{0}Ext_{B}^{1}(I_{A/B}, A) = 0. One may also prove this surjectivity in a more elementary manner by dualizing the exact sequence 0 → I_{B} ⊗_{R} A ≃ I_{B}/I_{A}I_{B} → I_{A} ⊗_{R} B → I_{A/B} → 0.

Remark 3. Let R → B be a graded quotient. Let (T, 𝔪_{T}) → (S, 𝔪_{S}) be a small Artinian 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. A graded deformation B_{T} of B_{S} to T is T -flat and satisfies B_{T} ⊗_{T} S ≃ B_{S}.

A flag (i.e. a surjection) of graded quotients of R, B → A, is said to be unobstructed if for every small Artinian surjection T ↠ S and for every graded deformation B_{S} → A_{S} of B → A to S, there exists a graded deformation B_{T} → A_{T} (of T-flat quotients) reducing to B_{S} → A_{S} via (−) ⊗_{T} S. Similarly a quotient B of R is unobstructed if every graded deformation B_{S} deforms further to T

The obstruction, o_{0}(B_{S}), to deforming B_{S} to T sits in _{0}H^{2}(R, B, B)⊗_{k}𝔞 and _{0}H^{1}(R, B, B)⊗_{k} 𝔞 corresponds to the set of graded deformations (hence _{0}Hom_{R}(I_{B}, B) ≃ _{0}H^{1}(R, B, B) is the tangent space of GradAlg^{H_{B}}(R)). More precisely if we fix a graded deformation B_{T} of B_{S} to T, then B_{T} defines a bijection from _{0}H^{1}(R, B, B) ⊗_{k} 𝔞 onto the set of graded deformations of B_{S} to T, allowing us to interpret B_{T} + λ as another graded deformation of B_{S} to T for every non-trivial λ ∈ _{0}H^{1}(R, B, B) ⊗_{k} 𝔞, cf. [31], Thm. 2.2.5 or [24], Thm. 1.5. Similarly the obstruction to deforming B_{S} → A_{S} to a given graded deformation B_{T} of B_{S} sits in _{0}H^{2}(B, A, A) ⊗_{k} 𝔞 and _{0}H^{1}(B, A, A) ⊗_{k} 𝔞 ≃ _{0}Hom_{B}(I_{A/B}, A) ⊗_{k} 𝔞 corresponds to the set of such deformations. Note that, by definition, the vanishing of the obstruction is equivalent to the existence of the corresponding desired deformation. Hence the vanishing of the obstruction groups (“spaces”) leads to the unobstructedness of the corresponding objects.

Finally let ϕ_{S} : B_{S} → A_{S} be a graded deformation of ϕ : B → A to S, and let B_{T} and A_{T} be given graded deformations of B_{S} and A_{S} respectively to T. By [31], Thm. 2.3.3 or [24], Thm. 1.6 there is an obstruction o_{0}(ϕ_{S}; B_{T} , A_{T}) ∈_{0}H^{1}(R, B, A) ⊗_{k} 𝔞 which vanishes if and only if there exists a graded morphism ϕ_{T} : B_{T} → A_{T} such that ϕ_{T} ⊗_{T} id_{S} = ϕ_{S}. Looking closer to Laudal’s proof of [31], Thm. 2.3.3, one may see

o_{0}(ϕ_{S}; B′_{T} , A_{T}) − o_{0}(ϕ_{S}; B_{T} , A_{T}) = (ϕ_{∗} ⊗ id_{𝔞})(λ)

where B_{T} + λ = B′_{T} and ϕ_{∗} is the morphism _{0}H^{1}(R, B, B) → _{0}H^{1}(R, B, A) induced by ϕ.

One may consider non-graded deformations of e.g. a quotient B of R in which case the whole algebra cohomology group Hom_{R}(I_{B}, B) ≃ H^{1}(R, B, B) (resp. H^{2}(R, B, B)) serves as tangent (resp. “obstruction”) space of the deformations of B as an R-algebra. For graded deformations it is the degree zero piece of the cohomology groups which counts. For a generically complete intersection, R ↠ B, (or more generally, for a quotient B which satisfies 0Hom_{B}(H_{2}(R, B, B), B) = 0) its obstructions sit in _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ⊗ 𝔞 by (5).

The following Proposition is a main result of the author’s unpublished thesis. It (as well as the contents of (10)) is just an application, with a few new ideas, of the general deep results of Laudal ([31] and [32], Sect. 2) on deformations of categories as briefly explained in [25], Sect. 2 and its appendix (see also [26], Sect. 1). Since the proofs of [25] or [26] are not in the generality we need in this paper, we include a proof.

Proposition 4. Let B be a graded quotient of R and let ϕ : B → A be a graded surjection, inducing morphisms _{0}Hom_{R}(I_{B}, B) → _{0}Hom_{R}(I_{B}, A) and _{0}H^{2}(R, B, B) → _{0}H^{2}(R, B, A).

(i) If _{0}Ext_{B}^{1}(I_{A/B}, A) = 0 (or if _{0}H^{2}(B, A, A) = 0), then the first projection q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R) is smooth with fiber dimension _{0}hom_{B}(I_{A/B}, A) at (B → A).

(ii) If _{0}Hom_{R}(I_{B}, B) → _{0}Hom_{R}(I_{B}, A) is surjective and _{0}H^{2}(R, B, B) → _{0}H^{2}(R, B, A) is injective, then the second projection p : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R) is smooth with fiber dimension _{0}hom_{R}(I_{B}, I_{A/B}) at (B → A).

(iii) If _{0}Hom_{R}(I_{B}, B) → _{0}Hom_{R}(I_{B}, A) is surjective and B is unobstructed as a graded R-algebra, then p is again smooth with fiber dimension _{0}hom_{R}(I_{B}, I_{A/B}) at (B → A).

(iv) If both _{0}Hom_{R}(I_{B}, B) → _{0}Hom_{R}(I_{B}, A) and the connecting homomorphism 0Hom_{R}(I_{B}, A) → _{0}H^{2}(B, A, A) (cf. (10)) are surjective and if B is unobstructed as a graded R-algebra, then A is unobstructed as a graded R-algebra.

Proof. Using the definition of smoothness of a morphism (e.g. [24], p.212) and the results of Remark 3, it is straightforward to get Proposition 4(i) from _{0}H^{2}(B, A, A) = 0 and the injection _{0}H^{2}(B, A, A) ↪ _{0}Ext_{B}^{1}(I_{A/B}, A).

To see (ii) and (iii) let ϕ_{S} : B_{S} → A_{S} be any graded deformation of ϕ to S, and let A_{T} be a graded deformations of A_{S} to T. By the definition of smoothness, it suffices to prove the existence of a graded deformation B_{T} of B_{S} to T such that o_{0}(ϕ_{S}; B_{T} , A_{T}) = 0. In (iii) the existence of some deformation B′_{T} of B_{S} follows by assumption. In (ii) we utilize that the two morphisms _{0}H^{2}(R, B, B) ⊗_{k} 𝔞 → _{0}H^{2}(R, B, A) ⊗_{k} 𝔞 and _{0}H^{2}(R, A, A) ⊗_{k} 𝔞 → 0H^{2}(R, B, A)⊗_{k}𝔞 induced by ϕ map the obstructions o_{0}(B_{S}) and o_{0}(A_{S}) onto the same element in _{0}H^{2}(R, B, A) ⊗_{k} 𝔞 by [31], proof of Thm. 4.1.14. Since o_{0}(A_{S}) = 0 and the first morphism is injective by assumption, we get o_{0}(B_{S}) = 0 and hence the existence of some deformation B′_{T} in this case as well. Now since o_{0}(ϕ_{S}; B′_{T} , A_{T}) ∈_{0}H^{1}(R, B, A) ⊗_{k} 𝔞 ≃ Hom_{R}(I_{B}, A) ⊗_{k} 𝔞 and since ϕ_{∗} is surjective, there is an element such that (ϕ_{∗} ⊗ id_{𝔞})(λ) = o_{0}(ϕ_{S}; B′_{T} , A_{T}). We get the smoothness of p by the displayed formula of Remark 3. Since smooth morphisms have smooth fibers, the fiber dimension is as claimed by [24], Thm. 1.6.

Finally (iv) follows from (iii), Remark 3 and the fact that _{0}H^{2}(B, A, A)⊗_{k}𝔞 → _{0}H^{2}(R, A, A)⊗_{k} 𝔞 maps obstructions to obstructions (by [31], Cor. 4.1.15). We leave the details as an exercise since we don’t fully need (iv) in this paper.

Let ϕ : B → A be a graded surjection. Using Proposition 4, we get the following Theorem which, at least in the geometric case and with various assumptions on Proj(B), is not new ([25], Sect.2, [26], Sect.1, [29], Ch.9, e.g. Thm. 9.4 and Prop. 9.14). Note that part B of the Theorem has as a consequence that the generic member of an irreducible family of quotients R ↠ A comes from a member of a family of quotients R ↠ B ↠ A. Since it is in general not true that a general member factors via a deformation of B even though a special member does, in part A of the Theorem below we determine the codimension of the stratum of quotients which factors. To be precise, let U ⊂ GradAlg^{H_{A}}(R) be a sufficiently small open subset containing (A). 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 exists some factorization B′ → A′ such that B′ has Hilbert function H_{B}. We will call p(p^{−1}(U)) 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 codimension of the H_{B}-stratum of A at (B → A) is defined to be dim U − dim p(U′) where U′ ⊂ p^{−1}(U) is the intersection of the irreducible components of p^{−1}(U) which contains (B → A). Using small letters for the k-dimension of _{0}Hom(−, −) and _{0}Ext_{B}^{1}(−, −) we have

Theorem 5. Let R be a graded polynomial k-algebra and let B ≃ R/I_{B} ↠ A ≃ B/I_{A/B} be a graded morphism of quotients of R such that _{0}Ext_{B}^{1}(I_{A/B}, A) = 0.

A) If _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, A) = 0 and (I_{B})_{℘} is syzygetic for any graded prime ℘ of Ass(A), then A is unobstructed as a graded R-algebra. Moreover, if depth_{𝔪} A ≥ min(1, dim A), then dim_{(A)} GradAlg^{H_{A}}(R) =

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

Furthermore let B be unobstructed as a graded R-algebra and let k be of characteristic zero. 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 if depth_{𝔪} A ≥ min(1, dim A) and depth_{𝔪} B ≥ min(1, dim B), then

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}) .

Proof. A) By the definition of a syzygetic ideal we have H_{2}(R, B, B)_{℘} = 0 for any ℘ of Ass(A). It follows that _{0}Hom_{B}(H_{2}(R, B, B), A) = 0 and hence that _{0}H^{2}(R, B, A) = 0 by (5). Since we have in general an injection _{0}H^{2}(B, A, A) ↪ _{0}Ext_{B}^{1}(I_{A/B}, A) ([14], exp. VI), we get 0H^{2}(B, A, A) = 0 by the general assumption of Theorem 5. It follows that _{0}H^{2}(R, A, A) = 0 by the long exact sequence of algebra cohomology (e.g. [31], Thm. 3.3.4)

→ _{0}H^{2}(B, A, A) → _{0}H^{2}(R, A, A) → _{0}H^{2}(R, B, A) →  (11)

This proves the unobstructedness of A and dim_{(A)} GradAlg^{H_{A}}(R) = _{0}hom_{R}(I_{A}, A). Now, looking to (10), we see that the vanishing of_{0}Ext_{B}^{1}(I_{A/B}, A) translates to a certain surjectivity in the cartesian square of (10). Since the vertical sequence of Hom-maps in (10) may be continued by

→ _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) → _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) → _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, A) →  (12)

it is straightforward to get the dimension formula from the vanishing assumption of A).

To see that _{0} ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B})− _{0} ext_{B}^{1}(I_{B}/I_{B}^{2}, B) measures the codimension, we observe that B ↠ A is unobstructed and hence that dim_{(B→A)} GradAlg(H_{A}, H_{B}) = dim _{0}A_{B→A}^{1} by Proposition 4(i). It follows that p : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{A}}(R) is a morphism between smooth schemes (i.e. smooth at (B → A) and (A) respectively) and with fiber dimension_{0}hom_{B}(I_{B}/I_{B}^{2}, I_{A/B}), from which we may conclude by using the Theorem of generic smoothness to p at (B → A), cf. [29], first part of the proof of Prop. 9.14 for more details from a similar proof.

B) Since H_{2}(R, B, B)_{℘} = 0 for any ℘ ∈ Ass(I_{A/B}) by assumption, we get_{0}H^{2}(R, B, I_{A/B}) = 0 from (5). By the long exact sequence of algebra cohomology associated to 0 → I_{A/B} → B → A → 0, we get precisely that the assumptions of Proposition 4(ii) are satisfied. By Proposition 4(i) and (ii), both projection morphisms p and q are smooth, and we deduce the H_{B}-genericity of A and the equivalence of the unobstructedness of A and B. Looking to (10) and the statement about the fiber dimensions of p and q in Proposition 4(i) and (ii), we get the dimension formulas of Theorem 5A as well.

