Families of Artinian and one-dimensional algebras

Jan O. Kleppe^{∗}

Abstract

The purpose of this paper is to study families of Artinian or one dimensional quotients of a polynomial ring R with a special look to level algebras. Let GradAlg^{H}(R) be the scheme parametrizing graded quotients of R with Hilbert function H. Let B → A be any graded surjection of quotients of R with Hilbert function H_{B} = (1, h_{1}, ..., h_{j}, ...) and H_{A} respectively. If dim A = 0 (resp. dim A = depth A = 1) and A is a “truncation” of B in the sense that H_{A} = (1, h_{1}, ..., h_{j−1}, α, 0, 0, ...) (resp. H_{A} = (1, h_{1}, ..., h_{j−1}, α, α, α, ...)) for some α ≤ h_{j}, then we show there is a close relationship between GradAlg^{H_{A}}(R) and GradAlg^{H_{B}}(R) concerning e.g. smoothness and dimension at the points (A) and (B) respectively, provided B is a complete intersection or provided the Castelnuovo-Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to “non-truncated” Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of deformations in a manageable form which we make rather explicit for level algebras of Cohen-Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg^{H}(R), H = (1, 3, 6, 10, 14, 10, 6, 2), whose general elements are Artinian level algebras of type 2. AMS Subject Classification. 14C05, 13D10, 13D03, 13D07, 13C40, 13D02. Keywords. Parametrization, Artinian algebra, level algebra, Gorenstein algebra, licci, Hilbert scheme, duality, algebra (co)homology, canonical module, normal module.

1 Introduction

The main goal of this paper is to contribute to the classification of Artinian and one dimensional graded quotients of a polynomial ring R in n variables (of degree one) over an algebraically closed field k. In particular we study the scheme GradAlg^{H}(R) = GradAlg(H) which parametrizes graded quotients A of R of depth A ≥ min(1, dim A) and with Hilbert function H. GradAlg^{H}(R) is the representing object of a correspondingly defined functor of flat families and it may be non-reduced. Thus GradAlg^{H}(R) may be different from the parameter spaces studied by Iarrobino, Gotzmann and others who study the “same” scheme with the reduced scheme structure. In our approach we try to benefit from having a well described tangent and obstruction space of GradAlg^{H}(R) at (A) at our disposal.

An important technique in determining GradAlg(H_{A}) is to take a graded surjection B → A of quotients of R with Hilbert functions H_{B} and H_{A} respectively, and, under certain conditions, make the relationship between GradAlg(H_{A}) and GradAlg(H_{B}) as tight as possible. We review some results of this technique in Section 1. If B = R/I_{B}, let N_{B} := Hom_{R}(I_{B}, B) and let reg(B) = reg(I_{B}) − 1 be the Castelnuovo-Mumford regularity of B. Let

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

be the minimal resolution and let ϵ(A/B) = ∑_{i=1}^{r_{1}}[H_{B}(n_{1,i})−H_{A}(n_{1,i})]. Our main results in Section 2 apply to GradAlg(H_{A}) where A is one-dimensional. We prove (cf. Theorem 12)

∗Partially supported by Research Council of Norway, project no. 154077/420.

Theorem 1. Let R be a polynomial k-algebra and let B = R/I_{B} → A = R/I_{A}, I_{A} ≠ 0, be a graded morphism such that A is Cohen-Macaulay of dimension one and depth B ≥ 1 and such that X := Proj(A) ↪ Y := Proj(B) is a local complete intersection of codimension r ≥ 0. Let H_{A}(v) = s for v >> 0 and suppose either

(a) I_{B} is generated by a regular sequence (allowing R = B), or

(b) B_{v} → A_{v} is an isomorphism for all v ≤ max_{i}{n_{2,i}} and dim R − dim B ≥ 2. Moreover suppose there is an integer j such that B_{v} ≃ A_{v} for all v ≤ j − 1 and such that I_{A} is (j + 1)-regular (i.e. reg(A) ≤ j, or equivalently, H_{A}(v) = s for v ≥ j). Then dim(N_{A})_{0} = dim(N_{B})_{0} + rs − ϵ(A/B) , and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + rs − ϵ(A/B) .

In particular A is unobstructed as a graded R-algebra (i.e. GradAlg^{H_{A}}(R) is smooth at (A)) if and only if B is unobstructed as a graded R-algebra.

One may look upon the conditions on j above as assuming the minimal free resolution of I_{A/B} := I_{A}/I_{B} to be semi-linear (close to being linear, cf. (5)), and the conditions of (b) and (a) as requiring this j to be large enough in the case Y is not a complete intersection (CI).

Theorem 1 is what we need to treat the case where X consists of s points in generic position on Y in the sense that H_{A} is the truncation of H_{B} at the level s and the points are distinct. Indeed Geramita et al. ([13]) defines such a truncation by

H_{A}(i) = inf{H_{B}(i), s} for i ≥ 0 ,

and they show that there exists a reduced scheme X on Y with truncated Hilbert function H_{A} provided Y is reduced and consists of more than s points. We prove (cf. Corollary 14)

Corollary 2. Let Y = Proj(B), B = R/I_{B}, be a reduced scheme consisting of more than s points, and let X = Proj(A) be s points (avoid SingY) of codimension r in generic position on Y. Let j be the smallest number such that H_{A}(j) ≠ H_{B}(j). If Y is not a CI, suppose j ≥ reg(I_{B}) + 2. Then dim(N_{A})_{0} = dim(N_{B})_{0} + rs − ϵ(A/B) , and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + rs − ϵ(A/B) .

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

Moreover in Corollary 2 (and Theorem 1) we may allow the codimension r to vary along the s points, say such that the i-th point has codimension c_{i} in Y (Y need not be equidimensional). Then Corollary 2 holds if we replace rs by ∑_{i=1}^{s} c_{i}.

The analogue of Theorem 1 for Artinian algebras is the main result of Section 3 (cf. Theorem 29).

Theorem 3. Let R be a polynomial k-algebra and let B = R/I_{B} → A = R/I_{A} be a graded morphism such that A is Artinian and depth B ≥ min(1, dim B), and suppose either

(a) I_{B} is generated by a regular sequence (allowing R = B), or

(b) B_{v} → A_{v} is an isomorphism for all v ≤ max_{i}{n_{2,i}} and dim R − dim B ≥ 2. Let F be a free B-module such that F → I_{A/B} is surjective and minimal, and suppose there is an integer j such that the degrees of minimal generators of the B-module ker(F → I_{A/B}) are strictly greater than j (e.g. B_{v} ≃ A_{v} for all v ≤ j − 1) and such that I_{A} is (j + 1)-regular (i.e. A_{j+1} = 0). Then dim(N_{A})_{0} = dim(N_{B})_{0} + dim _{0}Hom_{B}(F, A) − ϵ(A/B) , and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + dim _{0}Hom_{B}(F, A) − ϵ(A/B) .

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

In Proposition 42 we show an improvement of Theorem 3(a) in the case “B_{v} ≃ A_{v} for all v ≤ j−1”. In this case we can skip the condition A_{j+1} = 0, or equivalently (K_{A})_{−j−1} = 0 (K_{A} the canonical module) provided the minimal resolution of K_{A} has no relations in degree greater or equal to j. This generalization applies to algebras which are compressed or close to being compressed. In the compressed case the dimension and the smoothness of GradAlg(H_{A}) coincide with the results of [24].

Theorem 3 applies nicely to Artinian truncations and more generally to Artinian quotients A with Hilbert function H_{A} = (1, h_{1}, h_{2}, ..., h_{j−1}, α, 0, 0, ..) where H_{B} = (1, h_{1}, h_{2}, ..., h_{j−1}, h_{j}, h_{j+1}, ...) and α ≤ h_{j}. In that case the relationship between GradAlg(H_{A}) and the open subset GradAlg(H_{B})_{η} of GradAlg(H_{B}) consisting of points (B) where reg(I_{B}) ≤ η, may be described by an incidence correspondence

GradAlg(H_{B}, H_{A})_{η} ⟶^{q} GradAlg(H_{B})_{η} ⊂ GradAlg(H_{B})
↓^{p}
GradAlg(H_{A})

(1)

where p and q are the natural projections (cf. (8) and Proposition 33 for details).

Proposition 4. Let H_{B} = (1, h_{1}, h_{2}, ...) be the Hilbert function of an algebra B ≠ R satisfying depth B ≥ 1, and let j, η ≤ j − 2 and α ≤ h_{j} be non-negative integers. Let H_{A} = (1, h_{1}, ..., h_{j−1}, α, 0, 0, ..) and look to the maps p and q in (1). Then

(i) q is smooth and surjective with connected fibers, of fiber dimension α(h_{j} − α), and (ii) p is an isomorphism onto an open subscheme of GradAlg(H_{A}).

In particular the incidence correspondence (1) determines a well-defined injective application π from the set of irreducible components W of GradAlg(H_{B})_{η}, to the set of irreducible components V of GradAlg(H_{A}) whose general elements satify the Weak Lefschetz property. In this application the generically smooth components correspond. Indeed V = π(W) is the closure of p(q^{−1}(W)), and we have

dim V = dim W + α(h_{j} − α) .

Also Theorem 1 allows a corollary very similar to Proposition 4 in the one dimensional case (cf. Proposition 19).

In Section 4 we characterize the tangent and obstruction space of GradAlg^{H}(R) at an Artinian algebra (A). Note that if A is Gorenstein with socle degree j and S_{2}I_{A} is the second symmetric power of I_{A}, then the k-dual of the obstruction space is by [32], Thm. 11 isomorphic to the kernel of the natural map (S_{2}I_{A})_{j} → (I_{A}^{2})_{j}, or equivalently, to the cokernel of (Λ^{2}I_{A})_{j} → (I_{A}⊗I_{A})_{j} ≃ Tor_{1}^{R}(I_{A}, K_{A})_{0}. This result generalizes to the following result (cf. Theorem 36), in which H_{2}(R, A, K_{A}) is the algebra homology, cf. (2).

Theorem 5. Let R → A = R/I_{A} be a graded Artinian quotient with Hilbert function H. Then dim(I_{A} ⊗_{R} K_{A})_{0} is the dimension of the tangent space of GradAlg^{H}(R) at (A), and the dual of _{0}H_{2}(R, A, K_{A}) contains the obstructions of deforming A as a graded R-algebra. In particular GradAlg^{H}(R) is smooth at (A) provided the natural “antisymmetrization” map

I_{A} ⊗_{R} I_{A} ⊗_{R} K_{A} → Tor_{1}^{R}(I_{A}, K_{A})

(cf. (17)) is surjective in degree zero.

Since we in Theorem 44 show that the parameter space L( H) of level algebras, introduced in [9] through the ideas of [26], is sufficiently close to being an open subscheme of GradAlg^{H}(R), we get that Theorem 5 holds if we everywhere replace GradAlg^{H}(R) by L( H) at a level algebra (A). Note that the tangent space of L( H) is already well described in [9]. Moreover if A is Artinian of codimension 3 in R, we show that the obstruction space is contained in the dual of (N_{A})_{−3} and we make explicit a formula which is a lower bound for the dimension of any irreducible component of GradAlg^{H}(R) (Proposition 39 and Corollary 40).

Finally we look to type 2 level algebras A = R/ann(F_{1}, F_{2}) where F_{1} and F_{2} are forms of the same degree s in the “dual” polynomial algebra of R. Such algebras are studied in [25], and an extended draft of [25] determines the tangent space of GradAlg^{H}(R) at (A). If H_{A}(i) = min{dim R_{i}, H_{A_{1}}(i)+ H_{A_{2}}(i)} for any i, then {F_{1}, F_{2}} is said to be complementary [25]. Using Theorem 5 we describe the tangent and obstruction space of GradAlg^{H}(R) at (A) in the following way (see Proposition 45).

Proposition 6. Let {F_{1}, F_{2}} be complementary forms of degree s, and let A = R/I_{A} be the Artinian level quotient with Hilbert function H given by I_{A} = ann(F_{1}, F_{2}). Let I_{A_{i}} = ann(F_{i}). Then (I_{A}/I_{A} · I_{A_{1}})_{s} ⊕ (I_{A}/I_{A} · I_{A_{1}})_{s} is the dual of the tangent space of GradAlg^{H}(R) at (A), and _{s}H_{2}(R, A, A_{1}) ⊕ sH_{2}(R, A, A_{2}) is the dual of a space containing the obstructions of deforming A as a graded R-algebra. In particular if the sequences

I_{A} ⊗_{R} I_{A} ⟶^{λ} I_{A} ⊗ I_{A_{i}} → I_{A} · I_{A_{i}}

where λ(x ⊗ y) = x ⊗ y − y ⊗ x, are exact for i = 1 and 2, then GradAlg(H_{A}) is unobstructed at (A) and we have dim_{(A)} GradAlg^{H_{A}}(R) = ∑_{i=1}^{2} dim(I_{A}/I_{A} · I_{A_{i}})_{s} .