Remark 6. a) Note that the proof of Theorem 5A leads to _{0}H^{2}(R, A, A) = 0 (and in fact to vH^{2}(R, A, A) = 0 by twisting the two Ext_{B}^{1} vanishing assumptions by v).

b) It is possible to weaken the assumption_{0}Ext_{B}^{1}(I_{A/B}, A) = 0 in Theorem 5 to “ _{0}H^{2}(B, A, A) = 0” and still conclude as in Theorem 5A and 5B. One may also weaken “(I_{B})_{℘} is syzygetic for any ℘ of Ass(A)” in part A to “_{0}Hom_{B}(H_{2}(R, B, B), A) = 0” and still conclude as in Theorem 5A. Similarly we may in part B replace “(I_{B})_{℘} is syzygetic for any ℘ of Ass(I_{A/B})” by “_{0}Hom_{B}(H_{2}(R, B, B), I_{A/B}) = 0” \underline{or} the entire assumption “_{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})” by “_{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, I_{A/B}) → 0Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) is injective and B is unobstructed as a graded R-algebra” and still get the conclusion of Theorem 5B. There is no real change in the proofs, except when B is unobstructed in which case we need to use Proposition 4(iii) in the proof instead of Proposition 4(ii). These variations turn out to have an application notably to the case A is Artinian.

c) Continuing (12), we see that the condition _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, A) = 0 of part A is satisfied provided _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 and _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, I_{A/B}) ↪ _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) is injective.

In applying Theorem 5A the codimension of the H_{B}-stratum of A at (B → A) is often just the codimension of the H_{B}-stratum of A, due to

Lemma 7. Let B ≃ R/I_{B} ↠ A ≃ B/I_{A/B} be a graded morphism of quotients of R and let m(B) be the largest degree of the minimal generators of I_{B}.

a) If (I_{A/B})_{v} = 0 for all v ≤ m(B), then _{0}Hom_{R}(I_{B}, I_{A/B}) = 0 and p^{−1}((A)) consists of a single k-point. (i.e. if B′ → A is a quotient such that H_{B′} = H_{B}, then B′ = B.)

b) If _{0}Ext_{B}^{1}(I_{A/B}, A) = 0 (or if _{0}H^{2}(B, A, A) = 0) and (I_{A/B})_{v} = 0 for all v ≤ m(B), then there is an open subset U of GradAlg(H_{B}, H_{A}) containing (B → A) such that the restriction of the first projection q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}}(R) to U is smooth, and the restriction of the second projection p to U is unramified and injective. Moreover U = p^{−1}(p(U)). In particular if B → A satisfies the assumptions of Theorem 5A (except for char(k) = 0), then the codimension of the H_{B}-stratum of A is

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

Proof. a) Clearly _{0}Hom_{R}(I_{B}, I_{A/B}) = 0 and (I_{B})_{v} = (I_{A})_{v} for all v ≤ m(B) by assumption. Since (I_{B′})_{v} ⊂ (I_{A})_{v}, we get (I_{B})_{v} = (I_{B′})_{v} for v ≤ m(B) because H_{B} = H_{B′}. Hence I_{B′} contains all minimal generators of I_{B}, i.e. I_{B} ⊂ I_{B′}, and we get equality by using H_{B} = H_{B′}.

b) By Proposition 4(i) there is an open subset U of GradAlg(H_{B}, H_{A}) such that the restriction of q to U is smooth. By possibly shrinking U, we get m(B′) ≤ m(B) for any k-point (B′) ∈ q(U) by semicontinuity. Hence for any (B′ → A′) ∈ U we get (I_{A′/B′})_{v} = 0 for all v ≤ m(B′) because H_{B} = H_{B′} and H_{A} = H_{A′}, and we see by (i) that p is unramified on U and injective on the k-points of p^{−1}(U). Now we do not need to use the Theorem of generic smoothness because the codimension of the stratum must be dim_{(A)} GradAlg^{H_{A}} − dim_{(B→A)} GradAlg(H_{A}, H_{B}), and we conclude by (10) and (12) and the unobstructedness of A and B → A.

We will in the next section need to combine (9) with the following result which in fact is a corollary ([29], Cor. 9.10) of a geometric variation ([29], Thm. 9.4) of Theorem 5B. Below s(C/F) is the minimal degree of the generators of the homogeneous ideal I_{C/F}.

Proposition 8. Let C_{0} ⊂ ℙ^{c+1} be an arithmetically Cohen-Macaulay curve of degree d_{0} and arithmetic genus g_{0}, sitting on a smooth rational surface F ⊂ ℙ^{c+1} of degree f. Let K be the canonical divisor of F. If t is an integer such that t ≥ 5 − s(C_{0}/F) and t > (C_{0} · K)/d_{0}, then any member C = Proj(B) of the linear system |C_{0} + tH| is unobstructed and

dim_{(C)} Hilb^{p}(ℙ^{c+1}) = h^{0}(N_{F}) + g_{0} − C_{0} · K + t(d_{0} + c + 1 − f/2) + t^{2}f/2 − 1

Moreover, if F is nondegenerate, then h^{0}(N_{F}) = c^{2} + 4c + 13 − 2K^{2}.

We finish this section by looking more closely at deformations of an Artinian algebra A. Firstly we recall the representability of the functor (the sheaf) GRADALG^{H}(−), locally given by letting GRADALG^{H}(Spec(S)) (S a noetherian k-algebra) be the set of graded S-flat quotients R_{S} → A_{S} such that H_{A_{s}} = H and depth_{𝔪} A_{s} ≥ min(1, dim A_{s}) for any s ∈ Spec(S). Here R_{S} = R ⊗_{k} S and A_{s} is the “fiber” of Spec(A_{S}) → Spec(S) at s ∈ Spec(S). To simplify and clarify [27] (being more careful with the topologies involved), we sketch proofs. Note that the representability below ([27], Thm. 1.1) has later been generalized by Haiman and Sturmfels ([19]).

Proposition 9. The functor GRADALG^{H}(−) is representable. Its representing object, GradAlg^{H}(R), is a k-scheme of finite type.

Proof. Here we sketch a proof which is slightly different from that in [27] in the Artinian case (i.e. when H(v) = 0 for v >> 0). Indeed the flattening stratification described in [37] and Grothendieck’s representability Theorems of the Hilbert functor are the results we need to conclude. In the non-Artinian case, H(v) is a polynomial p(v) for v >> 0 and GradAlg^{H}(R) is the stratum of Hilb^{p}(ℙ^{n+c−1}) of constant postulation (cf. [27], Thm. 1.1 for details). If H(v) = 0 for v >> 0 and t = ∑_{v} H(v), then there exists a scheme Hilb := Hilb^{t}(𝔸^{n+c}) of finite type which parametrizes quotients R → A of length t. The restriction (pullback) R_{S} → A_{S} of the universal quotient to Spec(S) → Hilb defines the subscheme GradAlg^{H}(R) of Hilb by requiring ker((R_{S})_{v} → A_{S}) to be flat over S, of rank dim R_{v} − H(v) for every 0 ≤ v ≤ t at each fiber of Spec(S). Since such a subscheme GradAlg^{H}(R) of finite type exists by [37], Lect. 8 or [33], VI, Prop. 1.1, we are done.

Remark 10. We may in the case H(v) = 0 for v >> 0 use flattening stratification to show the existence of the scheme Z(H) of not necessarily graded Artinian quotients A of R with Hilbert function H (by requiring gr(A_{S}) to be S-flat, of Hilbert function H at each fiber of Spec(S)), to get Z(H) as a locally closed subscheme (stratum) of Hilb^{t}(𝔸^{n+c}). Its local deformation functor at a quotient A ≃ R/I_{A} (with \sqrt{I_{A}} = 𝔪) of R, defined on the category of local Artinian k-algebras with residue field k, consists of quotients A_{S} of R_{S} = S[X_{1}...., X_{n+c}] (or equivalently, of S[[X_{1}...., X_{n+c}]]) such that A_{S} and gr(A_{S}) are deformations of A and gr(A) respectively. If R → A is graded and satisfies _{v}H^{2}(R, A, A) = 0 for v ≥ 0, then [27], Thm. 1.10 and Rem. 1.11 shows that Z(H) is smooth at (R → A) and dim_{(A)} Z(H) = ∑_{v≥0} _{v}hom_{R}(I_{A}, A).

Inside GradAlg^{H}(R) there is an open set (and hence a scheme Gor_{c}^{H}(R) with induced scheme structure) consisting of graded Gorenstein quotients R → A with Hilbert function H. Gor_{c}^{H}(R) is empty unless the difference Δ^{dim A}H, i.e. the h-vector of A, is a symmetric Gorenstein sequence. In the Artinian case there is another known scheme parametrizing graded Gorenstein quotients with suitable Hilbert function H, namely the determinantal loci PGor(H) parametrizing forms F of degree s in ℙ^{N}, N = \binom{s+n+c−1}{n+c−1} − 1, given by requiring the “catalecticant matrices” of F to have ranks given by the Hilbert function (see Iarrobino-Kanev’s book [21], Sect. 1.1). We denote the restriction of the generic s-form of ℙ^{N} to PGor(H) by F_{PGor(H)} ([21], Def. 1.3). Then the underlying sets of closed points of Gor_{c}^{H}(R) and PGor(H) are the same by apolarity (the Macaulay correspondence), and one may see from the proof of our next result that they are in fact isomorphic as topological spaces. Moreover since Iarrobino and Kanev have proved that the tangent space of PGor(H) at A is isomorphic to _{0}Hom_{A}(I_{A}/I_{A}^{2}, A) ([21], Thm. 3.9) and it is well known that the tangent space of GradAlg^{H}(R) is the same group [24], we get

Theorem 11. Let R → A be a graded Artinian Gorenstein quotient with Hilbert function H. Then dim_{(A)} GradAlg^{H}(R) = dim_{(A)} PGor(H). Hence PGor(H) is smooth at (R → A) if and only if GradAlg^{H}(R) is smooth at (R → A). In particular PGor(H) is smooth at (R → A) provided _{0}H^{2}(R, A, A) = 0, i.e. provided the natural surjection (S_{2}I_{A})_{s} → (I_{A}^{2})_{s} from the second symmetric power to the second power of I_{A} is an isomorphism in the socle degree s of A.

Proof. By duality, _{0}H^{2}(R, A, A) = 0 is equivalent to _{s}H_{2}(R, A, A) = 0 which again is equivalent to (S_{2}I_{A})_{s} ≃ (I_{A}^{2})_{s} by (28) below. Hence we only need to prove dim_{(A)} GradAlg^{H}(R) = dim_{(A)} PGor(H), because we know their tangent spaces are isomorphic. Let V ⊂ PGor(H) be a closed irreducible subset (e.g. of the form V = Spec(T)), and let V have the reduced scheme structure. By the definition of PGor(H), the restriction F_{V} of the “universal” s-form F_{PGor(H)} to V defines via apolarity a family of graded Artinian Gorenstein quotients over V (e.g. the family A_{T} ≃ R_{T} ∘ F_{V} over V = Spec(T) where R_{T} is interpreted as the T -algebra of contractions, “partial derivations without coefficients”) with constant Hilbert function H. Since V is integral, it follows that the family (e.g. the morphism Spec(A_{T}) → Spec(T)) is flat ([37], Lect. 6). Hence we have a morphism π : V → GradAlg^{H}(R) by the universal property of GradAlg^{H}(R). π(V) is irreducible and closed in Gor_{c}^{H}(R) (it is closed because an “inverse” (Gor_{c}^{H}(R))_{red} → PGor(H) on closed points exists by [21], Ch. 8). So chains of closed irreducible subsets in PGor(H) and Gor_{c}^{H}(R) correspond, and we are done.

Problem 12. It would be interesting to prove that PGor(H) and Gor_{c}^{H}(R) are isomorphic as schemes, cf. [27], Rem. 1.9. The arguments in the proof above, which easily lead to an isomorphism of PGor(H) and Gor_{c}^{H}(R) as topological spaces, are mainly contained in [27], Rem. 1.9. In that Remark we also claimed that these arguments showed the existence of a natural morphism Π : PGor(H) → Gor_{c}^{H}(R). Unfortunately an index “red” is missing in Rem. 1.9, i.e. we can from the arguments only be sure of the existence of a natural morphism π : (PGor(H))_{red} → Gor_{c}^{H}(R), and the existence of Π above remains to be proved.

2 Families of codimension one Gorenstein quotients of Cohen-Macaulay algebras

The main purpose of this paper is to study families of graded Gorenstein quotients A obtained by taking a regular section σ of the anticanonical system (K_{B}^{∗})_{s}, i.e. quotients A given by

0 → K_{B}(−s) →^{σ^{∗}} B → A → 0  (13)

for some integer s. Our main results apply to Artinian Gorenstein algebras as well as to Gorenstein quotients of higher dimension. In what follows B is Cohen-Macaulay. In order to make the description of A in (13) meaningful, we must suppose that B = R/I_{B} is generically Gorenstein (cf. [29], Lemma 5.2 and 5.4). Then we say “A is a codimension one quotient of B, well-defined by a twist of the canonical module”. If A is such a quotient of B, then a stratum of quotients given by (13) around (A) is just a maximal (flat) family consisting of quotients (A′) of GradAlg^{H_{A}}(R) for which there exists a factorization B′ → A′ as in (13) where (B′) (resp. the sections) belong to some open neighbourhood of (B) in GradAlg^{H_{B}}(R) (resp. of σ in the linear system (K_{B′}^{∗})_{s} ⊂ H^{0}(Proj(B′), K̃_{B′}^{∗}(s))), making each A′ well-defined in the sense above. Looking to the proof of Proposition 13, one may see that such a stratum is just the image p(U) of some open U of GradAlg(H_{B}, H_{A}) containing (B → A) via the second projection p, provided _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0.

The following result determines dim_{(A)} GradAlg^{H_{A}}(R) in terms of invariants of B, such as 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}) (here S_{2}(K_{B}) is the second symmetric power of K_{B}), and leads to the main theorems of this section. Since Proposition 13 is a variation of Theorem 5 with I_{A/B} = K_{B}(−s), Remark 6 (except the first sentence of Remark 6(b)) will apply as a remark to Proposition 13 as well.

Proposition 13. Let B be a generically Gorenstein, graded Cohen-Macaulay quotient of a polynomial ring R, let A be a codimension one quotient of B, defined by (13) for some s, and let _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0.

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

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim(K_{B}^{∗})_{s} − 1 − δ(B)_{−s} − _{0}ext_{B}^{1}(I_{B}/I_{B}^{2}, B)

where δ(B)_{−s} = _{−s}hom_{R}(I_{B}, K_{B}) − _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}). Moreover if B is unobstructed as a graded R-algebra and char(k) = 0, then the codimension of the H_{B}-stratum of A at (B → A) is

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

This number also equals the codimension of the stratum of quotients given by (13) at (A).

B) If _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = 0 and I_{B} is generically syzygetic, then A is Gorenstein and 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) + dim(K_{B}^{∗})_{s} − 1 − _{−s}hom_{R}(I_{B}, K_{B}) .

Remark 14. (a) With the notations, 0 → ⊕_{i}R(−n_{c,i}) → ... → ⊕_{i}R(−n_{1,i}) → I_{B} → 0, of (6), recalling dim R = n + c and dim B = n, we claim that

δ(B)_{−s} = 0 provided s > 2 reg(I_{B}) − n

By (7) it suffices to prove _{−s}Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}) = 0 for i = 0 and 1 provided s > max n_{c,i} + max n_{2,i} − n − c. Indeed dualizing (6) we get a surjection ⊕_{i}R(n_{c,i}) ↠ K_{B}(n + c), and it follows that (K_{B})_{−v} = 0 for v > max n_{c,i} − n − c. We conclude by applying _{0}Hom_{B}(−, K_{B}) to the right-exact sequence ⊕_{i}B(−n_{2,i}) → ⊕_{i}B(−n_{1,i}) → I_{B}/I_{B}^{2} → 0 and the definiton of Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}).

(b) If dim B = 1, we have Ext_{B}^{i}(−, K_{B}) = 0 for i ≥ 2 by Gorenstein duality. In this case _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 implies _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, A) = 0 by Remark 6(c). Moreover looking to Remark 6(b) and using that A ≃ K_{A}(−s) and Gorenstein duality, we may weaken “I_{B} is syzygetic for any graded prime ℘ of Ass(A)” to “(H_{2}(R, B, B) ⊗_{B} A)_{s} = 0”. Hence if we suppose

0 Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = (H_{2}(R, B, B) ⊗_{B} A)_{s} = 0,

we still have the conclusion concerning the unobstructedness of A and the formula for dim_{(A)} GradAlg^{H_{A}}(R) of Proposition 13A. Furthermore if I_{B} is generically syzygetic (this is always true if c ≤ 3 since B is generically Gorenstein, cf. the proof of Theorem 16), then B is unobstructed as a graded algebra and the codimension formula of Proposition 13A holds as well (in the characteristic zero case).

(c) Slightly reformulating Proposition 13A (by replacing the vanishing of the two _{0}Ext_{B}^{1} groups by the corresponding vanishing of Ext_{B}^{1}) one may use the proof below to show that A is unobstructed as an R-algebra in the sense H^{2}(R, A, A) = 0, cf. Remark 6. A further slight reformulation (replacing Proj by Spec and skipping the twists) allows us to prove a corresponding result for non-graded objects as well. If B is licci then the three Ext_{B} groups, Ext_{B}^{1}(S_{2}(K_{B}), K_{B}), Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) and Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}), vanish ([5], [20], [38] Thm. 4.2.6, or Corollary 37 of this paper), and we get the unobstructedness of the Gorenstein quotient A by Remark 6(c). Indeed S_{2}(K_{B}) is Cohen-Macaulay if B is licci. In the interesting codimension c = 3 case, the Cohen-Macaulayness of S_{2}(K_{B}) and of the Koszul homology group, H_{1} = H_{1}(I_{B}), are equivalent [38], Thm. 4.2.9. This Cohen-Macaulayness holds in particular if the Cohen-Macaulay type of B is 2 [38], Cor. 4.2.11.

(d) If I_{B} is generically syzygetic, then Hom_{B}(H_{2}(R, B, B), K_{B}) = 0. Hence applying Hom_{B}(−, K_{B}) to (4), we see that _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ↪_{−s} Ext_{B}^{1}(H_{1}, K_{B}) is injective and that

δ(B)_{−s} = dim(G_{1}^{∗} ⊗_{R} K_{B})_{−s} − _{−s}hom_{B}(H_{1}, K_{B})

allowing us to restate the dimension formula in terms of H_{1} and G_{1}. If B is licci, then −sHom_{B}(H_{1}, K_{B}) ≃ (H_{r−1})_{−t−s} by [30], Prop. 18, where r and t are given by ∧^{r}H̃_{1}|_{U} ≃ K̃_{B}(t)|_{U}.

Proof. Note that we can show A ≃ K_{A}(−s) and hence that A is Gorenstein by applying the mapping cone construction to (13), (or see [29], Lemma 5.2 and 5.4). We claim that Proposition 13 above follows from Theorem 5 and Remark 6 by letting I_{A/B} = K_{B}(−s) and by paying some extra attention to the codimension of the strata. Indeed it is straightforward to obtain the assumptions of Theorem 5 from the assumptions of Proposition 13, except possibly the assumption _{0}Ext_{B}^{1}(I_{A/B}, A) = 0 of Theorem 5. Suppose temporarily that B is Gorenstein in codimension ≤ 1. Then _{0}Ext_{B}^{1}(K_{B}(−s), B) ≃ _{0}Ext_{B}^{1}(K_{B} ⊗ K_{B}(−s), K_{B}) ≃ 0Ext_{B}^{1}(S_{2}(K_{B})(−s), K_{B}) = 0 by a well known spectral sequence ([18], Satz 1.2). Moreover, since K_{B} is a maximal Cohen-Macaulay B-module, one knows _{0}Ext_{B}^{i}(K_{B}, K_{B}) = 0 for i > 0 and we get _{0}Ext_{B}^{1}(K_{B}(−s), A) = 0 by applying _{0}Hom(K_{B}(−s), −) to (13), i.e. we get the claim. In the general case where B is generically Gorenstein we get the conclusion of Theorem 5 by combining Lemma 15 and Remark 6(b), recalling K_{B}(−s) ≃ I_{A/B}. Furthermore to get the dimension formulas we note that _{0}Hom(K_{B}(−s), K_{B}(−s)) is one-dimensional. Hence by again applying _{0}Hom(K_{B}(−s), −) to (13), we get the dimension formulas from the corresponding formulas of Theorem 5.

It remains to see that the codimension may also be computed from the stratum of quotients given by (13). Now looking to the definition of the codimension of the H_{B}-stratum of A at (B → A) before Theorem 5, 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 (13). 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 a certain functor of graded flat quotients, there exists a universal quotient whose pullback to Spec(S) is denoted by B_{S} → A_{S}. Its kernel I_{A_{S}/B_{S}} is a flat deformation of the B-module I_{A/B} ≃ K_{B}(−s) to S. By the theory of deformations of the module I_{A/B} its tangent and obstruction spaces are given by _{0}Ext_{B}^{i}(K_{B}(−s), K_{B}(−s)) for i = 1 and 2 respectively. Since both these Ext-groups vanish, there is a unique deformation of I_{A/B} to B_{S_{i}}, i.e. we get isomorphisms l_{i} : I_{A_{S}/B_{S}} ⊗_{S} S_{i} ≃ K_{B_{S}}(−s) ⊗_{S} S_{i} which commute with l_{i−1} because K_{B_{S}} := Ext_{R_{S}}^{c}(B_{S}, R_{S}(−n−c)) where R_{S} := R ⊗_{k} S, is S-flat. (cf. [23], Prop.A1). It suffices to show that there exists a morphism l : I_{A_{S}/B_{S}} → K_{B_{S}}(−s) which reduces to l_{i} mod m_{S}^{i} because l must then be an isomorphism which extends to an isomorphism in an open neighbourhood of t in G. To show the existence of l, we claim that K_{B_{S}}^{∗}(s) → K_{B}^{∗}(s) is surjective. By the proof of [26], Prop. 2.4, it suffices to show that η_{i} : K_{B_{S_{i}}}^{∗}(s) → K_{B_{S_{i−1}}}^{∗}(s) is surjective. If _{0}Ext_{B}^{1}(K_{B}(−s), B) = 0 (e.g. B is Gorenstein in codimension ≤ 1, cf. the first part of the proof), this follows by applying _{0}Hom(K_{B_{S_{i}}}(−s), −) onto 0 → B ⊗_{k} 𝔞 → B_{S_{i}} → B_{S_{i−1}} → 0 where 𝔞 := m_{S}^{i−1}/m_{S}^{i}. In the general case, replacing every B by A in the latter sequence and applying _{0}Hom(K_{B_{S_{i}}}(−s), −) ≃ _{0}Hom(I_{A_{S_{i}}/B_{S_{i}}} ⊗_{B_{S_{i}}} A_{S_{i}}, −) to that sequence, we get the surjectivity of η_{i} from _{0}Ext_{A}^{1}(I_{A/B}/I_{A/B}^{2}, A) = 0 and _{0}Ext_{B}^{1}(K_{B}(−s), K_{B}(−s)) = 0. Note that we have the last mentioned vanishings by (5), Lemma 15 and _{s}Ext^{1}(S_{2}(K_{B}), K_{B}) = 0, and the claim is proved. Now using the proven claim we get a morphism K_{B_{S}}(−s) → B_{S} whose cokernel is an S-flat deformation of A. By the universal property of G, there is a morphism Spec(S) → Spec(S) whose pullback induces the desired map I_{A_{S}/B_{S}} → K_{B_{S}}(−s) and the proof is complete.

Lemma 15. Let B be a graded Cohen-Macaulay quotient of R and let A ≃ B/I_{A/B} be a graded Gorenstein quotient such that A ≃ K_{A}(−s) and dim B − dim A = r. If r > 0 then 0H^{2}(B, A, A) and _{−s}Ext_{B}^{r}(S_{2}(I_{A/B}), K_{B}) vanish simultaneously.

Proof. Using (5) we see that the vanishing of_{0}H^{2}(B, A, A) is equivalent to_{0}Ext_{A}^{1}(I_{A/B}/I_{A/B}^{2}, A) = 0 and the injectivity of the natural map _{0}Hom_{A}(H_{2}(B, A, A), A) ↪ _{0}Ext_{A}^{2}(I_{A/B}/I_{A/B}^{2}, A). Moreover since A ≃ K_{A}(−s) we have by Gorenstein duality (applied to both A and B) that

0Ext_{A}^{i}(I_{A/B}/I_{A/B}^{2}, A) ≃ _{s}H_{𝔪}^{dim A−i}(I_{A/B}/I_{A/B}^{2})^{∨} ≃ _{−s}Ext_{B}^{i+r−1}(I_{A/B}^{2}, K_{B})

for i ≥ 1 because the middle term is isomorphic to _{s}H_{𝔪}^{dim A+1−i}(I_{A/B}^{2}) by the long exact sequence of local cohomology applied to 0 → I_{A/B}^{2} → I_{A/B} → I_{A/B}/I_{A/B}^{2} → 0. Correspondingly 0Hom_{A}(H_{2}(B, A, A), A) ≃ _{−s}Ext_{B}^{r}(H_{2}(B, A, A), K_{B}). Hence _{0}H^{2}(B, A, A) = 0 is equivalent to_{−s}Ext_{B}^{r}(I_{A/B}^{2}, K_{B}) = 0 and the injectivity of of the natural map_{−s}Ext_{B}^{r}(H_{2}(B, A, A), K_{B}) ↪ −sExt_{B}^{r+1}(I_{A/B}^{2}, K_{B}). Recalling the exact sequence

0 → H_{2}(B, A, A) → S_{2}(I_{A/B}) → I_{A/B}^{2} → 0  (14)

(cf. [1], p.106 or [38], Sect. 2.1) and that _{−s}Ext_{B}^{r−1}(H_{2}(B, A, A), K_{B}) = 0 since H_{2}(B, A, A) is supported at the codimension r quotient A of B, we get the conclusion of Lemma 15 by the long exact sequence of _{−s}Ext_{B}^{i}(−, K_{B}) applied to (14).

Theorem 16. Let B = R/I_{B} be a graded Cohen-Macaulay quotient of a polynomial ring R, and let A be a graded codimension one quotient of B, well-defined by a twist of the canonical module K_{B}.

A) If B is licci, then A is unobstructed as a graded R-algebra, A is Gorenstein and,

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

where δ(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, then the codimension of the H_{B}-stratum of A at (B → A) is _{−s} ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}). This number also equals the codimension of the stratum of quotients given by (13) around (A).

B) If s >> 0 and the sheaf Ext_{O_{Proj(B)}}^{1}({ \atop S_{2}(K_{B})}, K̃_{B}) = 0, then A is H_{B}-generic and Gorenstein, 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 if and only if B is unobstructed as a graded R-algebra.