Then we use Proposition 6 and a linkage theorem (Theorem 24) to prove a conjecture of A. Iarrobino, appearing in the draft of [25], namely that L( H) with H = (1, 3, 6, 10, 14, 10, 6, 2) contains at least two irreducible components whose general elements are level quotients of type 2 (Example 49). Once having one example of such a phenomena, we produce infinitely many by liaison (Remark 50). Even though this conjecture was open until now, Iarrobino and Boij have in a joint work already constructed other examples of reducible L( H) whose general elements are type 2 level quotients, one with H = (1, 3, 6, 10, 14, 18, 20, 20, 12, 6, 2), and they have got a doubly infinite series of such components [5].

In this paper we give many examples to illustrate our results, some of them with the use of Macaulay 2. Among examples of particular interest, in addition to the proven conjecture, we mention Example 21 of two irreducible components of GradAlg(H) (and of PGor(H)) whose intersection contains Artinian Gorenstein algebras, and Example 41 of two components of GradAlg(H) with H = (1, 3, 6, 6, 3, 1) whose general elements of both components are Artinian licci algebras.

Together with co-authors we have in several papers ([30], [32], [34], [35], [36] and [33] which makes a correction to [36], Ch. 10) studied the scheme GradAlg^{H}(R) and its subset PGor(H) which parametrizes Gorenstein quotients. The latter is essentially an open subscheme of the former (cf. [32], Thm. 11, or Theorem 44 of this paper). One will see from [34], [36] and [33] that our cohomological methods often require the quotients to have depth at least two (in which case GradAlg^{H}(R) is locally isomorphic to the usual Hilbert scheme, see Propostion 8). In [32] and partially in [34], we were, however, able to treat Gorenstein algebras A = B/I_{B} of any dimension satisfactorily, utilizing properties of the canonical module K_{B}. In the present paper we take a major further step in generalizing our notable cohomological and infinitesimal approach to treat non-Gorenstein low dimensional algebras.

I thank Anthony Iarrobino for interesting discussion on the subject and for introducing me to type 2 level algebras and his conjecture by giving me the extended draft of [25]. I also thank him and the Northeastern University for their hospitality during my visit to Boston in February 2005. Moreover I thank Arvid Siqueland for a thorough reading of the manuscript and the referee for useful and clarifying comments to the exposition.

1.1 Preliminaries

Let R be a polynomial k-algebra in n variables of degree 1 where k is algebraically closed. In the following we focus on the scheme parametrizing Artinian graded quotients B of R, as well as the scheme parametrizing closed schemes Y = Proj(B) in ℙ = ℙ^{n−1}, with fixed Hilbert function H. Both schemes are denoted by GradAlg^{H}(R). When we write Y = Proj(B), we always take B as the homogeneous coordinate ring of Y and the Hilbert function of B, or Y, as H_{B}(v) = H_{Y} (v) := dim B_{v}. Now if H(v) = dim B_{v} does not vanish for v >> 0, we call GradAlg^{H}(R) the postulation Hilbert scheme because this name seems to be most common, at least when it is endowed with its reduced scheme structure and dim B = 1 (cf. [15], [26]). Since GradAlg^{H}(R) is the representing object of a certain functor of flat deformations, it may be non-reduced. We continue denoting it by GradAlg^{H}(R), to make it clear that it may be non-reduced.

Now we recall the definition of GradAlg^{H}(R). Let Hilb^{p}(ℙ) be the Hilbert scheme ([17]) parametrizing closed subschemes Y of ℙ = Proj(R) with Hilbert polynomial p ∈ ℚ[t]. The k-point of Hilb^{p}(ℙ) which corresponds to Y is denoted by (Y). A closed subscheme Y of ℙ is called unobstructed if Hilb^{p}(ℙ) is smooth at (Y).

Let GradAlg(H) := GradAlg^{H}(R) be the stratum of Hilb^{p}(ℙ) given by deforming Y = Proj(B) ⊂ ℙ with constant Hilbert function H_{B} = H (more precisely its functor deforms the homogeneous coordinate ring, B = R/I_{B}, of Y flatly), cf. [30] or [32]. GradAlg^{H}(R) allows a natural scheme structure whose tangent (resp. “obstruction”) space at (Y) 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 ([28], Thm. 1.5). 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 [32], Prop. 9 and Thm. 11). B is called unobstructed as a graded R-algebra if and only if (iff) GradAlg(H_{B}) is smooth at (B), i.e. at (Y).

Remark 7. (a) It follows from a theorem of Pardue (Thm. 34 of [47], cf. [15] for the codimension 2 case) that GradAlg(H) is a connected scheme (see also [40]).

(b) Note also that Ragusa-Zappala’s result for zero-schemes ([49]), that different minima of the set of graded Betti numbers yield different components of GradAlg(H), is valid for any H and R. This follows from the representability of the functor which defines GradAlg(H) ([32], Prop. 9), the semicontinuity of the graded Betti numbers and Ragusa-Zappala’s proof because we may avoid the “very flatness” argument in their proof by using the flatness of the representing object of the mentioned functor. Note that the set of graded Betti numbers is partially ordered because each Betti number (i.e. the number of minimal generators of a fixed degree of some finitely generated syzygy R-module) obey semicontinuity and may decrease under generization. Thus different minima may occur. In particular incomparable sets (i.e. sets without a common minimum) of graded Betti numbers lead to different components in general (see [43] for a discussion).

The following comparison result is due to Ellingsrud ([12]) in the case depth_{𝔪} B ≥ 2, see [28], Thm. 3.6 and Rem. 3.7 for the general case. Below s(I_{B}) is the minimal degree of the minimal generators of I_{B}. Note that the openness statements follow easily from the first isomorphism by the semicontinuity of dim H^{1}(Y, Ĩ_{B}(v)).

Proposition 8. Let B = R/I_{B} satisfy depth_{𝔪} B ≥ 1 and let Y = Proj(B). Then

GradAlg^{H}(R) ≃ Hilb^{p}(ℙ) at (Y) ,

provided _{0}Hom_{R}(I_{B}, H_{𝔪}^{1}(B)) = 0 (e.g. provided depth_{𝔪} B ≥ 2). In particular the open sets

U(H) := {(B) ∈ GradAlg^{H}(R)| _{v}H_{𝔪}^{1}(B) = 0 for every v ≥ s(I_{B})}

and {(B) ∈ GradAlg^{H}(R)| depth_{𝔪} B ≥ 2} of GradAlg^{H}(R) are also open in Hilb^{p}(ℙ).

Here depth_{𝔪} M (or just depth M, M finitely generated) denotes the length of a maximal Msequence in the irrelevant maximal ideal 𝔪, and H_{𝔪}^{i}(−) is the right derived functor of the functor of sections with support in Spec(B/𝔪). Note that depth_{𝔪} M ≥ r iff H_{𝔪}^{i}(M) = 0 for i < r, cf. [19]. A Cohen-Macaulay B-module M satisfies depth M = dim M by definition. If B is Cohen-Macaulay of codimension c in R and K_{B} = Ext_{R}^{c}(B, R(−n)) is the canonical module of B, we know by Gorenstein duality that the v-graded piece of H_{𝔪}^{i}(M) satisfies (cf. [19])

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

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. [44] for a survey).

The algebraic (co)homology groups H_{2}(R, B, M) and H^{2}(R, B, M) may be described as follows. The former group is given by an exact sequence

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

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} [39]. For the graded group H^{2}(R, B, M) we only remark that by [1], Prop. 16.1, and [39], there are injections

0Ext_{B}^{1}(I_{B}/I_{B}^{2}, M) ↪ _{0}H^{2}(R, B, M) ↪ _{0}Ext_{R}^{1}(I_{B}, M)  (3)

A 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. [11])

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

and B is Cohen-Macaulay (CM) iff G_{c+1} = 0. The function max_{i}{n_{j,i}}−j is increasing as a function in j if B is CM. If B is Artinian (i.e. c = n), then max_{i}{n_{c,i}} − c is the socle degree of B. More generally the Castelnuovo-Mumford regularity of I_{B} is given by reg(I_{B}) = max_{{j,i}} {n_{j,i} − j + 1} and reg(B) = reg(I_{B}) − 1 (cf. [44], p. 8). In particular

max_{i}{n_{j,i}} ≤ reg(I_{B}) + j − 1 for any j.

If G_{c+1} = 0 and G_{c} has rank 1 (resp. G_{c} = R(−s)^{t}), then B is Gorenstein (resp. level of type t). In these cases B is a compressed Artinian R-algebra if H_{B} (i.e. H_{B}(v) for any v) is as large as possible for a fixed socle degree and fixed type (cf. [24] for existence).

An R-module M of projective dimension t − 1 is said to have a semi-linear (resp. linear) resolution provided the minimal resolution of M has the following form

0 → R(−j − t)^{β_{t}} ⊕ R(−j − t + 1)^{α_{t}} → ... → R(−j − 1)^{β_{1}} ⊕ R(−j)^{α_{1}} → M → 0  (5)

(resp. with α_{i} = 0 for all i). With B as in (4) and B → A ≃ B/I_{A/B} a graded surjection, we define

ϵ = ϵ(A/B) = ∑_{i=1}^{r_{1}} dim(I_{A/B})_{n_{1,i}} = ∑_{i=1}^{r_{1}}[H_{B}(n_{1,i}) − H_{A}(n_{1,i})]  (6)

where H_{B} and H_{A} are the Hilbert functions of B and A. If B is a complete intersection (CI), allowing R = B, then I_{B}/I_{B}^{2} and the normal module N_{B} = (I_{B}/I_{B}^{2})^{∗} are R-free of rank r_{1} ≥ 0, and

dim_{(B)} GradAlg^{H_{B}}(R) = dim(N_{B})_{0} = ∑_{i=1}^{r_{1}} H_{B}(n_{1,i})

In Lemma 9 and Theorem 12 of the next section we look upon the special case B = R as a CI with r_{1} = 0. Throughout we pass to small letters to denote the k-vector space dimension of the (co)homology groups involved, e.g. for any i ≥ 0,

h^{i}(M̃) = dim H^{i}(M̃), _{v}h^{i}(R, B, M) = dim _{v}H^{i}(R, B, M), _{v}ext_{B}^{i}(M, N) = dim _{v}Ext_{B}^{i}(M, N).

Lemma 9. Let R → B = R/I_{B} → A ≃ B/I_{A/B} be graded morphisms, let c = dim R − dim B and suppose either

(a) I_{B} is generated by a regular sequence (allowing R = B), or

(b) c ≥ 2 and B_{v} → A_{v} is an isomorphism for all v ≤ max_{i}{n_{2,i}}. Then _{0}H^{2}(R, B, I_{A/B}) = 0 and _{0}hom_{R}(I_{B}, I_{A/B}) = ϵ(A/B). Moreover ϵ(A/B) = 0 if (b) holds.

Proof. If B is a CI, then it is well known that _{0}H^{2}(R, B, I_{A/B}) = 0, and moreover that _{0}H^{1}(R, B, I_{A/B}) ≃ 0Hom_{R}(I_{B}/I_{B}^{2}, I_{A/B}) ≃ (⊕_{i}I_{A/B}(n_{1,i}))_{0} and we get the lemma in this case. In (b) it suffices by (3) to show _{0}Ext_{R}^{i}(I_{B}, I_{A/B}) = 0 for i ≤ 1. Applying _{0}Hom_{R}(−, I_{A/B}) to the minimal resolution of I_{B} deduced from (4), we conclude by the assumptions of (b).

The following Proposition is a part of Prop. 4 of [32] and is used quite often in this paper. Below GradAlg(H_{B}, H_{A}) is the representing object of the functor deforming 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. Then there exist natural projection morphisms p : GradAlg(H_{B}, H_{A}) → GradAlg(H_{A}), induced by p((B → A)) = (A), and q : GradAlg(H_{B}, H_{A}) → GradAlg(H_{B}), induced by q((B → A)) = (B), which under the assumptions of Prop. 4 of [32] have nice properties. Recall that A is called H_{B}-generic if there is an open subset U ∋ (A) of GradAlg^{H_{A}}(R) such that every (A′) ∈ U belongs to im p. Now since the surjectivity of the natural map _{0}Hom_{B}(I_{B}, B) → 0Hom_{R}(I_{B}, A) together with the injectivity of _{0}H^{2}(R, B, B) → _{0}H^{2}(R, B, A) is equivalent to

0H^{2}(R, B, I_{A/B}) = 0

by the long exact sequence of algebra cohomology, we may state [32], Prop. 4 (i), resp. (ii) as (i), resp. (ii) of the Proposition below.

Proposition 10. Let B = R/I_{B} → A ≃ B/I_{A/B} be a graded morphism of quotients of R.