Note that if Y = Proj(B) is locally licci, then the sheaf { \atop S_{2}(K_{B})} is maximally Cohen-Macaulay by Remark 14(c), i.e. Ext_{O_{Y}}^{1} ({ \atop S_{2}(K_{B})}, K̃_{B}) = 0, and we get Theorem 1. The sheaf also vanishes if Y is (locally) Gorenstein, e.g. if dim Y = 0.

Proof. A) Since B is licci and generically Gorenstein, I_{B} is generically syzygetic by [29], Prop. 6.17 which implies H^{2}(R, B, B)_{℘} = 0 provided dim B_{℘} = 0. This in turn is equivalent to H_{2}(R, B, B)_{℘} = 0 by Gorenstein duality and the spectral sequence relating algebra-homology to algebra-cohomology, i.e. I_{B} is generically syzygetic. Then I_{B} is in fact syzygetic by (4) because H_{1} is a maximal Cohen-Macaulay module in the licci case. Since all Ext_{B}^{1} groups of Proposition 13A vanish by Remark 14(c) and Remark 6(c), we conclude by Proposition 13.

B) If Ext_{O_{Y}}^{1} ({ \atop S_{2}(K_{B})}, K̃_{B}) = 0, we see that _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) vanishes for large s because of the exact sequence

→ _{s}Hom_{B}(S_{2}(K_{B}), H_{𝔪}^{1}(K_{B})) → _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) → H^{1}(Y, Hom_{B}({ \atop S_{2}(K_{B})}, K̃_{B})(s))

(which is analogous to (3)). Since the vanishing of _{−s}Hom_{B}(H_{2}(R, B, B), K_{B}) = 0 and −sExt_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}), i = 0, 1, are obviously true for s >> 0, we conclude by Proposition 13B and Remark 6(b).

To apply Proposition 13 and Theorem 16, we need to compute dim(K_{B}^{∗})_{s} and verify its assumptions. In several cases it is easy to compute dim(K_{B}^{∗})_{s} at least for large s. For instance if B is 2-dimensional and C = Proj(B) is smooth, then K̃_{B}^{∗} is just the tangent sheaf (sheaf of derivations), θ_{C}, of C, and we can get the information we need from the vanishing of cohomology and the Riemann-Roch theorem. In this case the most difficult assumption to verify is perhaps the unobstructedness of B (or _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0), for which we point out a couple of important cases where the unobstructedness is known;

1. _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 provided C is linked (in possibly several steps) to a generically Gorenstein curve C′ = Proj(B′) satisfying _{0}Ext_{B′}^{1}(I_{B′}/I_{B′}^{2}, B′) = 0 (cf. next section).

2. B is unobstructed if C = Proj(B) sits on a smooth rational surface in ℙ^{4} and the degree of C is ≥ 24 ( [29], Prop. 9.13; the statement follows also from Proposition 8).

By 1) we get _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 if B is licci (e.g. if C′ above is a complete intersection), or if C′ is a local complete intersection curve and H^{1}(N_{C′}) = 0 (because_{0}Ext_{B′}^{1}(I_{B′}/I_{B′}^{2}, B′) ↪ H^{1}(N_{C′}) is injective by (3)). Indeed for a local complete intersection curve C = Proj(B) we have H^{1}(N_{C}(v)) ≃ Ext_{O_{C}}^{1}({ \atop I_{B}/I_{B}^{2}}, B̃(v)) and by (3) an exact sequence

vExt_{B}^{1}(I_{B}/I_{B}^{2}, B) ↪ H^{1}(N_{C}(v)) → _{v}Hom_{B}(I_{B}/I_{B}^{2}, H_{𝔪}^{2}(B)) → _{v}Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) → 0 (15)

We illustrate Theorem 16 by taking a relatively simple example, remarking that far more complex examples can be given by the same methods, by increasing s.

Example 17. Let B be the homogeneous coordinate ring of a smooth rational arithmetically Cohen-Macaulay curve C of degree d = 4 in ℙ^{4} = Proj(R), with minimal resolution

0 → R(−4)^{3} → R(−3)^{8} → R(−2)^{6} → R → B → 0,  (16)

and let A be defined by a regular section of (K_{B}^{∗})_{s} for s ≥ 0. Then C is the determinantal curve given by the maximal minors of a 2 by 4 matrix with linear entries, and it is straightforward to see that the Hilbert function H_{A} of A is given by dim A_{v} = 4v + 1 for 0 ≤ v ≤ s and dim A_{v} = 4s + 2 for v > s. Applying Hom_{R}(−, R(−5)) onto (16) we get the minimal resolution of K_{B} which combined with (13) and the mapping cone construction yields the resolution

0 → R(−s−5) → R(−s−3)^{6}⊕R(−4)^{3} → R(−s−2)^{8}⊕R(−3)^{8} → R(−s−1)^{3}⊕R(−2)^{6} → I_{A} → 0.

Even though B is not licci, Theorem 16 applies for s >> 0. Indeed, looking to Proposition 13 and Remark 14(a), we claim that the conclusions of Theorem 16B hold for s ≥ 3 (and those of Theorem 16A for s ≥ 0, see Example 35 in the next section). Indeed we need only to verify its assumptions. Since C is smooth and θ_{C} := K̃_{B}^{∗} is of degree deg θ_{C} = − deg K̃_{B} = 2 − 2g (g = the genus of C), we get H^{1}(θ_{C}(v)) = 0 and h^{0}(θ_{C}(v)) = dv + 3 − 3g = 4v + 3 for v ≥ 0 ([37], Lect. 11). Since we have an exact sequence

0 → θ_{C} → θ_{ℙ}|_{C} → N_{C} → 0

where θ_{ℙ}|_{C} = coker(O_{C} → O_{C}(1)^{⊕5}) is the tangent sheaf of ℙ = ℙ^{4} restricted to C, we get _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 from H^{1}(N_{C}) = 0 and (15). Moreover _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B})^{∨} ≃ −vH_{𝔪}^{1}(S_{2}(K_{B})) and since H_{∗}^{0}(C, { \atop S_{2}(K_{B})}) → H_{𝔪}^{1}(S_{2}(K_{B})) is surjective and H^{0}(C, { \atop S_{2}(K_{B})}(−v))^{∨} ≃ H^{1}(θ_{C}(v)) we get

vExt_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 for v ≥ 0.

By Remark 14(a), we know _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = 0 for s > 2 and the claim is proved for s ≥ 3. In conclusion the quotient A is an unobstructed Gorenstein algebra for s ≥ 3 and

dim_{(A)} GradAlg^{H_{A}}(R) = h^{0}(N_{C}) + h^{0}(θ_{C}(s)) − 1 = 5d + 1 − g + ds + 3 − 3g − 1 = 4s + 23

Remark 18. This example shows that if Proj(B) is a smooth curve of degree d and genus g, and if we let v be such that 2 − 2g + vd > 2g − 2, i.e. v > (4g − 4)/d, then H^{1}(θ_{C}(v)) = 0 and we get both _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 and dim(K_{B}^{∗})_{v} = h^{0}(θ_{C}(v)) = dv + 3 − 3g. In particular if s > max{2 reg(I_{B}) − 2, (4g − 4)/d} and B is unobstructed, then A is unobstructed and H_{B}-generic by Proposition 13B and Remark 14(a) and

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + ds + 2 − 3g.

Hence we may replace “s >> 0” in Theorem 1 and Theorem 16 by “s > max{2 reg(I_{B}) − 2, (4g − 4)/d}” in this case.

Now we consider Artinian Gorenstein quotients. In particular we will concentrate on Artinian reductions obtained from algebras A similar to those in Example 17, through modding out by a general linear form L (cf. Example 21 below). More generally let B ≃ R/I_{B} be an n-dimensional Cohen-Macaulay algebra, let A ≃ B/K_{B}(−s), and let R_{L} := R/(L) → B_{L} = B/(L) → A_{L} = A/(L) be their quotients modulo (L). The Hilbert functions of A_{L} and B_{L} are given by the first differences of the Hilbert functions of A and B respectively, e.g. H_{B_{L}}(v) = H_{B}(v) − H_{B}(v − 1) = ΔH_{B}(v). Moreover an R_{L}-free minimal resolution of B_{L} has the same graded Betti numbers as the R-free minimal resolution of B ([34], p.28), i.e. their numbers n_{j,i} given in (6) coincide. Dualizing (6) we see that the minimal resolutions of K_{B}(n + c) and K_{B_{L}}(n + c − 1) also coincide, i.e. K_{B} ⊗ B_{L} ≃ K_{B_{L}}(−1). Looking to (13) it follows that A_{L} is given by

0 → K_{B_{L}}(−s − 1) → B_{L} → A_{L} → 0,  (17)

(note −s − 1 instead of −s !) and A_{L} is an Gorenstein quotient, well defined by a twist of the canonical module. Hence we may apply Proposition 13 and Theorem 16. To compute dim(K_{B_{L}}^{∗})_{s+1} and verify its assumptions, we see, thanks to the arguments of Remark 14(a), that δ(B_{L})_{−s−1} = 0 and _{−s−1}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, K_{B_{L}}) = 0 (where I_{B_{L}} = ker(R_{L} → B_{L})), provided s + 1 > max n_{c,i} + max n_{2,i} − (n − 1) − c which is equivalent to the inequality of that Remark, used to show δ(B)_{−s} = 0. To check the other assumptions, we claim that Lemma 19. (i) If _{v−1}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0, then dim(K_{B_{L}}^{∗})_{v+1} = dim(K_{B}^{∗})_{v}−dim(K_{B}^{∗})_{v−1}.

(ii) If _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = _{v−1}Ext_{B}^{2}(S_{2}(K_{B}), K_{B}) = 0, then

v+1Ext_{B_{L}}^{1}(S_{2}(K_{B_{L}}), K_{B_{L}}) = 0.

(iii) If B is an unobstructed graded R-algebra and the map _{−1}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ↪ 0Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) induced by L is injective, then B_{L} is an unobstructed graded R_{L}-algebra and

dim(N_{B_{L}})_{0} = dim(N_{B})_{0} − dim(N_{B})_{−1}.

(iv) If _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = _{−1}Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) = 0, then _{0}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, B_{L}) = 0.

Proof. (i) and (ii). Apply Hom_{B}(S_{2}(K_{B}), −) to the exact sequence 0 → K_{B} → K_{B}(1) → K_{B_{L}} → 0 and use that S_{2}(K_{B}) ⊗ B_{L} ≃ S_{2}(K_{B_{L}})(−2) and that K_{B}^{∗} = Hom_{B}(S_{2}(K_{B}), K_{B}).

(iii) The unobstructedness is just a very particular case of Theorem 5B and Remark 6(b) letting I_{A/B} ≃ B(−1). To compute N_{B_{L}}, just apply _{0}Hom_{B}(I_{B}/I_{B}^{2}, −) to 0 → B(−1) → B → B_{L} → 0. Continuing this sequence into a long exact sequence of Ext-groups, we prove (iv) as well because _{0}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, B_{L}) ≃ _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B_{L}).

Remark 20. (a) In the case that Proj(B) is a curve (n = 2), we have by duality

vExt_{B}^{2}(S_{2}(K_{B}), K_{B})^{∨} ≃ _{−v}H_{𝔪}^{0}(S_{2}(K_{B})) ↪ S_{2}(K_{B})_{−v}. By Remark 14(a) there is a surjection S_{2}(⊗R(n_{c,i})) ↠ S_{2}(K_{B}(n+c)). Hence S_{2}(K_{B})_{−v} vanishes if v > 2 max n_{c,i} − 2(n + c), i.e.

vExt_{B}^{2}(S_{2}(K_{B}), K_{B}) = 0 provided v > 2 reg(I_{B}) − 2n − 2 = 2 reg(I_{B}) − 6

(b) If the curve C = Proj(B) is a local complete intersection, then (15) shows that −1Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) = 0 provided H^{1}(O_{C}(n_{1,i} − 1)) ≃ H^{0}(K̃_{B}(−n_{1,i} + 1))^{∨} = 0, i.e.

−1Ext_{B}^{2}(I_{B}/I_{B}^{2}, B) = 0 provided min n_{1,i} > reg(I_{B}) − n = reg(I_{B}) − 2.

Example 21. We consider the Artinian reduction A_{L} of the Gorenstein algebra of Example 17. Since B is 2-dimensional we get the vanishing of _{0}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, B_{L}), and hence the unobstructedness of B_{L}, from Lemma 19(iii) and Remark 20(b). Moreover since we have seen that Remark 14(a) also applies to the Artinian reduction, we get _{−s−1}Ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, K_{B_{L}}) = 0 for s > 2. Finally Lemma 19(ii) and Remark 20(a) shows that _{v+1}Ext_{B_{L}}^{1}(S_{2}(K_{B_{L}}), K_{B_{L}}) = 0 for v ≥ 0. By Remark 14(b) all assumptions of Proposition 13B concerning B_{L} are satisfied. It follows that A_{L} is an unobstructed Artinian Gorenstein algebra for s ≥ 3, i.e. PGor(H_{L}) is smooth at (A_{L}) by Theorem 11 and

dim_{(A_{L})} PGor(H_{L}) = dim(N_{B_{L}})_{0} + dim(K_{B_{L}}^{∗})_{s+1} − 1 = 3d + d − 1 = 15

Note that the Hilbert function of B_{L} is given by (1, 4, 4, 4, ..). Since the h-vector of A_{L} is the (s + 2)-tuple (1, 4, 4, ..., 4, 1), we get that the value of dim_{(A_{L})} PGor(H_{L}) above coincides with the dimension given in [21], Thm. 2.6 and Thm. 4.10A.