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

(ii) If _{0}H^{2}(R, B, I_{A/B}) = 0, then the projection p : GradAlg(H_{B}, H_{A}) → GradAlg(H_{A}) induced by p((B → A)) = (A) is smooth with fiber dimension _{0}hom_{R}(I_{B}, I_{A/B}) at (B → A). In particular A is H_{B}-generic.

Corollary 11. Let B → A be a graded surjection of quotients of R. If _{0}H^{2}(B, A, A) = 0 and 0H^{2}(R, B, I_{A/B}) = 0, then A is H_{B}-generic, and we have

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

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

Proof. Using Proposition 10(i) we get

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

while (ii) implies dim_{(B→A)} GradAlg(H_{B}, H_{A}) = dim_{(A)} GradAlg(H_{A}) +_{0} hom_{R}(I_{B}, I_{A/B}) which gives one of the dimension formulas. Since smooth morphisms imply surjective tangent maps of their tangent spaces and since the Hom-groups of Proposition 10 are the tangent spaces of the fibers, we can argue as above to get the other dimension formula.

2 Families of one dimensional quotients of R.

In this section we focus on families of zero schemes in ℙ = ℙ^{n−1} with fixed Hilbert function H, i.e. we study the (possibly non-reduced) postulation Hilbert scheme GradAlg^{H}(R) where H(v) is a constant for v >> 0.

If Y ⊂ ℙ = Proj R is a closed subscheme and X = Proj(A) is obtained by choosing s points in generic position on Y = Proj(B) (see the paragraph before Corollary 2 for a definition), the main theorem of this section implies that A and B are simultaneously unobstructed as graded algebras and dim_{(A)} GradAlg^{H_{A}}(R) and dim_{(B)} GradAlg^{H_{B}}(R) are closely related (Theorem 12, Corollary 14). Even though this result may seem new as stated, it is a straightforward consequence of Theorem 9.16 of [36] if Y is a curve. In this section we generalize the result to any scheme Y. In Proposition 19 we extend the result to families, and we finish by a theorem on linkage of families (Theorem 24).

A zero-dimensional closed scheme X ↪ Y is said to be a local complete intersection (l.c.i) of codimension (r_{1}, ..., r_{t}) with respect to X = X_{1} ∪ ... ∪ X_{t} if X can be written as a disjoint union X = X_{1} ∪ ... ∪ X_{t} where, for each i, the ideal (J_{X/Y})_{x} is generated by an O_{Y,x}-regular sequence of length r_{i} for every x ∈ X_{i}. If r_{i} are equal for all i, say r_{i} = r, we simply say X ↪ Y is an l.c.i of codimension r. Note that in the case r_{i} = 0, then X_{i} is mapped isomorphically onto an open subscheme of Y. Below N_{B} := Hom_{B}(I_{B}/I_{B}^{2}, B) is the normal module of B in R, ϵ(A/B) is defined in (6), n_{2,j} in (4) and the H_{B}-genericity of A is defined in the text before Proposition 10.

Theorem 12. Let R be a polynomial k-algebra and let B = R/I_{B} → A = R/I_{A}, I_{A} ≠ 0, be a graded morphism such that A is Cohen-Macaulay of dimension one and depth_{𝔪} B ≥ 1, and such that X := Proj(A) ↪ Y := Proj(B) is a local complete intersection of codimension (r_{1}, ..., r_{t}) with respect to X = X_{1} ∪ ... ∪ X_{t}. Let H_{A}(v) = s and H_{X_{i}}(v) = s_{i} for v >> 0 (so s = ∑_{i} s_{i}) and suppose either

(a) I_{B} is generated by a regular sequence (allowing R = B), or

(b) B_{v} → A_{v} is an isomorphism for all v ≤ max_{i}{n_{2,i}} and dim R − dim B ≥ 2. Moreover suppose there is an integer j such that B_{v} ≃ A_{v} for all v ≤ j − 1 and such that I_{A} is (j + 1)-regular (or equivalently, such that H_{A}(v) = H_{B}(v) for v ≤ j − 1 and H_{A}(v) = s for v ≥ j). Then A is H_{B}-generic, dim(N_{A})_{0} = dim(N_{B})_{0} + ∑_{i} r_{i}s_{i} − ϵ(A/B) , and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + ∑_{i}^{t} r_{i}s_{i} − ϵ(A/B) .

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

Remark 13. Theorem 12 applies to quotients B → A ≃ B/I_{A/B} where the mapping cone construction produces the minimal resolution of A from the free resolution of B and a semi-linear resolution of I_{A/B} (modulo redundant terms). For instance if M := I_{A/B} and t = r − 1 in (5) (i.e. depth I_{A/B} = 2), then B_{v} ≃ A_{v} for all v ≤ j − 1 and I_{A} is (j + 1)-regular, and Theorem 12 applies. Also in the case depth I_{A/B} = 1 in which the “G_{r−1} → ... → G_{1}”-part of the minimal resolution 0 → G_{r} → ... → G_{1} → I_{A/B} → 0 is semi-linear, then Theorem 12 applies because the contribution from G_{r} becomes redundant in the minimal resolution of A. Thus the condition on j of Theorem 12 essentially requires I_{A/B} to have a semi-linear resolution; in the non-CI case j must be large enough to have (b) fulfilled (e.g. j ≥ reg(I_{B}) + 2).

Proof. It is enough to prove that A is H_{B}-generic and the two dimension formulas. Due to Corollary 11 it suffices to show _{0}H^{2}(B, A, A) = 0 and_{0}hom_{B}(I_{A/B}, A) = ∑_{i} r_{i}s_{i}, as well as _{0}H^{2}(R, B, I_{A/B}) = 0 and _{0}hom_{R}(I_{B}, I_{A/B}) = ϵ(A/B). The latter follows from Lemma 9. Let N_{X/Y} the normal sheaf of X ↪ Y. Since dim X = 0 and the composition x ↪ X ↪ Y is a local complete intersection for any x ∈ X, then the sequence (9.6) in the proof of Thm. 9.16 of [36] is still exact (cf. [28], Lem. 3.5 and (3.3)) and may be written as

0 → _{0}H^{1}(B, A, A) → H^{0}(N_{X/Y}) → _{0}Hom_{R}(I_{A/B}, H_{𝔪}^{1}(A)) → _{0}H^{2}(B, A, A) → 0 .

Hence _{0}H^{2}(B, A, A) = 0 and _{0}Hom_{B}(I_{A/B}, A) ≃ _{0}H^{1}(B, A, A) ≃ H^{0}(N_{X/Y}) provided

0Hom_{R}(I_{A/B}, H_{𝔪}^{1}(A)) = 0 .

Since H_{𝔪}^{1}(A)_{v} ≃ H_{𝔪}^{2}(I_{A})_{v} ≃ _{−v}Ext_{R}^{n−2}(I_{A}, R(−n))^{∨}, we get that H_{𝔪}^{1}(A)_{v} = 0 for v ≥ j by the (j + 1)-regularity of I_{A}. Note that since 0 → A_{v} → H^{0}(O_{X}(v)) → H_{𝔪}^{1}(A)_{v} → 0 is exact, it follows that the vanishing of H_{𝔪}^{1}(A)_{v} = 0 for v ≥ j is equivalent to H_{A}(v) = dim A_{v} = s for v ≥ j. Now, using (I_{A/B})_{j−1} = 0 we conclude that _{0}Hom_{R}(I_{A/B}, H_{𝔪}^{1}(A)) = 0.

Hence it suffices to show dim H^{0}(N_{X/Y}) = ∑_{i} r_{i}s_{i}. Since Supp(X) is finite, we know that h^{0}(O_{X_{i}}) = ∑_{x∈Supp(X_{i})} length(O_{X_{i},x}) = s_{i}. Using that N_{X/Y,x} is a free O_{X,x}-module of rank r_{i} for any x ∈ Supp(X_{i}), we conclude by

h^{0}(N_{X/Y}) = ∑_{i}^{t} ∑_{x∈Supp(X_{i})} length(N_{X/Y,x}) = ∑_{i}^{t} ∑_{x∈Supp(X_{i})} r_{i} · length(O_{X_{i},x}) = ∑_{i}^{t} r_{i}s_{i} .

Theorem 12 is precisely what we need to treat the case where X consists of s points in generic position on Y (i.e. H_{A} is the truncation of H_{B} at the level s and the points are distinct). Indeed if we define the truncation of H_{B} at the level s by

H_{A}(i) = inf{H_{B}(i), s} for i ≥ 0 ,

then a theorem of Geramita-Maroscia-Roberts ([13]) show that there exists a reduced scheme X on Y with truncated Hilbert function H_{A} as above provided Y is reduced and consists of more than s points. Denoting the singular locus of Y by SingY, we get

Corollary 14. Let Y = Proj(B), B = R/I_{B}, be a reduced scheme consisting of more than s points, and let X = Proj(A) be s points (avoid SingY) in generic position on Y. Let j be the smallest number such that H_{A}(j) ≠ H_{B}(j). If Y is not a CI, suppose j ≥ max_{i}{n_{2,i}}+1 (e.g. j ≥ reg(I_{B})+2). Then X ↪ Y is an l.c.i of codimension (r_{1}, ..., r_{t}) with respect to some decomposition X = X_{1} ∪ ... ∪ X_{t}. Moreover A is H_{B}-generic, dim(N_{A})_{0} = dim(N_{B})_{0} + ∑_{i} r_{i}s_{i} − ϵ(A/B) , and

dim_{(A)} GradAlg^{H_{A}}(R) = dim_{(B)} GradAlg^{H_{B}}(R) + ∑_{i}^{t} r_{i}s_{i} − ϵ(A/B) .

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

Proof. Since O_{Y,x} and O_{X,x} are regular local rings for any x ∈ X, it follows that X ↪ Y is an l.c.i of codimension as in Corollary 14. By the definition of s generic points, (I_{A/B})_{j−1} = 0. Since H_{A}(v) ≠ H_{A}(j) for v ≥ j, it follows that I_{A} is (j +1)-regular (and j-regular if s = H_{A}(j −1)). Then Theorem 12 applies supposing j large enough.

Remark 15. It is well known that the Hilbert polynomial p_{B}(x) equals H_{B}(x) for all x ≥ reg(I_{B})−1. Thus the number j ≥ reg(I_{B}) + 2 of Corollary 14 is so large that p_{B}(x) = H_{B}(x) for x ≥ j − 3. In particular using Corollary 14 with say j = reg(I_{B}) + 2, we get an algebra A with Hilbert function H_{A}(x) = H_{B}(x) = p_{B}(x) for x ∈ {j − 3, j − 2, j − 1} and H_{A}(x) = s for x ≥ j.

Example 16. (an obstructed one-dimensional level algebra A with H_{A} = (1, 5, 9, 13, 13, 13, ...)).

The subset of the Hilbert scheme Hilb^{dx+1−g}(ℙ^{4}) consisting of rational normal curves of degree d = 4 has been thoroughly studied ([42], [46]). Indeed this subset forms a smooth, irreducible open subscheme of Hilb^{4x+1}(ℙ^{4}) whose closure V is an irreducible component of dimension 5d + 1 = 21. All arithmetically CM (ACM) curves are contained in the component V by [42]. Moreover the normal sheaf of the general curve Y_{g} of V satisfies H^{1}(N_{Y_{g}}) = 0, while for instance Y = Proj(B), the union of four lines meeting at a point, belongs to the same component V and satisfies dim H^{1}(N_{Y}) = 3 (cf. [36], Rem. 9.9), i.e. Y is an obstructed reduced ACM curve. Both curves have the same graded Betti numbers, e.g.

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

Since the locus of ACM curves in GradAlg(H) is open in Hilb^{4x+1}(ℙ^{4}) by Proposition 8, then V corresponds to an irreducible component of GradAlg(H) to which (Y_{g}) and (Y) belong. Let X = Proj(A) (resp. X_{g} = Proj(A_{g})) be obtained by choosing s ≥ 13 generic points on Y (resp. Y_{g}). Since dim B_{v} = 4v + 1 for v ≥ 0, we see that Corollary 14 applies for j ≥ 4. It follows that A_{g} is unobstructed while A is obstructed as graded R-algebras and

dim_{(A_{g})} GradAlg(H_{A_{g}}) = dim(N_{A_{g}})_{0} = h^{0}(N_{Y_{g}}) + s = 21 + s

(resp. dim(N_{A})_{0} = dim(N_{B})_{0} + s = 24 + s). In particular if s = 13, then H_{A} = H_{A_{g}} = (1, 5, 9, 13, 13, 13, ...), and it is straightforward to see that A_{g} and A are level algebras with the same graded Betti numbers, e.g. the minimal resolution of I_{A} is

0 → R(−7)^{4} → R(−6)^{12} ⊕ R(−4)^{3} → R(−5)^{12} ⊕ R(−3)^{8} → R(−4)^{4} ⊕ R(−2)^{6} → I_{A} → 0.

Corollary 14 applies also to families of reduced schemes Y which are not necessarily equidimensional.

Example 17. Let H(x) = 3x + 1 for x ≥ 0, so H = (1, 4, 7, 10, 13, ...). If Y_{1} = Proj(B_{1}) ⊂ ℙ^{3} is a twisted cubic curve and Y_{2} = Proj(B_{2}) is the union of a plane space curve C of degree 3 and a point P outside the plane containing C, then it is easy to see that both curves belong to the same stratum GradAlg(H) of the Hilbert scheme Hilb^{3x+1}(ℙ^{3}). We claim they belong to two different components of GradAlg(H). Indeed (Y_{1}) belongs to a 12-dimensional irreducible component of GradAlg(H), and using N_{Y_{2}} ≃ N_{C} ⊕ N_{P} and that C ↪ ℙ^{3} and P ↪ ℙ^{3} are CI, we easily get h^{0}(N_{Y_{2}}) = 15 and H^{1}(N_{Y_{2}}) = 0. Invoking Proposition 8 we see that (Y_{2}) belongs to a 15-dimensional irreducible component of GradAlg(H), cf. [48] for a complete description of Hilb^{3x+1}(ℙ^{3}). The minimal resolution of I_{B_{2}} (resp. I_{B_{1}}) is of the form

0 → R(−4) → R(−4) ⊕ R(−3)^{3} → R(−3) ⊕ R(−2)^{3} → I_{B_{2}} → 0  (7)

(resp. of the form (7) where both R(−4) and two of R(−3) are removed). Since Corollary 14 applies for j ≥ 5, let X_{1} = Proj(A_{1}) (resp. X_{2} = Proj(A_{2})) be obtained by choosing s ≥ 13 generic points on Y_{1} (resp. Y_{2}); on Y_{2} we must choose P as one of the s generic points to get the right Hilbert function. It follows that A_{i} are unobstructed as graded R-algebras for i = 1 and 2 and that dim_{(A_{1})} GradAlg(H′) = 12 + s where H′ = (1, 4, 7, ..., 3j − 2, s, s, ...), 3j − 2 ≤ s < 3j + 1. Since X_{2} ↪ Y_{2} is an l.c.i of codimension (1, 0) with respect to the decomposition X_{2} = C_{2} ∪ P where C_{2} consists of s − 1 points, we get

dim_{(A_{2})} GradAlg(H′) = 15 + s − 1 = 14 + s .

Hence we get two different components of GradAlg(H′). Finally if s = 13 it is straightforward to see that A_{2} has the minimal resolution

0 → R(−7)^{3} ⊕ R(−4) → R(−6)^{6} ⊕ R(−4) ⊕ R(−3)^{3} → R(−5)^{3} ⊕ R(−3) ⊕ R(−2)^{3} → I_{A_{2}} → 0.

Once the connection between GradAlg(H_{A}) and GradAlg(H_{B}) for s generic points X on Y is as nice as described in Corollary 14, one may also ask if their irreducible components correspond exactly and similar questions. E.g., may we look upon A_{g} of Example 16 as the general element of an irreducible component of GradAlg(H_{A_{g}})? To see that the answer is yes we use some ideas of [32].

Definition 18. Inside GradAlg^{H}(R), H ≠ H_{R}, we look to the following open subsets, Smc(H) (resp. SmCM(H)), consisting of points (R/I) such that Proj(R/I) is a smooth geometrically connected scheme (resp. smooth and arithmetically CM). Here “points” should be considered as “Ω-points” where Ω is an overfield of k. Moreover let Smc(H)_{η} be the open subset of Smc(H) consisting of points (R/I) where the Castelnuovo-Mumford regularity satisfies reg(I) ≤ η. Similarly we let CI(H) (resp. CM(H)) consist of points (R/I) where I is generated by a regular sequence (resp. R/I is CM).

Now let

SmCM(H_{B}, H_{A})_{η} := p^{−1}(SmCM(H_{A})) ∩ q^{−1}(Smc(H_{B})_{η})

where q : GradAlg(H_{B}, H_{A}) → GradAlg(H_{B}) and p : GradAlg(H_{B}, H_{A}) → GradAlg(H_{A}) are the two natural projection morphisms. (e.g. q((B → A)) = (B)). Denoting the following restrictions of p and q by the same letters, we get a diagram (incidence correspondence)

SmCM(H_{B}, H_{A})_{η} ⟶^{q} Smc(H_{B})_{η} ⊂ GradAlg(H_{B})
↓^{p}
GradAlg(H_{A})

(8)

Proposition 19. Let H_{B} be the Hilbert function of some smooth connected curve Proj(B), B ≠ R, and let H_{A} be its truncated Hilbert function at the level s, i.e. H_{A}(i) = inf{H_{B}(i), s} for i ≥ 0. Let j = min{i|H_{A}(i) ≠ H_{B}(i)}, let η ≤ j − 2 and look to the maps p and q in (8). Then

(i) q is smooth and surjective and its fibers are geometrically connected, of fiber dimension s, and

(ii) p is an isomorphism onto an open subscheme of GradAlg(H_{A}). In particular the correspondence (8) determines a well-defined injective application π from the set of irreducible components W of Smc(H_{B})_{η}, to the set of irreducible components V of GradAlg(H_{A}), in which generically smooth components correspond. Indeed V = π(W) is the closure of p(q^{−1}(W)), and we have

dim V = dim W + s .

Proof. (i) By Geramita et al. [13] we get the surjectivity of q. Since we showed _{0}H^{2}(B, A, A) = 0 in Theorem 12, the smoothness of q follows immediately from Proposition 10(i). To show that the fibers of q are (geometrically) connected, one may simply look at the fiber as the variation of s generic points on a fixed Y, i.e. as a non-empty dense set of Y^{s}. This set is irreducible since Y is irreducible, and we conclude as claimed.

(ii) In Lemma 9 we showed _{0}Ext_{R}^{i}(I_{B}, I_{A/B}) = 0 for i ≤ 1 assuming j ≥ reg(I_{B}) + 2. By Proposition 10(ii) this implies that p is smooth and unramified. It is easy to see that j ≥ reg(I_{B})+1 implies that p is injective (in fact, universally injective or “radiciel”), cf. Lemma 7(a) of [32]. Hence we get (ii) by [18], Thm. 17.9.1. Now combining (i) and [20], Prop. 1.8, we get that q^{−1}(W) is an irreducible component of SmCM(H_{B}, H_{A})_{η}. The application π is therefore well defined, and it is injective by (ii). Finally since q is smooth and p is an open immersion, we easily get the dimension formulas.

Remark 20. If we in Proposition 19 drop the assumption dim Proj(B) = 1 and maintain the other assumptions, we still get that q is smooth and that p is an isomorphism onto an open subscheme (but the irreducibility of q^{−1}(W) may fail).

Now we consider an example of several components of GradAlg(H_{A}), which one may, as in [43], distinguish by the incomparability of the set of graded Betti numbers (Remark 7). Applying, however, Proposition 19 to our example we can also describe as well the graded Betti numbers of some algebras in the intersection of the two components. Below ΔH is the first difference of H, i.e. ΔH(v) = H(v)−H(v −1) for any integer v, in which case ΔH_{A} is of the form (1, c_{1}, c_{2}, ..., c_{t}, 0, 0, ...) (with c_{t} ≠ 0) if A is one-dimensional. In this case we often write ΔH_{A} as

ΔH_{A} = (1, c_{1}, c_{2}, ..., c_{t}) ,

i.e. as the so-called h-vector of A. In the Artinian case, the h-vector of A coincides with the Hilbert function H_{A} provided we write H_{A} in the form; H_{A} = (1, h_{1}, h_{2}, ..., h_{t}), with h_{t} ≠ 0 and h_{j} = 0 for j > t.

Example 21. In [50] C. Walter gives examples of infinitely many Hilbert schemes of space curves containing obstructed smooth curves of maximal rank. Indeed his example of a smooth space curve Y of the lowest degree (i.e. the curve with Hilbert polynomial p(x) = 33x−116 which we consider below) was independently discovered by Bolondi et al [6] and it was the first example of an obstructed curve of maximal rank which was detected. In [6] we showed that Hilb^{33x−116}(ℙ^{3}) contains at least two irreducible components whose intersection contains (Y). Since the curve Y = Proj(B) is of maximal rank, we have _{0}Hom_{R}(I_{B}, H_{𝔪}^{1}(B)) = 0, and Proposition 8 applies. It follows that the corresponding algebra B is obstructed as a graded algebra since Hilb^{33x−116}(ℙ^{3}) is not smooth at (Y). Indeed (B) sits in the intersection of two irreducible components W_{1} and W_{2}, both of dimension 4d = 132, of the postulation Hilbert scheme of space curves GradAlg(H_{B}), cf. [35], ex. 35.

In [35] we also considered the minimal resolution of B as well as the minimal resolution of the general elements B_{1} and B_{2} of W_{1} and W_{2} respectively. Indeed

0 → G_{3} = R(−9) → R(−10)^{2} ⊕ R(−9) ⊕ R(−8)^{4} → R(−9) ⊕ R(−8) ⊕ R(−7)^{5} → I_{B} → 0

is exact and we get the minimal resolution of B_{1} (resp. B_{2}) by making the factor R(−9) redundant in two different ways, i.e. by removing this factor from the leftmost (G_{3}) and the middle term (G_{2}), (resp. from G_{2} and rightmost term G_{1}). The Castelnuovo-Mumford regularity for all three curves satisfies reg(I) = 9, and the Hilbert function of all algebras is

(1, 4, 10, 20, 35, 56, 84, 115, 148, 181, 214, ...) .

Thus taking s ≥ 214 points X = Proj(A) on Y in general position and correspondingly for the others, then Proposition 19 applies with j ≥ 11. Or more precisely, both W_{1} and W_{2} and its intersection essentially belong to Smc(H_{B})_{9} ⊂ GradAlg(H_{B}), and Proposition 19 applies to (every element Proj(B′) of) Smc(H_{B})_{9} and an s-dimensional linear space of choices of s generic points on Proj(B′). Hence for each s ≥ 214 it follows that X is in the intersection of two irreducible components V_{1} and V_{2} of the postulation Hilbert scheme GradAlg(H_{A}) of dimension dim V_{i} = 132+s for i = 1, 2. In the special case s = 214 we have ΔH_{A} = (1, 3, 6, 10, 15, 21, 28, 31, 33, 33, 33, 0, ...), and it is not difficult to see that the minimal resolution of I_{A} is

0 → R(−13)^{33} ⊕ G_{3} → R(−12)^{66} ⊕ G_{2} → R(−11)^{33} ⊕ G_{1} → I_{A} → 0.

and that the minimal resolutions of the corresponding general elements Proj(A_{i}) of V_{i} are obtained by removing the free factor R(−9) from G_{3} and G_{2} (resp. from G_{2} and G_{1}). Looking to the corresponding sets of graded Betti numbers of A_{1} and A_{2} we see that they are incomparable.

We finish this section by recalling some known results about the postulation Hilbert scheme GradAlg^{H}(R), consisting of zero-dimensional schemes Proj(A) of degree s. Since we have observed that _{0}H^{2}(R, A, A) = 0 implies the smoothness of GradAlg^{H}(R) at (A), we remark that the smoothness results of Remark 22(i) (when A is generically a CI) and of Remark 22(ii) also follow from works of Herzog, Buchweitz-Ulrich and Huneke ([21], [8] and [22]). Moreover for Remark 22(iii) we remark that Buchweitz’s thesis [7], or [8], show that a generically CI licci quotient is unobstructed. Now in addition to Theorem 12 and Proposition 19, we have

Remark 22. (i) If Proj(R) = ℙ^{2}, then Gotzmann ([15]) shows that GradAlg^{H}(R) is irreducible and he finds its dimension ([26], Thm. 5.21 and Thm. 5.51). It is smooth by licciness and say (iii) below (or by [15] provided GradAlg^{H}(R) is reduced). As indicated by Iarrobino-Kanev ([26], Remark to Thm. 5.51), the dimension formula given in [33], Rem. 4.4, holds in this case ([33], Rem. 4.6).

(ii) If Proj(R) = ℙ^{3}, then the the open part of GradAlg^{H}(R) consisting of Gorenstein quotients is irreducible (cf. [10]), of known dimension by ([31], Remark to Thm. 2.6) and smooth by say (iii) below. This dimension formula is included in [30], Thm. 2.3 with a proof (which also takes care of the Artinian case). [30], Prop. 3.1 contains a second “dual” dimension formula for the same parameter space.