Finally we claim that this conclusion about A_{L} holds for any Artinian Gorenstein quotient A (well-defined by a twist of the canonical module of B) with the same graded Betti numbers as A_{L}. Indeed to use Remark 14(a) we only need the graded Betti numbers of the minimal resolution. Since we may prove Remark 20 for a one-dimensional B by replacing Ext_{B}^{2}(−, −) by the corresponding Ext_{B}^{1}(−, −) (in (b) we use _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) ≃ _{0}Ext_{B}^{1}(I_{B} ⊗_{R}K_{B}, K_{B}) ≃ 0H_{𝔪}^{0}(I_{B} ⊗_{R} K_{B})^{∨} in order to see that this group vanishes for min n_{1,i} > reg(I_{B}) − 1), we get the claim.

Remark 22. Pushing the arguments of this example a little further, we get that if Y = Proj(B) is a zero-scheme of degree d and if v ≥ 2 reg(I_{B}) − 3, then _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 and dim(K_{B}^{∗})_{v} = d. Indeed with v as above we have S_{2}(K_{B})_{−v} = 0 and _{−v}H_{𝔪}^{0}(S_{2}(K_{B})) = 0 (cf. Remark 20(a)) and it follows that _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 by Gorenstein duality. Hence it suffices to show (K_{B}^{∗})_{v} ≃ H^{0}(Y, K̃_{B}^{∗}(v)), i.e. to show _{v}H_{𝔪}^{1}(K_{B}^{∗}) = 0. By Gorenstein duality this group is dual to _{−v}Hom_{B}(K_{B}^{∗}, K_{B}) = _{−v}Hom_{B}(Hom_{B}(S_{2}(K_{B}), K_{B}), K_{B}) ≃ S_{2}(K_{B})_{−v} = 0. In this case note that (K_{B}^{∗})_{v} contains a regular section. In particular if s ≥ 2 reg(I_{B}) and B is unobstructed, we have by Proposition 13B and Remark 14(a) that A is unobstructed and H_{B}-generic and that

dim_{(A)} PGor(H_{A}) = dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + d − 1

cf. Theorem 11. Hence we may replace “s >> 0” in Theorem 1 (and in Theorem 16 provided B is generically syzygetic) by “s ≥ 2 reg(I_{B})”.

Now we will study the case s >> 0 of Theorem 16 to see that (13) actually determines a well-defined injective application from the set of irreducible components of GradAlg^{H_{B}}(R) containing locally licci, Cohen-Macaulay codimension c quotients of R (not necessarily unobstructed !), to the set of irreducible components of GradAlg^{H_{A}}(R) containing Gorenstein codimension c + 1 quotients of R. To be precise let U_{s} ⊂ GradAlg^{H′}(R) be an open (e.g. the largest open) subscheme whose corresponding k-points (R → B) are such that B is Cohen-Macaulay and generically Gorenstein, Proj(B) is locally licci and satisfies H_{B} = H′, sExt_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 and (K_{B}^{∗})_{s} contains a regular section. When we apply our results we usually take U_{s} to be the open set in which “B is generically Gorenstein and Proj(B) is locally licci” is replaced by “Proj(B) is locally a complete intersection” or “ Proj(B) is smooth” or “ Proj(B) is locally Gorenstein and the codimension of B is c = 3”, and with s so large such that every such Cohen-Macaulay quotient B satisfies _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 and (K_{B}^{∗})_{s} contains a regular section (cf. Remark 18 and 22). If we take a point (R → B) of U_{s} and two particular quotients B → A_{1} and B → A_{2} (corresponding to two regular sections σ_{i} of (K_{B}^{∗})_{s}), we have I_{A_{1}/B} ≃ I_{A_{2}/B} as graded B-modules. Recalling that the _{0}H^{2}(B, A, A)- group of Lemma 7 is given by _{s}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) (Lemma 15), we can take the open set U of Lemma 7 to contain (B → A_{2}) provided it contains (B → A_{1}). Hence we can let U be q^{−1}(U_{s}) intersected by the space of those quotients (B → A) which correspond to the regular sections of (K_{B}^{∗})_{s} where q : GradAlg(H_{B}, H_{A}) → GradAlg^{H_{B}} is the first projection. Calling this intersection q^{−1}(U_{s})_{reg}, we have an incidence correspondence (cf. the text before (10))

q^{−1}(U_{s})_{reg} ⟶^{q_{res}} U_{s} ⊂ GradAlg^{H_{B}}(R)

↓^{p_{res}}  (18)

GradAlg^{H_{A}}(R)

of restricted projection morphisms in which q_{res} and p_{res} have nice properties. Note that v = n − reg(I_{B}), n = dim B, is the largest integer such that (K_{B})_{v} = 0 by Remark 14(a). Letting n_{j,i} be the number appearing in the minimal resolution (6) of B, we get Proposition 23. With notations as above, we have

i) The morphism q_{res} in (18) is smooth and its fibers are geometrically connected, of fiber dimension dim(K_{B}^{∗})_{s} − 1.

ii) If U_{n_{1i}} is the open subset of U_{s} whose corresponding k-points (B) satisfy max n_{1,i} ≤ n − reg(I_{B}) + s, then p^{−1}(p_{res}(q_{res}^{−1}(U_{n_{1i}}))) = q_{res}^{−1}(U_{n_{1i}}) and p_{res} is unramified and injective on q_{res}^{−1}(U_{n_{1i}}). Moreover if s ≥ 2 reg(I_{B}) − n, then U_{n_{1i}} = U_{s}.

iii) If U_{n_{2i}} is the open subset of U_{s} whose k-points (B) satisfy max n_{2,i} ≤ n − reg(I_{B}) + s, then p^{−1}(p_{res}(q_{res}^{−1}(U_{n_{2i}}))) = q_{res}^{−1}(U_{n_{2i}}) and the restriction of p_{res} to q_{res}^{−1}(U_{n_{2i}}) is an isomorphism onto an open subscheme of GradAlg^{H_{A}}(R). In particular if s ≥ 2 reg(I_{B}) − n + 1, then U_{n_{2i}} = U_{s} and the morphism p_{res} in (18) is an isomorphism onto an open subscheme of GradAlg^{H_{A}}(R).

Proof. i) To see the smoothness of q_{res}, we combine Proposition 4(i) and Lemma 15 (see also the proof of Proposition 13). To show that the fibers of q_{res} are (geometrically) connected, it suffices to see that two quotients B → A_{1} and B → A_{2}, given as in (13) by two regular sections σ_{i} of (K_{B}^{∗})_{s}, correspond to two k-points of a flat irreducible family U′ ⊂ 𝔸^{1}(k) = Spec(k[t]) contained in the fiber q_{res}^{−1}((B)). This is, however, quite trivial because we can easily show that tσ_{1} + (1 − t)σ_{2} is a regular section of (K_{B[t]}^{∗})_{s} for any t in some open set of 𝔸^{1}(k) containing t = 0 and t = 1. ii) It is straightforward to get ii) from Lemma 7(b) because K_{B}(−s)_{max n_{1,i}} = 0 by assumption. iii) Note that p is universally injective (“radiciel”) on the set U of Lemma 7 because the proof of the injectivity of p|_{U} in Lemma 7(b) extends to F -valued points where F is any field extension of k. Hence by [13], Thm. 17.9.1, it suffices to prove that p_{res} is etale on q_{res}^{−1}(U_{n_{2i}}). By (ii) above, it suffices to prove smoothness. Looking to Remark 14(a), we see that _{−s}Ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = 0 by assumption. The first sentence of the proof of Theorem 16 shows that I_{B} is generically syzygetic. Hence we get the smoothness of p_{res} by Proposition 4(ii) and the proof of Theorem 5B.

Recall that a quotient A of R has the Weak Lefschetz property if there is a linear form l and an Artinian reduction A_{L} := A/(L) of A such that the multiplication l : (A_{L})_{v} → (A_{L})_{v+1} is either injective or surjective for every v (cf. the text before (17)). Letting “general member or k-point” mean any member or k-point in a suitable open dense subset, we get

Theorem 24. Let s ≥ 2 reg(I_{B}) − n + 1. Then the correspondence (18) determines a welldefined injective application π from the set of irreducible components W of U_{s} (whose members B are codimension c Cohen-Macaulay quotients of R satisfying the defining conditions of U_{s} above), to the set of irreducible components V of GradAlg^{H_{A}}(R) whose general members are Gorenstein codimension c + 1 quotients A of R satisfying the Weak Lefschetz property. In this application the generically smooth components correspond. Indeed if m_{W} (resp. m_{V}) is the maximal ideal of the local ring of a general k-point of W (resp. of V = π(W)), then dim m_{V} /m_{V}^{2} − dim V = dim m_{W} /m_{W}^{2} − dim W, and we have

dim V = dim W + dim(K_{B}^{∗})_{s} − 1.

Proof. By Proposition 23(i) and [17], Prop. 1.8, q_{res}^{−1}(W) is an irreducible component of q^{−1}(U_{s})_{reg}. The application is therefore well-defined, and it is injective by Proposition 23(iii). Since q_{res} is smooth and p_{res} is an open embedding, we easily get the dimension formulas.

It remains to see that A has the Weak Lefschetz property. Indeed we claim that any A′ which corresponds to a point in the image p_{res}(q^{−1}(U_{s})_{reg}) has the Weak Lefschetz property. First suppose dim B′ = 1, with (B′ → A′) ∈ q^{−1}(U_{s})_{reg}. Since reg(I_{B′} = σ(B′) where σ(B′) is defined by min{i|ΔH_{B′}(i) = 0} (e.g. [35], Sect. 2), it follows that the multiplication by a regular element of B′ of degree one induces a bijection (B′)_{t−1} → (B′)_{t} for t ≥ reg(I_{B′}, hence (A′)_{t−1} → (A′)_{t} is surjective for t ≥ reg(I_{B′}. Let t < reg(I_{B′}. Since the assumption s ≥ 2 reg(I_{B}) − n + 1 leads to K_{B′}(−s)_{v} = 0 for v ≤ reg(I_{B′} + 1, we get (B′)_{v} ≃ (A′)_{v} for v ≤ reg(I_{B′} + 1 and certainly for v < reg(I_{B′}. The latter is precisely what we need to get the injectivity of (A′)_{t−1} → (A′)_{t} from the injectivity of (B′)_{t−1} → (B′)_{t}. Hence A′ has the Weak Lefschetz property. Moreover if n = dim B′ > 1 we know that the graded Betti numbers of B′ and of K_{B′}(−n) are unchanged by modding out by a (n − 1)-dimensional space L′ of general linear forms (cf. the explanation to (17)). Hence we may apply the argument above to the one dimensional quotient B′/(L′), and we are done.

Remark 25. (a) Let dim B = 1, i.e. let Proj(B) be a zero-scheme of degree d and let s ≥ 2 reg(I_{B}). Then we automatically get the vanishing of the Ext^{1}-group appearing in the definition of U_{s} by Remark 22. Since the quotients of U_{s} satisfy K_{B}(−s)_{reg(I_{B})+1} = 0, we get dim B_{reg(I_{B})+1} = dim A_{reg(I_{B})+1}. Hence we see that the Hilbert function of each A in Theorem 24 is an unimodal SI-sequence with at least three consecutive peaks, that is, H_{A} ⊃ (d, d, d) (because A has the Weak Lefschetz property, see e.g. [35]). Moreover

dim V = dim W + d − 1.

Indeed we really have by Proposition 23(iii) much more, namely a smooth morphism q_{res}∘p_{res}^{−1} of schemes defined on p_{res}(q^{−1}(U_{s})_{reg}) with values in GradAlg^{H_{B}}(R), allowing us to compare properties of GradAlg^{H_{B}}(R) at (B) with the corresponding properties of GradAlg^{H_{A}}(R) at (A) beyond that done in Theorem 24. Note that this correspondence is well studiet by Iarrobino-Kanev if the codimension of B is c = 2 ([21], Sect. 5.3.2), in which case they get a topological fibration defined on the whole parameter space PGor(H_{A}). In that case they also get the dimension formula above ([21], Cor. 5.50). Since the points of p_{res}(q^{−1}(U_{s})_{reg}) ⊂ PGor(H_{A}) satisfy the Weak Lefschetz property, we can not in general extend the morphism q_{res} ∘ p_{res}^{−1} to the whole of PGor(H_{A}) (cf.[3])!

(b) Let dim B = 1 and suppose s ≥ 2 reg(I_{B}) − 1. Using the arguments of (a) it follows that the h-vector of A is an SI-sequence with at least two peaks. By Proposition 23(ii) the correspondence (18) applies provided the points (B) of U_{s} (or of U_{n_{1i}} if we only assume max n_{1,i} ≤ 1 − reg(I_{B}) + s)) satisfy the assumptions of Proposition 13A. These assumptions are satisfied if B is licci (Remark 14(c)), \underline{o}r if I_{B} is generically syzygetic (e.g. if c ≤ 3) and satisfies

(H_{2}(R, B, B) ⊗_{B} A)_{s} = 0 and _{0}H_{𝔪}^{0}(I_{B} ⊗_{R} K_{B}) = 0

(Remark 14(b)) because the latter vanishing is equivalent to _{0}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 by Gorenstein duality. Hence we get by Proposition 13A and Theorem 11 that to each irreducible component W of say U_{s} whose general member is licci \underline{o}r as above, then (18) determines an irreducible closed subset V of codimension _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) in PGor(H_{A}) such that PGor(H_{A}) is generically smooth along V, and dim V = dim W + d − 1. If c = 2 this result is known ([21], Sect. 5.3.3 and Cor. 5.50).