(iii) Let Proj(R) = ℙ^{n} and let A and A′ be two graded CM quotients algebraically linked by a CI B of type (a_{1}, ..., a_{m}) with resolution (4). By [30], Prop. 1.7, then A and A′ are simultaneously unobstructed as graded algebras, and we have

dim_{(A)} GradAlg^{H_{A}}(R) − ∑_{i=1}^{m} H_{A}(a_{i}) = dim_{(A′)} GradAlg^{H_{A′}} (R) − ∑_{i=1}^{m} H_{A′}(a_{i}) .

(iv) Let B = R/I_{B} be a graded, generically Gorenstein CM quotient with canonical module K_{B} and let A be the Gorenstein algebra given by a regular section of σ ∈ (K_{B}^{∗})_{t} for some integer t, i.e. given by a graded exact sequence 0 → K_{B}(−t) ⟶^{σ} B → A → 0.

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

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

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

b) If Proj(B) is locally Gorenstein and t >> 0, then A and B are simultaneously unobstructed as graded algebras, and the dimension formula of a) holds (with δ(B)_{−t} = 0).

This theorem is true in arbitrary dimension of B. It is proved in [32], Thm. 16 and is a substantial generalization of some of the statements of (ii) above because, when we apply it to a CM B of codimension two (necessarily licci), we get the dimension formula of (ii) by [32], Ex. 26. The preprint [34] contains further generalizations of this theorem.

By (iii) we see that CI-linkage preserves the smoothness of the parameter spaces. Due to [29], Prop. 3.4 it also preserves the irreducibility of the linked family. To define the linked family, let CICM(H_{B}, H_{A}) consist of points (B → A) such that B is CI and A is CM, i.e.

CICM(H_{B}, H_{A}) := p^{−1}(CM(H_{A})) ∩ q^{−1}(CI(H_{B}))

where p : GradAlg(H_{B}, H_{A}) → GradAlg(H_{A}) (resp. q) is the second (resp. first) projection morphism (e.g. q((B → A)) = (B)). In the case dim A = dim B (not necessarily equal to one) and (B → A) ∈ CICM(H_{B}, H_{A}), the linked algebra is defined by A′ := B/ Hom_{B}(A, B). This definition extends to families and preserves flatness [29]. Indeed by [29], Thm. 2.6 there is an isomorphism τ of schemes and obvious second projection morphisms p and p′ fitting into

τ : CICM(H_{B}, H_{A}) ⟶^{≃} CICM(H_{B}, H_{A′})
↓^{p}  ↓^{p′}
GradAlg(H_{A})  GradAlg(H_{A′})

(9)

where τ is given by sending (B_{1} → A_{1}) to (B_{1} → A′_{1} := B_{1}/ Hom_{B_{1}}(A_{1}, B_{1})).

Definition 23. Let the Hilbert polynomials p_{B} and p_{A} (corresponding to H_{B} and H_{A} respectively) have the same degree (≥ −1) and let U be a locally closed subset of im p in (9). Then the H_{B}-linked family of U is

U′ := p′(τ(p^{−1}(U))

Theorem 24. In (9) the morphisms p and p′ are smooth and their fibers are geometrically connected, of fiber dimension ϵ(A/B) at (B → A) and ϵ(A′/B) at (B → A′) respectively. In particular the H_{B}- linked family U′ is irreducible (resp. open in GradAlg(H_{A′})) if and only if U is irreducible (resp. open in GradAlg(H_{A})).

Proof. The proof of [30], Prop. 1.7 takes care of the smoothness of p and p′ and their fiber dimension. It remains to prove the connectedness of the fibers since the other conclusions then follow easily. The connectedness is, however, a straightforward consequence of the proof of Theorem 1.16 of [29] (that part of the proof doesn’t require deg p > 0 and it is easy to reformulate it for the Artinian case as well), cf. [41], Ch. VII for similar results.

Of course Theorem 24 implies Remark 22(iii) above. We may use Theorem 24 to see that many other properties are by preserved by linkage. Indeed subsets of GradAlg(H) for which the members allow the same sequence of CI-linkages which ends in a CI, is irreducible. It does not mean that the subset of GradAlg(H) of licci quotients is irreducible, as the following example shows.

Example 25. We claim GradAlg^{H}(k[x, y, z, w]) with ΔH = (1, 3, 6, 6, 3, 1) contains (at least) two irreducible components whose general elements are licci. Of course, the general element of one of the components is an arithmetically Gorenstein scheme consisting of 20 points, with minimal resolution

0 → R(−8) → R(−5)^{4} ⊕ R(−4) → R(−4) ⊕ R(−3)^{4} → R → A_{1} → 0.

We get A_{1} by starting with a CI of type (1, 1, 2) and then perform general CI-linkages of type (2, 3, 3) and (4, 3, 3). It follows that the component is generically smooth of dimension 44 by using Remark 22(iii), or Theorem 24, twice.

To get the other component, we start with a point Proj(A), i.e. a CI of type (1, 1, 1), and we proceed by performing six general CI linkages of type (1, 2, 3), (2, 2, 4), (2, 3, 4), (3, 4, 4), (3, 4, 5), (3, 3, 5), in this order. (The first five linkages are the same as for the level algebra 64] in the appendix C of [14]; hence GradAlg(H) with ΔH = (1, 3, 6, 7, 6, 2) also contains two “licci” components. By experimenting with Macaulay 2 we have learned that the three linkages; (2, 3, 4), (3, 4, 4), (3, 4, 5) above may be reduced to one single linkage; (3, 3, 4)). We get in this way an open subset U of GradAlg(H) of algebras A_{2} with minimal resolution

0 → R(−8) ⊕ R(−7) ⊕ R(−6) → R(−7) ⊕ R(−6) ⊕ R(−5)^{5} → R(−5) ⊕ R(−3)^{4} → R → A_{2} → 0.

Since the Betti numbers do not coincide with the general element A_{1} of the other component, the closure of U must be a generically smooth component of dimension 44 by Theorem 24.

(This example holds correspondingly for codimension 3 quotients with h-vector (1, 3, 6, 6, 3, 1) in a polynomial ring of any dimension. For the Artinian case, see Example 41).

3 Families of Artinian R-quotients (possibly Gorenstein).

In this section we look to families of Artinian algebras A of Hilbert function H = H_{A}, i.e. we study the scheme GradAlg(H) in the Artinian case with a special look to level and Gorenstein Artinian quotients. In particular we give examples of codimension 4 (resp. 3) quotients where GradAlg(H) has at least two components with a Gorenstein (resp. level) algebra belonging to the intersection of the two components. Moreover we notice that almost all the results of the preceding section (cf. Remark 22) are known in the Artinian case, except Theorem 12 and Proposition 19, whose corresponding Artinian counterparts are the main new results of this section (Theorem 29 and Proposition 33). Of course there are a few changes to Remark 22, mostly concerned with references, and we include some further results. To summarize,

Remark 26. (i) Iarrobino shows that GradAlg^{H}(k[x, y]) is irreducible ([23], Thm. 2.9) and he finds the dimension ([23], Thm. 2.12 and Thm. 3.13). It is smooth by licciness (or by [23], Thm. 2.9 provided GradAlg^{H}(k[x, y]) is reduced). Also in this case, the dimension formula given in [33], Rem. 4.4, holds (by the indicated argument of [33], Rem. 4.6).

(ii) If R = k[x, y, z], then the open part of GradAlg^{H}(R) consisting of Gorenstein quotients is irreducible ([10]) and smooth of known dimension ([30], Thm. 2.3). See also [22], Cor 4.9 for the smoothness.

(iii) of Remark 22 holds as stated in Remark 22.

(iv) of Remark 22 holds as well. One may make a little progress to (iv,b) by stating it as:

b) If Proj(B) is a locally Gorenstein zero-scheme of degree s and if t ≥ 2 reg(I_{B}), then A and B are simultaneously unobstructed as graded algebras, and the dimension formula of (iv, a) holds (with δ(B)_{−t} = 0 and dim(K_{B}^{∗})_{t} = s, cf. [32], Rem. 22). We formulate this using the ideas of Proposition 19 as Theorem 27 below ([32], Prop. 23, cf. [34] for further generalizations).

(v) One may, via the Macaulay correspondence, consider the set P S(s, j, n) of Gorenstein quotients obtained from the set of homogeneous polynomials of degree j in the “dual” polynomial ring, of the form

f = L_{1}^{j} + ... + L_{s}^{j} ,

where L_{i} are general enough linear forms and s is fixed. If H_{A} (which we denote by H(s, j, n)) contains a subsequence of the form (s, s, s), then the closure of P S(s, j, n) in GradAlg^{H_{A}}(R) determines by Macaulay duality a generically smooth irreducible component of GradAlg^{H_{A}}(R) of known dimension ([26], Thm. 4.10A and Thm. 1.61, see Thm. 4.13 for similar results when H_{A} does not contain such a subsequence).

(vi) In the interesting Gorenstein Artinian codimension 4 case, there is a structure theorem when H_{A} = (1, 4, 7, h, i, ...) with 3h−i−17 ≥ 0, allowing us to describe well the corresponding (generically smooth) irreducible component of GradAlg^{H_{A}}(R) ([26]). In [37] Johannes Kleppe comes up with classes of generically smooth components of known dimension of a similar nature.

(vii) Compressed Artinian algebras of fixed socle degrees belong to an irreducible generically smooth component of known dimension by [24], Thm. IIB.

To accomplish Remark 26(iv,b), let U_{t} ⊂ GradAlg(H′) be an open subscheme consisting of points (B) such that B is CM and such that Proj(B) is a locally Gorenstein zero-scheme of degree s satisfying H_{B} = H′ and reg(I_{B}) ≤ t/2. Recall that a regular section of σ ∈ (K_{B}^{∗})_{t} defines a graded Gorenstein algebra A given by the exact sequence 0 → K_{B}(−t) ⟶^{σ} B → A → 0. Let q : GradAlg(H_{B}, H_{A}) → GradAlg(H_{B}) be the first projection and let q^{−1}(U_{t})_{reg} be the intersection of q^{−1}(U_{t}) by the space of those quotients (B → A) which correspond to regular sections of (K_{B}^{∗})_{t}. Then we have a diagram where we have restricted the two natural projection morphisms q and p : GradAlg(H_{B}, H_{A}) → GradAlg(H_{A}) to q^{−1}(U_{t})_{reg}:

q^{−1}(U_{t})_{reg}  ⟶^{q_{res}} U_{t} ⊂ GradAlg(H_{B})
↓^{p_{res}}
GradAlg(H_{A})

(10)

Theorem 27. With notations as above, then

(i) q_{res} : q^{−1}(U_{t})_{reg} → U_{t} is smooth and surjective, and its fibers are geometrically connected of fiber dimension s − 1, and

(ii) p_{res} is an isomorphism onto an open subscheme of GradAlg(H_{A}). In particular the correspondence (10) determines a well-defined injective application π from the set of irreducible components W of U_{t}, to the set of irreducible components V of GradAlg(H_{A}), in which generically smooth components correspond. Indeed V = π(W) is the closure of p_{res}(q_{res}^{−1}(W)), and we have

dim V = dim W + s − 1 .

Proof (also of Remark 26(iv,b)). These results are almost exactly the second part of Thm. 16 (cf. Rem. 22) and Thm. 24 (cf. Prop. 23) of [32] with a slight improvement for the case dim B = 1. Indeed in replacing “t >> 0” by “t ≥ 2 reg(I_{B})” we assumed in [32], Rem. 22 and Prop. 23(iii) that B was generically syzygetic (e.g. Proj(B) locally licci). This was needed in Prop. 23(iii) to show that p_{res} was smooth. Since, however, K_{B}(−t) ≃ I_{A/B} and the R-dual of (4) is a free resolution of K_{B}(1), we have (I_{A/B})_{v} = 0 for v ≥ t + 1 − reg(I_{B}). Hence we may use Lemma 9 and Proposition 10(ii) of this paper to see that p_{res} is smooth under the assumption t ≥ 2 reg(I_{B}) without requiring B to be generically syzygetic. Note also that q_{res} is surjective by [4], Thm. 3.2, (cf. [32], Rem. 22). Since the remaining part of Theorem 27 was proved in [32] (Prop. 23, Thm. 24, cf. Rem. 25(a)), we get the theorem. Moreover note that Remark 26(iv,b) follows from Theorem 27 since we may get δ(B)_{−t} = 0 and dim(K_{B}^{∗})_{t} = s from Proposition 10 (or by directly using [32], Rem. 14(a) and the first part of Rem. 22). Hence also in Remark 26(iv,b) it suffices to suppose t ≥ 2 reg(I_{B}) without requiring B to be generically syzygetic.

Now we illustrate Theorem 27. The benefit of using Theorem 27 instead of Remark 26(iv,b) is clear because it is a statement about the whole subscheme U_{t} ⊂ GradAlg(H′) and not only about a point in U_{t}. E.g. note that if we apply (iv,b) to the two components of GradAlg(H′) of Example 17, say with s = 13 and t ≥ 10 to simplify, we get two components of PGor(H), or of GradAlg(H), with

H = (1, 4, 7, 10, 13, 13, ..., 13, 10, 7, 4, 1)

of dimension 37 and 39 where the number 13 occurs t − 7 times. The existence of such components for PGor(H) is now well known ([3], see also [27]). Since, however, GradAlg(H) is connected there are graded Artinian quotients belonging to the intersection of the components of GradAlg(H). But are there Gorenstein quotients in this intersection? The answer would be yes if the intersection of the two components of GradAlg(H′) of Example 17 contains points (B) such that Proj(B) is a locally Gorenstein zero-scheme because we then could apply Theorem 27! We doubt that there exists such a quotient B, i.e. we expect that the intersection of the two mentioned components of PGor(H) is empty (cf. Piene-Schlessinger’s characterization of the intersection of the two components described in Example 34). Here is an example where we somehow control the intersection.

Example 28. (Two components of PGor(H) with non-empty intersection)

In Example 21 we showed the existence of an algebra, which we now call B whose corresponding point (B) of the postulation Hilbert scheme, GradAlg(H_{B}), sat in the intersection of two irreducible components V_{1} and V_{2} of GradAlg(H_{B}) of dimension dim V_{i} = 132 + s = 346 for i = 1, 2. The element (B) as well as the two general elements (B_{i}) of V_{i} were obtained by taking s = 214 generic points on certain curves of Hilb^{33x−116}(ℙ^{3}). One of the curves had minimal resolution

0 → G_{3} = R(−9) → G_{2} = R(−10)^{2} ⊕ R(−9) ⊕ R(−8)^{4} → G_{1} = R(−9) ⊕ R(−8) ⊕ R(−7)^{5} → I

Moreover H_{B} = H_{B_{i}} = (1, 4, 10, 20, 35, 56, 84, 115, 148, 181, 214, 214, 214, ...) and the minimal resolution of I_{B} (resp. of I_{B_{1}}, or I_{B_{2}}) was

0 → R(−13)^{33} ⊕ G_{3} → R(−12)^{66} ⊕ G_{2} → R(−11)^{33} ⊕ G_{1} → I_{B} → 0  (11)

(resp. (11) in which the factor R(−9) from G_{3} and G_{2}, or from G_{2} and G_{1}, were removed).

Since we have reg(I_{B}) = reg(I_{B_{i}}) = 11, we may use Theorem 27 to get, for every t ≥ 22, two generically smooth irreducible components of PGor(H_{A}) of dimension 132 + s + s − 1 = 559 whose intersection is non-empty, i.e. the intersection contains an obstructed Gorenstein Artinian algebra whose h-vector is the (t + 1)-tuple

H_{A} = (1, 4, 10, 20, 35, 56, 84, 115, 148, 181, 214, 214, ..., 214, 181, 148, ..., 4, 1)

where the number 214 occurs t − 19 times. The corresponding sets of graded Betti numbers of the general elements, A_{1} and A_{2}, of the two components turn out to be incomparable because the factors R(−9) (and R(−t + 5)) appearing in the resolution of I_{A} become redundant in different ways in the resolutions I_{A_{1}} and I_{A_{2}}. Of course, for every s ≥ 214 we can construct similar examples.

Now we prove the analogue of Theorem 12, which is the main result of this section.

Theorem 29. Let R be a polynomial k-algebra and let B = R/I_{B} → A = R/I_{A} be a graded morphism such that A is Artinian and depth_{𝔪} B ≥ min(1, dim B), and suppose either

(a) I_{B} is generated by a regular sequence (allowing R = B), or

(b) B_{v} → A_{v} is an isomorphism for all v ≤ max_{i}{n_{2,i}} and dim R − dim B ≥ 2. Let F be a free B-module such that F → I_{A/B} is surjective and minimal, and suppose there is an integer j such that the degrees of minimal generators of the B-module ker(F → I_{A/B}) are strictly greater than j (e.g. B_{v} ≃ A_{v} for all v ≤ j − 1) and such that I_{A} is (j + 1)-regular (i.e. A_{j+1} = 0). Then A is H_{B}-generic, dim(N_{A})_{0} = dim(N_{B})_{0} + _{0} hom_{B}(F, A) − ϵ(A/B) , and

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

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

At least in the case depth_{𝔪} B ≥ 1, the natural application of Theorem 29 is the same as for Theorem 12; the minimal resolution of A should be the one of B in addition to a semi-linear contribution coming from I_{A/B} via the mapping cone construction, cf. Remark 32.

Proof. All we need to prove is the two dimension formulas. Due to Corollary 11 it suffices to show 0H^{2}(R, B, I_{A/B}) = 0 and _{0}hom_{R}(I_{B}, I_{A/B}) = ϵ(A/B) together with

dim _{0}Hom_{B}(I_{A/B}, A) = _{0}hom_{B}(F, A) and _{0}Ext_{B}^{1}(I_{A/B}, A) = 0 ,  (12)

because the latter of (12) implies _{0}H^{2}(B, A, A) = 0. By Lemma 9 it suffices to prove (12). Let

F′ → F → I_{A/B} → 0  (13)

be the first terms of a B-free minimal resolution of I_{A/B}. Applying _{0}Hom_{B}(−, A) = 0 onto (13) and using A_{j+1} = 0, we get _{0}Hom_{B}(I_{A/B}, A) ≃ _{0}Hom_{B}(F, A) and _{0}Ext_{B}^{1}(I_{A/B}, A) = 0 by the assumption, and we are done.

Remark 30. By the long exact sequence of algebra cohomology, we have the exact sequence

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

Since it is well known that H^{2}(R, B, −) = 0 if Theorem 29(a) holds and since we have _{0}H^{2}(B, A, A) = 0 by the proof above, it follows that we in Theorem 29(a) have

0H^{2}(R, A, A) = 0 .

Remark 31. A natural choice of j in Theorem 29 such that (I_{A/B})_{j−1} = 0 and such that (b) holds, is j ≥ reg(I_{B})+2, in which case we have that I_{A} is (j +1)-regular iff I_{A/B} is (j +1)-regular, and that H_{A}(x) = H_{B}(x) = p_{B}(x) for x ≥ j − 3, cf. Remark 15. Since it then follows that B ≃ A provided B is Artinian, the (only) real application of Theorem 29(b) seems to be in the case depth_{𝔪} B ≥ 1. It is, however, natural to use Theorem 29(a) also when depth_{𝔪} B = 0.

Remark 32. Suppose depth_{𝔪} B ≥ 1 and that I_{A/B} is (j + 1)-regular, and look to

0 → I_{A/B} → B → A → 0 .  (14)

Since H_{𝔪}^{0}(I_{A/B}) = 0, we have depth_{𝔪} I_{A/B} ≥ 1, i.e. pd_{R}(I_{A/B}) ≤ n − 1 and in fact pd_{R}(I_{A/B}) = n − 1 since pd(A) = n. A mapping cone construction applied to (14) in which we use the minimal resolutions of I_{A/B} and B, leads easily to an R-free resolution of A. Moreover if I_{A/B} admits a semi-linear resolution, then (I_{A/B})_{j−1} = 0, and conversely provided reg(I_{A/B}) = j + 1. Note that A becomes a level algebra if I_{A/B} admits a linear resolution. In particular, the natural application of Theorem 29(b) is the same as for Theorem 12, cf. Remark 13, i.e. the minimal resolution of A should be the one of B in addition to a semi-linear contribution coming from I_{A/B} via the mapping cone construction.

Theorem 29 applies nicely to Artinian truncations and more generally to Artinian quotients A with Hilbert function H_{A} = (1, h_{1}, h_{2}, ..., h_{j−1}, α, 0, 0, ..) where H_{B} = (1, h_{1}, h_{2}, ..., h_{j−1}, h_{j}, h_{j+1}, ...) is the Hilbert function of B and α ≤ h_{j}. To see it let, in a very similar way to what we did in Definition 18 and Proposition 19, GradAlg(H)_{η} (resp. GradAlg(H_{B}, H_{A})_{η}) be the open subset of GradAlg(H) consisting of (R/I) (resp. (B = R/I → A)) where the Castelnuovo-Mumford regularity satisfies reg(I) ≤ η. Then we have a diagram as in (8) where we now restrict the natural projection morphism q : GradAlg(H_{B}, H_{A}) → GradAlg(H_{B}) and p to GradAlg(H_{B}, H_{A})_{η} ;

GradAlg(H_{B}, H_{A})_{η} ⟶^{q} GradAlg(H_{B})_{η} ⊂ GradAlg(H_{B})
↓^{p}
GradAlg(H_{A})

(15)

Below we restrict to the case B ≠ R, even though the proposition holds for B = R with the following small changes; skip the lower indices η in (15) and drop the assumption η ≤ j − 2 below, and note that GradAlg(H_{R}) is a smooth scheme consisting of one point. Recalling that an Artinian quotient A has the Weak Lefschetz property if the multiplication map A_{v} ⟶^{· l} A_{v+1} by a general linear form l is either injective or surjective for every v, we have

Proposition 33. Let H_{B} = (1, h_{1}, h_{2}, ...) be the Hilbert function of an algebra B ≠ R satisfying depth_{𝔪} B ≥ 1 and let j, η ≤ j − 2 and α ≤ h_{j} be non-negative integers. Let H_{A} = (1, h_{1}, h_{2}, ..., h_{j−1}, α, 0, 0, ..) and look to the maps p and q in (15). Then

(i) q is smooth and surjective with geometrically connected fibers, of fiber dimension α(h_{j} − α), and

(ii) p is an isomorphism onto an open subscheme of GradAlg(H_{A}). In particular the incidence correspondence (15) determines a well-defined injective application π from the set of irreducible components W of GradAlg(H_{B})_{η}, to the set of irreducible components V of GradAlg(H_{A}) whose general elements satisfy the Weak Lefschetz property. In this application the generically smooth components correspond. Indeed V = π(W) is the closure of p(q^{−1}(W)), and we have

dim V = dim W + α(h_{j} − α) .

Proof. (i) To any point (B′) of GradAlg(H_{B})_{η}, let A′ := ⊕_{i=0}^{j−1}B′_{i} ⊕ V_{j} where V_{j} is an α-dimensional quotient of B′_{j}. This shows that q is surjective. Moreover we get the smoothness of q from Proposition 9(i) since _{0}H^{2}(B, A, A) = 0 by the proof of Theorem 29. To show that the fibers of q are (geometrically) connected, one may look upon the fiber as the Grassmannian of α-dimensional quotients of B′_{j}. Since the Grassmannian is irreducible, we conclude easily. (ii) Since the proof of the Weak Lefschetz property is standard (depth_{𝔪} B ≥ 1), the proof is the same as for (ii) of Proposition 19.

We will call an Artinian algebra A with Hilbert function H_{A} as in Proposition 33 with α = 0 an Artinian truncation in degree j. Moreover, by Remark 31, we normally need j ≥ reg(I_{B}) + 2 for some B to use Proposition 33 with GradAlg(H_{B})_{j−2} non-empty. Having several irreducible components with a non-empty intersection in GradAlg(H_{B})_{η}, we get exactly the same type of irreducible components with a non-empty intersection for their Artinian truncations in a fixed degree j (for every j ≥ η + 2) in GradAlg(H_{A}) (for instance, the B and the components given by the B_{i} of Example 21, we leave the details to the reader). We finish this section by another example.

Example 34. (obstructed Artinian level algebra with Hilbert function (1, 4, 7, 10, 13, 0, 0, ...)).