Example 26. Here we will use Theorem 24 and Proposition 8 to give classes of examples in which the scheme GradAlg^{H}(R) has several irreducible components. To simplify the analysis, we will use the classification of ACM (arithmetically Cohen-Macaulay) curves on a general rational ACM surface appearing in [29]. So our examples will yield irreducible components of GradAlg^{H}(R) of Gorenstein codimension 4 quotients of dimension one. It is not obvious that the corresponding Artinian reductions lead to irreducible components of PGor(H) of Artinian Gorenstein codimension 4 quotients.

First we explicitly describe three irreducible components of Hilb^{p}(ℙ^{4}) with the same Hilbert function (in fact their graded Betti numbers are the same, and are the minimum possible consistent with the Hilbert function). Recalling that a general Castelnuovo surface F ⊂ ℙ^{4} is obtained by blowing up 8 general points in ℙ^{2}, thus that Pic(F) ≃ ℤ^{9}, we write the hyperplane section and the canonical divisor as H = (4; 2, 1^{7}) and K = (−3; −1^{8}) respectively. Consider the three families of curves, C ∈|C_{0} + tH|, of [29], p.73 given by

(4t + 1; 2t, t^{7}), (4t + 2; 2t, t + 1^{4}, t^{3}), (4t + 3; 2t, t + 2, t + 1^{6})

having C_{0} · K = (−3, −2, −1) respectively. We always have (d_{0}, g_{0}) := (deg(C_{0}), gen(C_{0})) = (4, 0) and H^{2} = 5. By Proposition 8 any such C is unobstructed provided t ≥ 3 and

dim_{(C)} Hilb^{p}(ℙ^{4}) = 31 − C_{0} · K + (11t + 5t^{2})/2 ,

where p is the Hilbert polynomial of C. Since the three families have different C_{0} · K, they must correspond to three different components of Hilb^{p}(ℙ^{4}).

We claim that all curves C = Proj(B) of the three families have the same Hilbert function, and hence that their quotients B belong to the same GradAlg^{H}(R). Indeed looking to the sequence

0 → J_{C/F} ≃ O_{F} (−C) → O_{F} → O_{C} → 0,

and the corresponding one for C_{0}, we get h^{1}(O_{C}(v)) = h^{1}(O_{C_{0}}(v −t)) = 0 for v ≥ t. It is well known that the homogeneous ideal of F has three minimal generators of degrees 2, 3, 3 and two relations of degrees 4, 4. Combining h^{0}(J_{C/F} (v)) = h^{0}(J_{C_{0}/F} (v − t)) and s(C_{0}/F) = 2 with h^{1}(O_{C}(t)) = 0, we see that the additional minimal generators of I_{B} sit all in degree t+2. Hence we get that all C of the three families have the same Hilbert function and the same regularity, reg(I_{B}) = t + 2, i.e. we get the claim. Indeed the members of the three families satisfy H_{B}(v) = h^{0}(O_{C}(v)) = dv + 1 − g for v ≥ t and H_{B}(v) = h^{0}(O_{F} (v)) for v < t, and we easily find the minimal resolution to be

0 → R(−t − 4)^{5} → R(−t − 3)^{10} ⊕ R(−4)^{2} → R(−t − 2)^{5} ⊕ R(−3)^{2} ⊕ R(−2) → I_{B} → 0.

Now, thanks to (9), the corresponding Cohen-Macaulay algebras B are unobstructed as graded algebras and dim_{(B)} GradAlg^{H}(R) = dim_{C} Hilb^{p}(ℙ^{4}). The three families must therefore correspond to three generically smooth irreducible components of GradAlg^{H_{B}}(R) of different dimensions whose general members B are two-dimensional Cohen-Macaulay codimension 3 quotients of R.

By Theorem 24, we get three generically smooth irreducible components V of GradAlg^{H_{A}}(R) whose general members are one-dimensional Gorenstein codimension 4 quotients A of R. Indeed Theorem 24 applies for s > 2 reg(I_{B}) − dim B = 2t + 2 provided we can show sExt_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 for such s. Looking to the arguments of Remark 18 it suffices to prove s > (4g − 4)/d which is straightforward since (d_{0}, g_{0}) = (4, 0), d = d_{0} + 5t and 2g = 2g_{0} + 2td_{0} + 5t^{2} − 3t by the adjunction formula 2g − 2 = C · (C + K). Now, inserting these values for d and g into the expression dim(K_{B}^{∗})_{s} = ds + 3 − 3g of Remark 18, we get, for every t ≥ 3 and every s > 2t + 2, three irreducible components V of GradAlg^{H_{A}}(R) satisfying

dim V = dim_{C} Hilb^{p}(ℙ^{4}) + dim(K_{B}^{∗})_{s} − 1 = 33 − 2t − 5t^{2} − C_{0} · K + s(4 + 5t).

It is not difficult to compute the Hilbert function H_{A} from the Hilbert function H_{B}. Indeed the h-vector of A, i.e. the h-vector of an Artinian reduction of A, is just

(1, 4, 9, 14, 19, ..., 4 + 5t, 4 + 5t, 4 + 5t, ..., 19, 14, 9, 4, 1).

The simplest case t = 3 and s = 9 yields the h-vector (1, 4, 9, 14, 19, 19, 19, 14, 9, 4, 1).

Remark 27. We can use the classification of ACM curves on a general rational ACM surface in ℙ^{4} in [29] to get many more reducible schemes GradAlg^{H_{A}}(R) whose general members are Gorenstein codimension 4 algebras. For instance on a Castelnuovo surface, look to the linear systems, C ∈ |C_{0} + tH|, of [29], p.73 where C_{0} is a minimal curve and t ≥ 0. For both (d_{0}, g_{0}) = (5, 1) and (6, 2) there are at least two linear systems with different C_{0} · K. Here s(C_{0}/F) = 2 and Proposition 8 applies for t ≥ 3. Since H^{1}(O_{C_{0}}(1)) = 0 one may see that reg(I_{B}) = t + 3 (C = Proj(B)) and that Theorem 24 applies for every s > 2t + 4. The h-vectors of the simplest case t = 3 and s = 11 of the two families are

(1, 4, 9, 14, 19, 20, 20, 20, 19, 14, 9, 4, 1) and (1, 4, 9, 14, 19, 21, 21, 21, 19, 14, 9, 4, 1).

For both (d_{0}, g_{0}) = (2, 0) and (3, 0) there are also two linear systems with different C_{0} · K. Now s(C_{0}/F) = 1 and H^{1}(O_{C_{0}}) = 0 and Proposition 8 applies for t ≥ 4 and Theorem 24 for s > 2 reg(I_{B}) − 2 = 2t + 2. The h-vectors of the simplest case t = 4 and s = 11 are

(1, 4, 9, 14, 19, 22, 22, 22, 19, 14, 9, 4, 1) and (1, 4, 9, 14, 19, 23, 23, 23, 19, 14, 9, 4, 1).

Putting the analysis on a Castelnuovo surface together we get that GradAlg^{H_{A}}(R) has at least two generically smooth irreducible components whose general members are one-dimensional Gorenstein codimension 4 quotients A of R in the following range of the h-vector;

(1, 4, 9, 14, ..., 5α − 1, β, β, ..., β, 5α − 1, 5α − 6, ..., 9, 4, 1)

where 5α−1 ≤ β ≤ 5α+4, α ≥ 3, β ≥ 19, and with at least three consecutive β′s (three peaks).

Of course the part (1, 4, 9, 14, ..., 5α − 1) of the h-vector comes from the Castelnuovo surface. On a cubic surface there are two linear systems with (d_{0}, g_{0}) = (1, 0) and with different C_{0} · K, giving rise to reducible GradAlg^{H_{A}}(R) for every t ≥ 4 and s > 2t. For (t, s) = (3, 7) the h-vector is (1, 4, 7, 10, 10, 10, 7, 4, 1). Increasing t to 4 and s to 10 we get precisely the hvector of Boij’s example [3], which is different from ours because our algebras have the Weak Lefschetz property (see also [22], Thm. 3.9). There are also quite a lot of linear systems on a Bordiga surface with which we could make a similar analysis. All examples we get in this case have a symmetric h-vector starting with (1, 4, 10, 16, 22, ...).

We end this section by using Theorem 16 to essentially reprove the smoothness and give a formula for the dimension of GradAlg^{H}(R) for (e.g. Artinian) Gorenstein codimension 3 quotients equivalent to those in [27], [28] or [6].

Example 28. Recall that if A ≃ R/I_{A} is Gorenstein of codimension 3 with minimal resolution

0 → R(−f) ≃ F_{3} → F_{2} → F_{1} → I_{A} → 0.  (19)

one knows that A is unobstructed and that N_{A} ≃ ∧^{2}I_{A}(f) (char(k) ≠ 2). It follows that

dim_{(A)} GradAlg^{H_{A}}(R) = dim(∧^{2}F_{1})_{f} − dim(F_{1} ⊗ F_{2})_{f} + dim(S_{2}(F_{2}))_{f}  (20)

cf. [28], Thm. 2.6 and its Remark. In (20) it is not really necessary to suppose (19) to be minimal because we easily see that (20) is invariant under adding a common free factor to both F_{1} and F_{2}.

If B ≃ R/I_{B} is Cohen-Macaulay of codimension 2 (and hence licci), and A is given as in Theorem 16, then we claim the dimension formula for dim_{(A)} GradAlg^{H_{A}}(R) in Theorem 16 coincides with (20). Note that since every irreducible component of the Hilbert scheme of Gorenstein codimension 3 quotients of R contains a quotient constructed via (13) by [10], this example proves the generic smoothness of GradAlg^{H}(R) and PGor(H), as well as (20). The argument holds in the case char(k) = 2 as well.

First we compute dim(K_{B}^{∗})_{s}. Indeed the dual of (8) (for c = 2) yields the exact sequence

0 → K_{B}^{∗}(−n − c) → G_{2} ⊗ B → G_{1} ⊗ B → I_{B}/I_{B}^{2} → 0  (21)

in which H_{1} is the kernel of G_{1} ⊗ B → I_{B}/I_{B}^{2} by (4). By [2] one knows that

0 → ∧^{i+1}G_{2} → ∧^{i+1}G_{1} → ∧^{i}G_{2} → H_{i} → 0  (22)

is exact for any i. Let f = s + n + c. Using the last mentioned sequence for i = 1, (21) and the R-free resolution of G_{2} ⊗ B which we easily deduce from (6), we get

dim(K_{B}^{∗})_{s} = dim(∧^{2}G_{1})_{f} − dim(G_{1} ⊗ G_{2})_{f} + dim(S_{2}(G_{2}))_{f}.  (23)

Next we will compute δ = δ(B)_{−s} = dim(G_{1}^{∗} ⊗_{R} K_{B})_{−s} − _{−s}hom_{B}(H_{1}, K_{B}), cf. Remark 14(d). If r = rank H_{1}, we have by (22) a resolution of H_{r−1} leading to

0 → G_{2}^{∗} → (∧^{2}G_{1})^{∗} → (∧^{2}G_{2})^{∗} → H_{r−1} ⊗(∧^{r+1}G_{2})^{∗} ≃ Hom(H_{1}, K_{B}(n + c)) → 0 (24)

We have an R-free resolution of K_{B} by dualizing (6) and hence a resolution of G_{1}^{∗} ⊗ K_{B}. Combining with (24), we get

−δ = dim(G_{1} ⊗ G_{2})_{−f}^{∗} − dim(S_{2}(G_{1})^{∗})_{−f} − dim(∧^{2}G_{2})_{−f}^{∗} + dim(G_{1})_{−f}^{∗} − dim(G_{2})_{−f}^{∗}. (25)

Note that dim(N_{B})_{0} is well known, e.g. one may easily deduce it from the exact sequence

0 → G_{1}^{∗} ⊗ G_{2} → ((G_{1}^{∗} ⊗ G_{1}) ⊕ (G_{2}^{∗} ⊗ G_{2}))/k → G_{2}^{∗} ⊗ G_{1} → N_{B} → 0  (26)

of [30], Remark 16 (k is a 1-dimensional subspace). Putting these formulas together, using Theorem 16, we get exactly (20) because we find the following resolution of I_{A},

0 → R(−f) → G_{1}^{∗}(−f) ⊕ G_{2} → G_{2}^{∗}(−f) ⊕ G_{1} → I_{A} → 0.  (27)

by applying the mapping cone construction onto (13).

Inside the scheme Z(H) of Remark 10 we have the open subscheme ZGor(H) consisting of not necessarily graded Gorenstein quotients R → A with Hilbert function H. ZGor(H) has been studied in different contexts (see [21], p.126 for references). Since we by Remark 14(c) easily see that _{v}H^{2}(R, A, A) = 0 for v ≥ 0 provided B is licci, we can use Remark 10 to prove the smoothness and find the dimension of the scheme ZGor(H) at (A). Indeed _{v}hom_{R}(I_{A}, A) is found using (10) as in the proof of Theorem 5. And if we apply _{v}Hom(K_{B}(−s), −) to (13), we get _{v}hom(K_{B}(−s), A) = dim(K_{B}^{∗})_{s+v} − dim B_{v}. Since A_{v} = 0 for v > s, we have

Proposition 29. Let B ≃ R/I_{B} be licci and generically Gorenstein, and let A be a graded codimension one quotient of B, defined by (13) for some s. Then ZGor(H) is smooth at (R → A) and

dim_{(A)} Z(H) = ∑_{v≥0}^{s} (dim(N_{B})_{v} − δ(B)_{−s+v}) + ∑_{v≥0}^{s} (dim(K_{B}^{∗})_{s+v} − dim B_{v}).

Invoking Remark 14(c), we take the opportunity to state the main result of this section in the non graded case. The proof is omitted because it is the same as in the graded case.

Theorem 30. Let B ≃ R/I_{B} be Cohen-Macaulay, generically Gorenstein and generically syzygetic of dimension ≥ 1 and suppose B satisfies

Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 , Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 and Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = 0.

Consider K_{B} as an ideal of B and let A := B/K_{B}. Then A is Gorenstein and strongly unobstructed in the sense H^{2}(R, A, A) = 0.

In particular let B ≃ R/I_{B} be licci and generically Gorenstein. Then A is Gorenstein and strongly unobstructed in the sense H^{2}(R, A, A) = 0. Indeed licci and generically Gorenstein imply Cohen-Macaulay and generically syzygetic as well as the vanishing of the three Extgroups above.

3 Computing the dimension via liaison

In this section we show how we can compute δ(B)_{−s} = _{−s}hom_{R}(I_{B}, K_{B})− _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) and dim(K_{B}^{∗})_{s}. For instance it turns out that the difference dim(K_{B}^{∗})_{s}−δ(B)_{−s} (which appears in Theorem 16) is quite easy to compute because, after adding a simple term to it, it becomes a liaison invariant among syzygetic ideals I_{B}. Moreover δ(B)_{−s} is computable in terms of dim(K_{B′}^{∗})_{s} of the linked algebra B′. To allow any complete intersection linkage of Cohen-Macaulay quotients and still get the liaison invariance described above, we need to define δ(B) also for Cohen-Macaulay quotients R → B which satisfy H_{2}(R, B, B) ≠ 0 generically. To do this we consider the exact sequence ([38], Sect. 2.1)

0 → H_{2}(R, B, B) → S_{2}I_{B} → I_{B} → I_{B}/I_{B}^{2} → 0.  (28)

Let ψ : H_{𝔪}^{n}(H_{2}(R, B, B)) → H_{𝔪}^{n}(S_{2}I_{B}), dim B = n, be the induced map.

Definition 31. Let B = R/I_{B} be Cohen-Macaulay of dimension n, not necessarily generically syzygetic. Then

δ^{∗}(B)_{v} = _{v}hom_{R}(I_{B}, K_{B}) − _{v}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) − dim(im ψ)_{−v}

Remark 32. If B is generically syzygetic, then _{v}Hom_{B}(H_{2}(R, B, B), K_{B}) = 0, and we get −vH_{𝔪}^{n}(H_{2}(R, B, B)) = 0 by duality and hence δ^{∗}(B)_{v} = δ(B)_{v}, i.e. the “old and the new” definition of δ coincides.

Using (28) and_{v}Ext_{B}^{i}(I_{B}/I_{B}^{2}, K_{B}) ≃ _{−v}H_{𝔪}^{n−i}(I_{B}/I_{B}^{2})^{∨} and letting h_{𝔪}^{j}(−)_{v} = dim(_{v}H_{𝔪}^{j}(−)), one shows

δ^{∗}(B)_{v} = h_{𝔪}^{n+1}(S_{2}I_{B})_{−v} − h_{𝔪}^{n}(S_{2}I_{B})_{−v} − h_{𝔪}^{n+1}(I_{B})_{−v}  (29)

Indeed splitting (28) into two short exact sequences we see that h_{𝔪}^{n−1}(I_{B}/I_{B}^{2})_{−v} = h_{𝔪}^{n}(I_{B}^{2})_{−v} and h_{𝔪}^{n}(I_{B}/I_{B}^{2})_{−v} = h_{𝔪}^{n+1}(I_{B}^{2})_{−v} − h_{𝔪}^{n+1}(I_{B})_{−v}. Moreover h_{𝔪}^{n+1}(I_{B}^{2})_{−v} = h_{𝔪}^{n+1}(S_{2}I_{B})_{−v} and h_{𝔪}^{n}(I_{B}^{2})_{−v} = h_{𝔪}^{n}(S_{2}I_{B})_{−v} − dim(im ψ)_{−v}; thus we get (29) from Definition 31.

Proposition 33. Let R be a finitely generated polynomial k-algebra with canonical module R(e) (so e = −n − c), and let B and B′ be graded Cohen-Macaulay quotients, algebraically linked by a complete intersection R ↠ D of dimension n ≥ 1, and with minimal resolution

0 → R(−a) → ... → ⊕_{i=1}^{c}R(−c_{i}) → I_{D} → 0.

Let β(B, D)_{v} := dim B_{v} + ∑_{i=1}^{c} dim(I_{B/D})_{v+c_{i}}. Then I_{B/D} ≃ K_{B′}(−e − a) and

(i) dim(N_{B})_{0} − ∑_{i=1}^{c} dim B_{c_{i}} = dim(N_{B′})_{0} − ∑_{i=1}^{c} dim B′_{c_{i}}

(ii) If B and B′ are generically Gorenstein (resp. Gorenstein in codimension ≤ 1), then vExt_{B}^{i}(I_{B}/I_{B}^{2}, B) and _{v}Ext_{B′}^{i}(I_{B′}/I_{B′}^{2}, B′) are isomorphic for i = 1 (resp. i = 2).

(iii) δ^{∗}(B)_{v−2e−2a} = β(B′, D)_{v−e−a} − dim(K_{B′}^{∗})_{v} for any v. Moreover if β(B′, D)_{v−e−a} = 0 and I_{B} is generically syzygetic, then _{v−2e−2a}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = dim(K_{B′}^{∗})_{v}.

(iv) dim(K_{B}^{∗})_{v} − δ^{∗}(B)_{−v} − β(B, D)_{v−e−a} = dim(K_{B′}^{∗})_{−v+2e+2a} − δ^{∗}(B′)_{v−2e−2a} − β(B′, D)_{−v+e+a}.

(v) If I_{B} is generically syzygetic (resp. syzygetic in codimension ≤ 1), then vExt_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ↪ _{v+2e+2a}Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) is injective (resp. an isomorphism).

(vi) The conclusion of (ii) holds if B is linked to B′ in possibly several steps. Here the algebras of the intermediate steps need not necessarily be generically Gorenstein.

Proof. (i) is proved in [27], Prop. 1.7. (ii) Also this part, i.e. the liaison invariance of I_{B}/I_{B}^{2} ⊗ K_{B}, is known ([5], or [29], Ch. 6). Here the Gorenstein assumption is required for having _{v}Ext_{B}^{i}(I_{B}/I_{B}^{2}, B) ≃ _{v}Ext_{B}^{i}(I_{B}/I_{B}^{2} ⊗ K_{B}, K_{B}). Note that the liaison invariance of I_{B}/I_{B}^{2} ⊗ K_{B} takes care of (vi) as well.

To prove (iii), recall that K_{D} ≃ D(e+a) and hence, I_{B/D} ≃ Hom_{D}(B, D) ≃ K_{B′}(−e−a), i.e. we get exact sequences

0 → I_{D} · I_{B} → S_{2}I_{B} → S_{2}(K_{B′})(−2e − 2a) → 0  (30)

0 → I_{D} · I_{B} → I_{D} → I_{D} ⊗_{R} R/I_{B} ≃ I_{D}/I_{D}^{2} ⊗_{D} B ≃ ⊕_{i=1}^{c}B(−c_{i}) → 0,  (31)

cf. [38], Lemma 4.2.7 to see (30). Since β(B′, D)_{v} := dim B′_{v} + ∑_{i=1}^{c} dim(I_{B′/D})_{v+c_{i}} and I_{B′/D} ≃ K_{B}(−e − a), we get β(B′, D)_{v} = h_{𝔪}^{n}(K_{B′})_{−v} + ∑_{i} h_{𝔪}^{n}(B)_{e−v+a−c_{i}} by Gorenstein duality. Using (29) and (30), we have dim(K_{B′}^{∗})_{v} = h_{𝔪}^{n}(S_{2}K_{B′})_{−v} = −δ^{∗}(B)_{−2e+v−2a} − h_{𝔪}^{n+1}(I_{B})_{2e−v+2a} + h_{𝔪}^{n+1}(I_{D}I_{B})_{2e−v+2a} while (31) and h_{𝔪}^{n}(K_{B′})_{−v+e+a} = h_{𝔪}^{n+1}(I_{D})_{2e−v+2a} − h_{𝔪}^{n+1}(I_{B})_{2e−v+2a} show −h_{𝔪}^{n+1}(I_{B})_{2e−v+2a} + h_{𝔪}^{n+1}(I_{D}I_{B})_{2e−v+2a} = β(B′, D)_{v−e−a}. Hence we have proved

dim(K_{B′}^{∗})_{v} = −δ^{∗}(B)_{−2e+v−2a} + β(B′, D)_{v−e−a}  (32)

Moreover if β(B′, D)_{v−e−a} = 0, then dim B′_{v−e−a} = 0, i.e. v − e − a < 0 and hence H_{𝔪}^{n+1}(I_{D})_{2e−v+2a} = 0 by duality (e.g. if dim R > n+1, then h_{𝔪}^{n+1}(I_{D})_{2e−v+2a} = h_{𝔪}^{n}(D)_{2e−v+2a} = h_{𝔪}^{n}(K_{D})_{−v+e+a} = 0 by duality). Since h_{𝔪}^{n}(K_{B′})_{−v+e+a} = 0, it follows that h_{𝔪}^{n+1}(I_{D}I_{B})_{2e−v+2a} = β(B′, D)_{v−e−a} = 0 which easily leads to the final conclusion of (iii).

Note that (32) also shows dim(K_{B}^{∗})_{v} = −δ^{∗}(B′)_{v−2e−2a} + β(B, D)_{v−e−a}. Moreover if we replace v by −v + 2e + 2a in (32), we get (iv) by combining these formulas.

To prove (v), we use Gorenstein duality. Indeed

vExt_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ≃ _{−v}H_{𝔪}^{n−2}(I_{B}/I_{B}^{2})^{∨} ≃ _{−v}H_{𝔪}^{n−1}(I_{B}^{2})^{∨} ↪ _{−v}H_{𝔪}^{n−1}(S_{2}I_{B})^{∨},

where the final injection follows from Gorenstein duality and _{v}Hom_{B}(H_{2}(R, B, B), K_{B}) = 0 because I_{B} is generically syzygetic. Note that the injection is an isomorphism if I_{B} is syzygetic in codimension ≤ 1. Now combining (30) and (31) we get

−vH_{𝔪}^{n−1}(S_{2}I_{B})^{∨} ≃ _{−v}H_{𝔪}^{n−1}(S_{2}(K_{B′})(−2e − 2a))^{∨} ≃ _{v+2e+2a}Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) (33)

and we conclude easily.

Remark 34. With notations as in Proposition 33, allowing also n = dim B ≥ 0, we have

dim(N_{B})_{v} − ∑_{i=1}^{c} dim B_{v+c_{i}} = dim(N_{B′})_{v} − ∑_{i=1}^{c} dim B′_{v+c_{i}} for every integer v .

This follows from [27], Prop. 1.7 for v = 0, and from [26], Cor. 2.12 and (2.19) or [29], Prop. 9.20 for n ≥ 2 and every v. Since the arguments of the proof of [27], Prop. 1.7 hold for the linkage appearing in the non-graded deformation functors, we may use the proof to get the displayed dimension formula by showing that the isomorphism of their tangent spaces, A_{D→B}^{1} ≃ A_{D→B′}^{1} (whose degree zero parts are given by (10)), is graded. This isomorphism is, however, concretely studied in [29], pp. 38-39, without using deformations. There, if we take the diagrams (6.11) and (6.12) without sheafification, we get a graded isomorphism A_{D→B}^{1} ≃ A_{D→B′}^{1} from which we get the displayed formula by e.g. the proof of [29], Prop.9.20.

Example 35. We now complete Example 17 for the remaining cases (0 ≤ s ≤ 2) by verifying the assumptions of Proposition 13A, using liaison. Since we in Example 17 proved 0Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0, it suffices by Remark 6(c) to show _{0}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}(−s)) = 0. Now if we link B to a B′ via a complete intersection of three hypersurfaces, all of degree 2, the linked curve C′ = Proj(B′) has degree d′ = 8 − 4 = 4 and genus g′ = 0, i.e. we see that C and C′ are essentially the same curve. Hence _{v}Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) = 0 for v ≥ 0 since B satisfies this vanishing. By Proposition 33 it follows that _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) ↪ −s+2Ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) = 0 for s ≤ 2, which precisely covers the cases of Example 17 which are not already analyzed. So, for 0 ≤ s ≤ 2, Proposition 13A applies! In Example 17 we saw that dim(K_{B}^{∗})_{s} = 4s + 3 for any s ≥ 0. It remains to compute δ(B)_{−s}. Since δ(B)_{v−2} = β(B′, D)_{v−1} − dim(K_{B′}^{∗})_{v} and β(B′, D)_{v} = dim B′_{v} + 3 dim(I_{B′/D})_{v+2} by Proposition 33, we get (δ(B)_{0}, δ(B)_{−1}, δ(B)_{−2}) to be equal to (15, 3, −3), and as previously

dim_{(A)} GradAlg^{H}(R) = h^{0}(N_{C}) + h^{0}(θ_{C}(s)) − 1 − δ(B)_{−s} = 4s + 23 − δ(B)_{−s}

Hence dim_{(A)} GradAlg^{H}(R) equals (8, 24, 34) for s = (0, 1, 2) respectively. Note that it is easy, for s = 2, to see that (K_{B})_{0} = 0 and hence that _{−2}Hom(I_{B}/I_{B}^{2}, K_{B}) = 0 and −2ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) = −δ(B)_{−2} = 3 by looking to the minimal resolution of I_{B} (or use Proposition 33(iii)). By the interpretation of _{−s}ext_{B}^{1}(I_{B}/I_{B}^{2}, K_{B}) of Proposition 13A we get that the codimension of the Gorenstein quotients given by (13) is 3.