We have seen that Y_{1} = Proj(B_{1}) ⊂ ℙ^{3}, a twisted cubic curve and Y_{2} = Proj(B_{2}), a union of a plane space curve C of degree 3 and a point P outside the plane containing C, correspond to two different irreducible components of the stratum GradAlg(H) of the Hilbert scheme Hilb^{3x+1}(ℙ^{3}) where H = (1, 4, 7, 10, 13, ...). Indeed (Y_{1}) belongs to a 12-dimensional irreducible component of GradAlg(H) while (Y_{2}) belongs to a 15-dimensional irreducible component of GradAlg(H). Using Piene-Schlessinger’s Theorem from [48] to see a complete description of Hilb^{3x+1}(ℙ^{3}), we also get that the general element of the intersection of the two components (which is 11-dimensional) is a curve Y = Proj(B) with an embedded point. The minimal resolution of I = I_{B} or I_{B_{2}} (resp. I_{B_{1}}) are of the form

0 → R(−4) → R(−4) ⊕ R(−3)^{3} → R(−3) ⊕ R(−2)^{3} → I → 0  (16)

(resp. of the form (16) where both R(−4) and two of R(−3) are removed). Hence the regularity of I_{B} and I_{B_{i}} for i = 1 and 2 is at most 3, i.e. the two components and its intersection essentially belong to GradAlg(H)_{3}. Applying Proposition 33 for j ≥ 5 and η = 3 and to any α ≤ 3j + 1, we get two irreducible components V_{i} of GradAlg(H_{A}) with a well described non-empty intersection. Indeed let X_{1} = Proj(A_{1}) and X_{2} = Proj(A_{2}) (resp. X = Proj(A)) be obtained by modding out by h_{j} − α linearly independent forms of (B_{1})_{j} and (B_{2})_{j} (resp. B_{j}) and all forms of degree j + 1. It follows that A_{i} are unobstructed as graded R-algebras for i = 1 and 2 and that dim_{(A_{1})} GradAlg(H′) = 12 + α(h_{j} −α) and dim_{(A_{2})} GradAlg(H′) = 15+α(h_{j} −α) where H′ = (1, 4, 7, ...3j −5, 3j −2, α, 0, 0, ...). Moreover (A) is a singular point of GradAlg(H′)and belongs to the 11 + α(h_{j} − α)-dimensional intersection of the components. Finally if α = 0 and j = 5 it is straightforward to see that the free terms of a minimal resolution of A_{2} (and A) are

0 → R(−8)^{13} → R(−7)^{42} ⊕ R(−4) → R(−6)^{45} ⊕ R(−4) ⊕ R(−3)^{3} → R(−5)^{16} ⊕ R(−3) ⊕ R(−2)^{3}

4 Tangent and obstruction spaces of Artinian families.

In this section we consider graded Artinian algebras with a special look to level quotients of k[x, y, z]. Note that, in most cases, results such as Theorem 29, Proposition 33 and Remark 26 do not apply because their assumptions limit their applications considerably. We can, however, still analyze GradAlg^{H}(R) at a point (A) infinitesimally by means of its tangent and obstruction spaces and a certain obstruction morphism, cf. [38]. In the following we make these spaces more explicit by duality (Theorem 36), and since we show that the parameter space of level schemes, L( H), of [9] is sufficiently close to being an open subscheme of GradAlg^{H}(R) (cf. Theorem 44), we can use our results to study L( H). In particular we study in detail the level type 2 algebras which correspond to a pencil of forms by apolarity ([25]), and we prove in Example 49 a conjecture of Iarrobino on the existence of several irreducible components of L( H) when H = (1, 3, 6, 10, 14, 10, 6, 2).

Indeed inside GradAlg^{H}(R) there is an open set, possibly empty, consisting of graded Artinian Gorenstein quotients R → A with Hilbert function H (which essentially is the scheme PGor(H), see [32]). An elementary way of finding the obstruction space of PGor(H) is to compute the kernel of the natural surjection

η_{j} : (S_{2}I_{A})_{j} → (I_{A}^{2})_{j}

from the second symmetric power to the second power of I_{A} in the socle degree j of A. Indeed, up to duality, this kernel is isomorphic to _{0}H^{2}(R, A, A), the obstruction space of PGor(H). To generalize this result to any Artinian A, we remark that ker η_{j} is isomorphic to the cokernel of the natural morphism (Λ_{2}I_{A})_{j} → Tor_{2}^{R}(A, A)_{j} (at least if char(k) ≠ 2). This formulation allows a generalization to any Artinian A. Indeed there is a special product, given as an antisymmetrization map ([1], Prop. 24.1),

Tor_{1}^{R}(A, A) ⊗_{A} Tor_{1}^{R}(A, K_{A}) → Tor_{2}^{R}(A, K_{A})  (17)

with cokernel H_{2}(R, A, K_{A}). Up to duality we will show that the zero degree piece of this cokernel is the obstruction space of GradAlg^{H}(R) at (A). To prove it we need a variation of the following spectral sequence

Ext_{A}^{p}(H_{q}(R, A, A), M) ⟹ H^{p+q}(R, A, M)

(cf. [1], Prop. 16.1). Keeping also the spectral sequence ([21], Satz 1.2)

Ext_{C}^{p}(Tor_{q}^{A}(M, K_{C}), K_{C}) ⟹ Ext_{A}^{p+q}(M, C)  (18)

in mind (C a CM quotient of A with canonical module K_{C}), the following result is not surprising

Proposition 35. If B → A → C are quotients of R of arbitrary dimension and if C is CM with canonical module K_{C}, then there is a spectral sequence converging to H^{∗}(B, A, C) ;

′E_{2}^{p,q} := Ext_{C}^{q}(H_{p}(B, A, K_{C}), K_{C}) ⟹ H^{p+q}(B, A, C) .  (19)

In particular if C is a graded Artinian algebra, then there is a degree-preserving isomorphism

Hom_{C}(H_{q}(B, A, K_{C}), K_{C}) ≃ H^{q}(B, A, C) .

Proof. One knows that Hom_{A}(M, C) ≃ Hom_{C}(M ⊗_{A} K_{C}, K_{C}), M an A-module. Using this we can prove our proposition in the usual way, i.e. by considering the double complex

K_{∗,∗} = Hom_{C}(Diff(B, A_{∗}, A) ⊗_{A} K_{C}, I_{∗})

where 0 → K_{C} → I_{∗} is an injective resolution of the C-module K_{C} and Diff(B, A_{∗}, A) := Ω_{A_{∗}/B} ⊗_{A_{∗}} A is the complex of Kähler differentials based on a simplicial resolution, A_{∗}, of the B-algebra A (as in [1], Prop. 17.1, so each A_{i} is a polynomial ring over B). If we in ′′E_{2}^{p,q} first take homology of K_{∗,∗} with respect to the second variable (i.e. I_{∗}), we get ′′E_{2}^{p,0} = H^{p}(B, A, C) and ′′E_{2}^{p,q} = 0 for q > 0 because Ext_{C}^{q}(K_{C}, K_{C}) = 0 for q > 0 by Cohen-Macaulayness and the fact that each Diff(B, A_{i}, A) ⊗_{A} C is C-free. Calculating ′E_{2}^{p,q} by reversing the order, i.e. by first taking homology with respect to the first variable, we get (19). Finally since K_{C} is an injective C-module in the Artinian case, we are done.

Theorem 36. Let R → A = R/I_{A} be a graded Artinian quotient with Hilbert function H. Then the dual of (I_{A} ⊗_{R} K_{A})_{0} is the tangent space of GradAlg^{H}(R) at (A), and the dual of _{0}H_{2}(R, A, K_{A}) contains the obstructions to deforming A as a graded R-algebra. Moreover

dim (I_{A} ⊗_{R} K_{A})_{0} − _{0}h_{2}(R, A, K_{A}) ≤ dim_{(A)}GradAlg(H) ≤ dim (I_{A} ⊗_{R} K_{A})_{0}.

In particular GradAlg^{H}(R) is smooth at (A) provided the natural map

I_{A} ⊗_{R} I_{A} ⊗_{R} K_{A} → Tor_{1}^{R}(I_{A}, K_{A})

of (17), i.e. the map ζ concretely described in (20) below, is surjective in degree zero.

Proof. Since it is known that the tangent (resp. “obstruction”) space of GradAlg(H_{A}) at (A) is 0H^{1}(R, A, A) = _{0}Hom_{A}(I_{A}/I_{A}^{2}, A) ≃ _{0}Hom_{R}(I_{A}, A) (resp. _{0}H^{2}(R, A, A)) by [28], Thm. 1.5, we get the description in Theorem 36 of these spaces by Proposition 35. Then the left inequality of the dimension formula follows rather easily from [38], Thm. 4.2.4 while the right inequality is trivial. Hence we get all conclusions of the theorem once we have shown that the surjectivity in (17) and the surjectivity of ζ in (20) are equivalent. Indeed Tor_{2}^{R}(A, K_{A}) ≃ Tor_{1}^{R}(I_{A}, K_{A}) and Tor_{1}^{R}(A, K_{A}) ≃ I_{A} ⊗_{R} K_{A} and the map of (17) is just the natural map ζ : I_{A} ⊗_{R} I_{A} ⊗_{R} K_{A} → Tor_{1}^{R}(I_{A}, K_{A}) uniquely described in the following way. Let 0 → N → F → K_{A} → 0 be a short exact sequence where F is A-free. Applying I_{A} ⊗_{R} (−) onto this sequence we get an injection Tor_{1}^{R}(I_{A}, K_{A}) ↪ I_{A} ⊗ N which together with the surjection F ↠ K_{A} lead to the composition

I_{A} ⊗_{R} I_{A} ⊗_{R} F ↠ I_{A} ⊗_{R} I_{A} ⊗_{R} K_{A} ⟶^{ζ} Tor_{1}^{R}(I_{A}, K_{A}) ↪ I_{A} ⊗_{R} N  (20)

given by x ⊗ y ⊗ ω → x ⊗ (yω) − y ⊗ (xω) ∈ I_{A} ⊗_{R} N (cf. [2], Prop. 9, p. 204 for details).

Remark 37. Let M be a graded A-module and let 0 → N → F → M → 0 be a graded exact sequence where F is A-free. Arguing as in the proof above, we see that _{v}H_{2}(R, A, M) is the homology in degree v of (21) below. Hence it vanishes if and only if the sequence

I_{A} ⊗_{R} I_{A} ⊗_{R} F ⟶^{λ} I_{A} ⊗_{R} N → I_{A} ⊗_{R} F ,  (21)

where λ(x ⊗ y ⊗ ω) = x ⊗ (yω) − y ⊗ (xω), is exact in degree v ([2], Prop. 9, p. 204).

Remark 38. Let A = R/I_{A} be a graded Artinian algebra and let M be a finitely generated R-module. Using (18) we get

Hom_{A}(Tor_{q}^{R}(M, K_{A}), K_{A}) ≃ Ext_{R}^{q}(M, A) .

Thus (I_{A} ⊗_{R} K_{A})_{v} (resp. _{v}Tor_{1}^{R}(I_{A}, K_{A})) is dual to _{−v}Hom_{R}(I_{A}, A) (resp. _{−v}Ext_{R}^{1}(I_{A}, A)) and the dual of the degree v part of (17) augmented by _{v}H_{2}(R, A, K_{A}) yields an exact sequence

−vH^{2}(R, A, A) ↪ _{−v}Ext_{R}^{1}(I_{A}, A) → _{−v}Hom_{R}(I_{A} ⊗_{R} I_{A}, A)

where the left injective map must be the right inclusion of (3) in degree −v.

In the codimension 3 case it turns out that _{−v}Ext_{R}^{1}(I_{A}, A) is also dual to _{v−3}Hom_{R}(I_{A}, A):

Proposition 39. Let R → A = R/I_{A}, R = k[x, y, z] be a graded Artinian quotient with Hilbert function H and minimal resolution

0 → ⊕_{i=1}^{r_{3}}R(−n_{3,i}) → ⊕_{i=1}^{r_{2}}R(−n_{2,i}) → ⊕_{i=1}^{r_{1}}R(−n_{1,i}) → R → A → 0 .  (22)

Then _{v}Ext_{R}^{i}(I_{A}, A) ≃ _{v}Tor_{1−i}^{R}(I_{A}, K_{A}(3)) for 0 ≤ i ≤ 1 and N_{A} := Hom_{R}(I_{A}, A) satisfies

dim(N_{A})_{v} − _{v}ext_{R}^{1}(I_{A}, A) = ∑_{j=1}^{3} ∑_{i=1}^{r_{j}}(−1)^{j−1}H(n_{j,i} + v) − H(−v − 3) .

Hence _{v}Ext_{R}^{1}(I_{A}, A) is dual to (N_{A})_{−v−3} for every v. Moreover if (N_{A})_{−3} = 0, then _{0}H^{2}(R, A, A) = 0 and GradAlg(H) is smooth at (A) of dimension ∑_{j=1}^{3} ∑_{i=1}^{r_{j}}(−1)^{j−1}H(n_{j,i}).

Proof. Applying _{0}Hom_{R}(−, R) to (22) we get an R-free resolution of K_{A}(3). Then we get the complex

0 → A → ⊕_{i=1}^{r_{1}}A(n_{1,i}) → ⊕_{i=1}^{r_{2}}A(n_{2,i}) → ⊕_{i=1}^{r_{1}}A(n_{3,i}) → K_{A}(3) → 0

by tensoring the resolution of K_{A}(3) by A. Note that the map A → ⊕_{i=1}^{r_{1}}A(n_{1,i}) is zero since its matrix is given by the generators of I_{A}. It follows that the homology groups of the complex

0 → ⊕_{i=1}^{r_{1}}A(n_{1,i}) → ⊕_{i=1}^{r_{2}}A(n_{2,i}) → ⊕_{i=1}^{r_{1}}A(n_{3,i}) → 0

are precisely Tor_{2−j}^{R}(A, K_{A}(3)) for 0 ≤ j ≤ 2 and that Tor_{0}^{R}(A, K_{A}(3)) ≃ K_{A}(3). Moreover by applying _{0}Hom_{R}(−, A) onto the minimal resolution of I_{A} deduced from (22), we get exactly the latter complex. Hence the homology groups of the complex are also Ext_{R}^{j}(I_{A}, A) for 0 ≤ j ≤ 2 by the definition of Ext. In particular _{v}Ext_{R}^{2}(I_{A}, A) ≃ K_{A}(3)_{v} and _{v}Ext_{R}^{j}(I_{A}, A) ≃ _{v}Tor_{2−j}^{R}(A, K_{A}(3)) ≃ vTor_{1−j}^{R}(I_{A}, K_{A}(3)) for 0 ≤ j ≤ 1 where the last isomorphism is easily proved by tensoring the exact sequence 0 → I_{A} → R → A → 0 by K_{A}(3). Since the alternating sum of the dimension of the terms in a complex equals the alternating sum of the dimension of its homology groups, we also get the double summation formula by combining with dim K_{A}(3)_{v} = dim A_{−v−3}.