Example 36. Now we consider the Artinian reduction A_{L} of the Example 21 of socle degree s + 1, to treat the remaining cases of interest (0 ≤ s ≤ 2). Since s ≤ 2 and I_{B} has no linear generator, _{s+1}H_{2}(R_{L}, B_{L}, B_{L}) ↪ (S_{2}I_{B_{L}})_{s+1} = 0. Looking to Example 21 and Remark 14(b) we see that all assumptions of Proposition 13A are satisfied. Now examining H^{i}(θ_{C}(v)) of Example 17 for v = −1 a little further and invoking (S_{2}K_{B})_{1} = 0 we get −1ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 1 in addition to _{v}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) = 0 for v ≥ 0. By Lemma 19(i) we get dim(K_{B_{L}}^{∗})_{v} = 4 for v ≥ 1. Moreover if we apply Hom_{B}(I_{B}/I_{B}^{2}, −) to the exact sequence 0 → K_{B} → K_{B}(1) → K_{B_{L}} → 0, we get

δ(B_{L})_{v−1} = δ(B)_{v} − δ(B)_{v−1} + _{v}ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) − _{v−1}ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}).

Since _{−3}ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = _{−1}ext_{B′}^{1}(S_{2}(K_{B′}), K_{B′}) = 1 and _{−s}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) = 0 for s ≤ 2, we get that (δ(B_{L})_{−1}, δ(B_{L})_{−2}, δ(B_{L})_{−3}) is equal to (12, 6, −4) from the values of δ(B)_{v} in Example 35. It follows that A_{L} is an unobstructed Artinian Gorenstein algebra for 0 ≤ s ≤ 2, i.e. PGor(H_{L}) is smooth at (A_{L}) by Theorem 11 and

dim_{(A_{L})} PGor(H_{L}) = dim(N_{B_{L}})_{0} + dim(K_{B_{L}}^{∗})_{s+1} − 1 − δ(B_{L})_{−s−1} = 15 − δ(B_{L})_{−s−1} .

Moreover, for s = 2, we see that −δ(B_{L})_{−3} = _{−3}ext_{B_{L}}^{1}(I_{B_{L}}/I_{B_{L}}^{2}, K_{B_{L}}) = 4 is the codimension of the stratum of Gorenstein quotients given by (13) around (A_{L}).

Finally we remark that the conclusions (about A_{L}) also hold for any Artinian Gorenstein quotient A, well-defined by a twist of the canonical module of B, with the same graded Betti numbers as A_{L}. In this case we can compute the numbers δ(B)_{v} of a one-dimensional B as we did in Example 35 by essentially the same linkage argument.

Mainly using Proposition 33 twice (for (i) below, we utilize (33)), we get

Corollary 37. Let B and B′ be Cohen-Macaulay quotients of R, algebraically linked by a complete intersection D of type (c_{1}, c_{2}, .., c_{c}), let B′ and B′′ be algebraically linked by a complete intersection D′ of type (c′_{1}, c′_{2}, .., c′_{c}), and let a = ∑ c_{i}, a′ = ∑ c′_{i} and dim B > 0. Then

(i) _{v+2a′}Ext_{B′′}^{1}(S_{2}(K_{B′′}), K_{B′′}) ≃ _{v+2a}Ext_{B}^{1}(S_{2}(K_{B}), K_{B}) for any v.

(ii) _{v−2a′}Ext_{B′′}^{2}(I_{B′′}/I_{B′′}^{2}, K_{B′′}) ≃ _{v−2a}Ext_{B}^{2}(I_{B}/I_{B}^{2}, K_{B}) provided I_{B} and I_{B′′} are syzygetic in codimension ≤ 1.

(iii) δ^{∗}(B′′)_{v−2e−2a′} = δ^{∗}(B)_{v−2e−2a} − β(B′, D)_{v−e−a} + β(B′, D′)_{v−e−a′}.

(iv) dim(K_{B′′}^{∗})_{v+2e+2a′} = dim(K_{B}^{∗})_{v+2e+2a} −β(B′, D)_{v+e+a} +β(B′′, D′)_{v+e+a′} or equivalently, dim(K_{B′′}^{∗})_{v+2e+2a′} = dim(K_{B}^{∗})_{v+2e+2a}+dim B′′_{v+e+a′} −dim B_{v+e+a}+∑_{i=1}^{c}(dim(K_{B′})_{v+c′_{i}} − dim(K_{B′})_{v+c_{i}}).

We include one example of biliaison starting with the smooth rational curve of Example 17. Of course starting with a licci curve, it is immediate to verify the assumptions of Proposition 13 (see Theorem 16). However, some part of the argument below is still needed to find dim(K_{B}^{∗})_{s} − δ^{∗}(B)_{−s} and to see how small we can let the number s in Theorem 16B be.

Example 38. If we link the homogeneous coordinate ring B of the smooth rational quartic curve of Example 17 having minimal resolution

0 → R(−4)^{3} → R(−3)^{8} → R(−2)^{6} → R → B → 0,

to a B′ using a complete intersection of type (2, 2, 3) and further link B′ to a B′′ via a complete intersection of type (2, 3, 4) (using the same hypersurface of degree 2 and 3 as in the first link) one may, notably due to I_{B′/D} ≃ K_{B}(−2) and the mapping cone construction used twice, see that the minimal resolution of B′′ is

0 → R(−7) ⊕ R(−6)^{3} → R(−5)^{10} → R(−4)^{5} ⊕ R(−3) ⊕ R(−2) → R → B′′ → 0.

If A is defined by a regular section of (K_{B′′}^{∗})_{s}, we can easily find the minimal resolution of A and its Hilbert function by using (13). For instance dim A_{v} = 16v − 19 for 3 ≤ v ≤ s − 3 and dim A_{v} = 16s − 38 for v > s. The h-vector of A is in fact

(1, 4, 8, 8, 4, 1), (1, 4, 9, 14, 9, 4, 1), (1, 4, 9, 15, 15, 9, 4, 1), (1, 4, 9, 15, 16, 15, 9, 4, 1)

for s = 4, 5, 6, 7 and (1, 4, 9, 15, 16, ..., 16, 15, 9, 4, 1) for s ≥ 8 respectively. Let us suppose s > 3 since otherwise Proj(A) is degenerate. To use Proposition 13, we may argue as in Example 17 to see that δ(B′′)_{−s} = 0 for s > 7 by Remark 14(a), and that H^{1}(θ_{C′′}(v)) = 0 and hence dim(K_{B′′}^{∗})_{v} = h^{0}(θ_{C′′}(v)) = 16v − 57 for v ≥ 5 because C′′ := Proj(B′′) is smooth curve of degree d′′ = 16 and genus g′′ = 20. Corollary 37 leads, however, to more precise information. Indeed it is rather straightforward to combine the Corollary and the computations of Example 17 and Example 35 to see that the conclusions of Theorem 16 hold for s ≥ 4, that dim(K_{B′′}^{∗})_{4} = 8, and that (δ(B′′)_{−4}, δ(B′′)_{−5}, δ(B′′)_{−6}) is equal to (7, −6, −6) and δ(B′′)_{−s} = 0 for s > 6. Finally patiently using Proposition 33(i) several times, one shows dim(N_{B′′})_{0} = 62 (It is, however, faster to show _{0}Ext_{B′′}^{i}(I_{B′′}/I_{B′′}^{2}, B′′) = 0 for i = 1 and 2 by Proposition 33(ii), using the computations of Example 17 and (15), and then use (15) once more to see h^{1}(N_{C′′}) = 1 and hence dim(N_{B′′})_{0} = 5d′′ +1−g′′ +h^{1}(N_{C′′}) = 62). Putting this together we have by Proposition 13 that A is an unobstructed Gorenstein algebra for s ≥ 4 and

dim_{(A)} GradAlg^{H}(R) = dim(N_{B′′})_{0} + dim(K_{B′′}^{∗})_{s} − 1 − δ(B′′)_{−s} = 16s + 4 for s ≥ 7, dim_{(A)} GradAlg^{H}(R) = 16s + 10 for 5 ≤ s ≤ 6 and dim_{(A)} GradAlg^{H}(R) = 62 for s = 4. By the interpretation of _{−s}ext_{B′′}^{1}(I_{B′′}/I_{B′′}^{2}, K_{B′′}) of Proposition 13, we get by Proposition 33(iii) that the codimension of the stratum of Gorenstein quotients given by (13) is 6 for 5 ≤ s ≤ 6.

We may treat the Artinian reduction of A ≃ B′′/K_{B′′}(−s) in Example 38 satisfactorily by remarking that _{s+1}H_{2}(R_{L}, B′′_{L}, B′′_{L}) = 0 for 4 ≤ s ≤ 6 (since one may show (S_{2}I_{B′′_{L}})_{s+1} ≃ (I_{B′′_{L}}^{2})_{s+1} by using that the generators of I_{B′′_{L}} of degree 2 and 3 form a regular sequence), and that B′′_{L} is unobstructed by Lemma 19(iii) for any s > 3. For the latter, we use the liaison invariance of _{v}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) and the fact that H^{1}(N_{C}(v)) = 0 for v ≥ −1 for a smooth rational curve C. Instead of working out these details, we will finish this subsection by considering two general classes of examples where B is licci. We concentrate on licci codimension 3 quotients, leading to unobstructed Gorenstein codimension 4 quotients by Theorem 16. In one of the cases the use of Proposition 33 is quite effective.

Example 39. Let B be a Gorenstein quotient of R of codimension 3 given by

0 → R(−f) ≃ F_{3} → F_{2} → F_{1} → I_{B} → 0.  (34)

(as A in Example 28). In this case e = −n − 3 and K_{B} ≃ B(−f + n + 3) and since B is licci ([38]) and therefore satisfies _{v}Ext_{B}^{1}(I_{B}/I_{B}^{2}, B) = 0 for any v by Proposition 33, we get δ(B)_{−s} = dim(N_{B})_{−(f+s−n−3)}. If A is defined by a section of (K_{B}^{∗})_{s} as in Theorem 16, then A is Gorenstein and unobstructed and

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B})_{0} + dim B_{f+s−n−3} − 1 − dim(N_{B})_{−f−s+n+3}

where dim(N_{B})_{v} is given by [28], Thm. 2.6, i.e. by modifying (20) to

dim(N_{B})_{v} = dim(∧^{2}F_{1})_{f+v} − dim(F_{1} ⊗ F_{2})_{f+v} + dim(S_{2}(F_{2}))_{f+v} + dim(F_{1})_{v} − dim(F_{2})_{v}

Example 40. Let B be an almost complete intersection of R of codimension 3 with a minimal resolution

0 → G_{3} → G_{2} → G_{1} = ⊕_{i=1}^{3}R(−c_{i}) ⊕ R(−h) → I_{B} → 0  (35)

and let B′ be the Gorenstein quotient obtained by linking B to B′ via a complete intersection whose minimal generators (of degree c_{i}) corresponds to three of those of I_{B}. A standard mapping cone construction and duality yields

0 → R(h − a) → G_{2}^{∗}(−a) → G_{3}^{∗}(−a) ⊕ (⊕_{i=1}^{3}R(−c_{i})) → I_{B′} → 0  (36)

and we see that K_{B′} ≃ B′(a − h − n − 3) where a = ∑ c_{i}. If A is defined by the zero locus of a section of (K_{B}^{∗})_{s} as in Theorem 16, then A is Gorenstein and unobstructed and invoking Proposition 33 and dim(K_{B′}^{∗})_{−s+2a−2n−6} − δ(B′)_{s−2a+2n+6} = dim B′_{−s+h+a−n−3} − dim(N_{B′})_{s−h−a+n+3} we get

dim_{(A)} GradAlg^{H_{A}}(R) = dim(N_{B′})_{0} + dim B′_{−s+h+a−n−3} − dim(N_{B′})_{s−h−a+n+3} − 1 + γ

where γ := ∑_{i=1}^{3}(dim B_{c_{i}}−dim B′_{c_{i}}+dim(I_{B/D})_{s−a+n+3+c_{i}}−dim(I_{B′/D})_{−s+a−n−3+c_{i}})+dim B_{s−a+n+3}− dim B′_{−s−n−3+a}. Using I_{B/D} ≃ K_{B′}(n + 3 − a) ≃ B′(−h), we easily express γ in terms of the Hilbert function of B′ (or of B) and D. Moreover dim(N_{B′})_{v} is given by the final formula of Example 39. See Example 2 for a more explicit example.

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OSLO UNIVERSITY COLLEGE, FACULTY OF ENGINEERING, CORT ADELERSGT. 30, N-0254 OSLO, NORWAY. E-mail address: JanOddvar.Kleppe@iu.hio.no