Finally we know that (I_{A} ⊗ K_{A}(3))_{v} is dual to _{−v−3}Hom_{R}(I_{A}, A) = (N_{A})_{−v−3} by Remark 38 and that _{v}Ext_{R}^{1}(I_{A}, A) ≃ _{v}Tor_{0}^{R}(I_{A}, K_{A}(3)) by the first part of the proof. Hence _{v}Ext_{R}^{1}(I_{A}, A) is dual to (N_{A})_{−v−3} and since _{0}Ext_{R}^{1}(I_{A}, A) contains _{0}H^{2}(R, A, A) which is dual to _{0}H_{2}(R, A, K_{A}) by Remark 38 we conclude by Theorem 36.

Corollary 40. With A as in Proposition 39 we have

∑_{j=1}^{3} ∑_{i=1}^{r_{j}}(−1)^{j−1}H(n_{j,i}) = 1 − ∑_{j=1}^{3} ∑_{i=1}^{r_{j}}(−1)^{j−1}H(n_{j,i} − 3) .

Moreover the sum on the left above, which we call ρ(H), depends only upon the Hilbert function H and not upon the graded Betti numbers. We have

dim_{(A)} GradAlg^{H}(k[x, y, z]) ≥ ρ(H) .

Indeed ρ(H) is a lower bound for the dimension of any irreducible (non-embedded) component of GradAlg^{H}(k[x, y, z]).

Proof. The duality of the proposition shows that

dim(N_{A})_{v} − _{v}ext_{R}^{1}(I_{A}, A) = _{−v−3}ext_{R}^{1}(I_{A}, A) − dim(N_{A})_{−v−3} .

Putting v = 0 we get the equality of the two expressions of ρ(H) of the corollary by using the corresponding formula of Proposition 39 for v = 0 and v = −3. Moreover, by Theorem 36 and Remark 38, the number dim(N_{A})_{0} − dim _{0}H^{2}(R, A, A) is a lower bound of dim_{(A)} GradAlg^{H}(k[x, y, z]). Using Remark 38 we get ρ(H) to be a possibly smaller lower bound. Finally let V be a non-embedded irreducible component of GradAlg^{H}(k[x, y, z]) and let (A′) ∈ V such that dim_{(A′)} GradAlg^{H}(k[x, y, z]) = dim V. Note that the sum which defines ρ(H) depends only upon the Hilbert function because the contribution from all ghost terms (i.e. common direct free factors in consecutive terms in the minimal resolution of A) or from incomparable sets of graded Betti numbers (Remark 7) sums to zero! Hence dim V ≥ ρ(H) by the first part of the proof, we are done.

Example 41. To illustrate Proposition 39, we consider H = (1, 3, 6, 6, 3, 1) and the two different irreducible components (now of GradAlg^{H}(k[x, y, z])) of Example 25 whose general elements are licci. Looking to the minimal resolutions of A_{i} of Example 25, we get

dim(N_{A_{i}})_{0} − _{0}ext_{R}^{1}(I_{A_{i}}, A_{i}) = 4H(3) − 4H(5) = 20 .

By Remark 38, _{0}Ext_{R}^{1}(I_{A_{1}}, A_{1}) = 0 since we get _{0}Tor_{1}^{R}(I_{A_{1}}, K_{A_{1}}) ↪ (I_{A_{1}} ⊗ I_{A_{1}})_{5} = 0 by using K_{A_{1}} ≃ A_{1}(5). (Indeed this _{0}Ext^{1} -group always vanishes in the compressed Gorenstein case.) Thus the “Gorenstein” component has dimension 20 (also well known by [24]), while Remark 26(iii) or Theorem 24 applied to the successive linkages (1, 2, 3), (2, 2, 4), (2, 3, 4), (3, 4, 4), (3, 4, 5), (3, 3, 5) obtained from a CI of type (1, 1, 1), shows that the other component is generically smooth of dimension 21. Thus _{0}ext_{R}^{1}(I_{A_{2}}, A_{2}) = 1.

An alternative way of describing A_{2} is to specify the three generators, F_{1}, F_{2} and F_{3}, of degree 3, 4 and 5 respectively, in the dual polynomial algebra of R which we will consider more closely later in this section. Indeed if we take F_{1} to be a general polynomial of degree 3 (i.e. an element of some open set of the irreducible parameter space of all forms of degree 3), F_{2} to be a sum of length 4 of general linear forms to the 4-th power and F_{3} to be a sum of length 2 of general linear forms to the 5-th power, we get precisely A_{2} as A_{2} = R/ann(F_{1}, F_{2}, F_{3}) (verifyed by using Macaulay 2).

Recall that if A itself admits a semi-linear R-free resolution (except possibly at the minimal generators of I_{A}), then

0H^{2}(R, A, A) = 0  (23)

by Remark 30 and Remark 32. This vanishing also follows from Theorem 36. Moreover using Theorem 36, we can prove a “dual” result. Indeed suppose I_{A} admits a semi-linear resolution except possibly at the left end of the resolution, i.e. suppose I_{A} has minimal generators only in degree j and j + 1 and that the resolution continues by

0 → G ⊕ R(−j − n + 1)^{α_{n}} → R(−j − n + 1)^{β_{n−1}} ⊕ R(−j − n + 2)^{α_{n−1}} → ... → F_{1} → I_{A} (24)

where G is any R-free module. Here F_{1} = R(−j − 1)^{β_{1}} ⊕ R(−j)^{α_{1}} and R is the polynomial ring k[x_{1}, ..., x_{n}] with n ≥ 2. Then (23) holds. Using (18) we can even replace F_{1} by

F_{1} = R(−j − 1)^{β_{1}} ⊕ R(−j)^{α_{1}} ⊕ (⊕_{i=1}^{m}R(−a_{i})) , a_{i} < j for all i  (25)

where the set of generators {f_{1}, ..., f_{m}} which correspond to {a_{1}, ..., a_{m}} form a regular sequence, and still get (23), i.e.;

Proposition 42. Let A = R/I_{A} be a graded Artinian quotient with Hilbert function H, whose minimal resolution is given by (24) and (25) where the generators {f_{1}, ..., f_{m}} of I_{A} which correspond to {a_{1}, ..., a_{m}} form a regular sequence. Let B = R/(f_{1}, ..., f_{m}) (and B = R if m = 0). Then 0H^{2}(R, A, A) = 0 and GradAlg(H) is smooth at (A). Moreover

dim_{(A)}GradAlg(H) = _{−n}hom_{R}(G, B) − _{−n}hom_{R}(G, A) + ∑_{i=1}^{m} H(a_{i}).

Proof. By the long exact sequence of algebra cohomology (Remark 30) and (3) we get _{0}H^{2}(R, A, A) = 0 provided we can show Ext_{B}^{1}(I_{A/B}, A) = 0. Continuing the long exact sequence of Remark 30 to the left we see that Ext_{B}^{1}(I_{A/B}, A) = 0 also leads to

dim(N_{A})_{0} =hom_{B}(I_{A/B}, A) + hom_{B}(I_{B}/I_{B}^{2}, A) .  (26)

To show Ext_{B}^{1}(I_{A/B}, A) = 0, we improve a little bit upon Theorem 36 by using (18). Indeed we have Hom_{A}(Tor_{q}^{B}(I_{A/B}, K_{A}), K_{A}) ≃ Ext_{B}^{q}(I_{A/B}, A). Hence it suffices to show _{0}Tor_{1}^{R}(I_{A/B}, K_{A}) = 0. Now look to the exact sequence

→ R(j + n − 1)^{β_{n−1}} ⊕ R(j + n − 2)^{α_{n−1}} → G^{∗} ⊕ R(j + n − 1)^{α_{n}} → K_{A}(n) → 0

which we tensor with I_{A/B}(−n). By the definition of Tor_{1}^{R}(I_{A/B}, K_{A}), it suffices to show (I_{A/B}(−n)(j+ n − 1))_{0} = 0 and (I_{A/B}(−n)(j + n − 2))_{0} = 0. This is true since (I_{A/B})_{j−1} = 0 by assumption. Moreover the argument also shows (I_{A/B} ⊗ K_{A})_{0} ≃ dim(G^{∗}(−n) ⊗ I_{A/B})_{0}. Hence we get

0hom_{B}(I_{A/B}, A) = dim Tor_{0}^{B}(I_{A/B}, K_{A}) = _{−n}hom_{R}(G, B) − _{−n}hom_{R}(G, A) ,

and we conclude by (26) and the fact that I_{B}/(I_{B})^{2} ≃ ⊕_{i=1}^{m}B(−a_{i}).

Proposition 42 with B = R applies nicely to compressed Artinian algebras. Indeed the number −nhom_{R}(G, B) −_{−n}hom_{R}(G, A) coincides with the dimension of the corresponding component given in Thm. IIB of [24]. If B ≠ R, Proposition 42 also applies to non-compressed algebras:

Example 43. As a special case of Proposition 42 we look to Artinian level quotients with “almost semi-linear” resolution. All level algebras below may be constructed as A = R/ann(F_{1}, F_{2}) where F_{1} and F_{2} are forms of degree 7 in the dual polynomial algebra of R (cf. later discussion). Indeed we easily construct in this way algebras A_{i} with Hilbert functions H_{A_{1}} = (1, 3, 6, 10, 15, 12, 6, 2), H_{A_{2}} = (1, 3, 6, 10, 14, 12, 6, 2), H_{A_{3}} = (1, 3, 6, 10, 13, 12, 6, 2) and corresponding minimal resolutions

0 → R(−10)^{2} → R(−7)^{5} ⊕ R(−6)^{5} → R(−5)^{9} → I_{A_{1}} → 0

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

Only A_{1} is compressed, but since one may show that the minimal generators of I_{A_{i}} of degree 4 (which we use to define B_{i}) form a regular sequence, Proposition 42 applies (we have used Macaulay 2 to check it and to find the minimal resolutions). Hence the algebras A_{i} are unobstructed and since −nhom_{R}(G, M) = 2 · dim M_{7} and dim(B_{i})_{7} = (i − 1) dim R_{3} for i = 2 and 3 (and B_{1} = R), we get the number

dim_{(A_{i})}GradAlg(H_{A_{i}}) = 2 · dim(B_{i})_{7} − 2 · dim H_{A_{i}}(7) + (i − 1)H_{A_{i}}(4)

to be 68, 62, 54 for i = 1, 2, 3 respectively.

To this end we consider level algebras of CM-type t. Let LevAlg(H) be the open set of GradAlg(H) (and hence open as a subscheme with its induced scheme structure) consisting of graded level quotients with Hilbert function H. Since we work with Artinian algebras there is another known scheme, L( H), parametrizing graded level quotients with suitable Hilbert function H, namely the determinantal loci in the Grassmannian G(t, j) of t-dimensional vector spaces of forms of degree j, cut out by requiring their “catalecticant matrices” to have ranks given by the Hilbert function (see [9], and [26], Sect. 1.1 for the Gorenstein case). Then the underlying sets of closed points of L( H) and LevAlg(H) are the same by apolarity (the Macaulay correspondence), and their tangent spaces are isomorphic ([9], Thm. 2.1 for L( H), and [28], Thm. 1.5 for GradAlg(H)). Since one may by the proof below see that LevAlg(H) and L( H) are in fact isomorphic as topological spaces (expected since they have the Zariski topologies and the bijection between them is natural), we have

Theorem 44. Let R → A be a graded Artinian level quotient with Hilbert function H. Then dim_{(A)} GradAlg^{H}(R) = dim_{(A)} L( H). Hence L( H) is smooth at (A) if and only if GradAlg^{H}(R) is smooth at (A). In particular L( H) is smooth at (A) provided _{0}H^{2}(R, A, A) = 0, i.e. provided the map of (17) is surjective in degree zero, or equivalently, the displayed sequence of Remark 37 with M = K_{A} is exact in degree zero.

Proof. Let V ⊂ L( H) be a closed irreducible subset, and let V have the reduced scheme structure. By the definition of L( H), the restriction of the “universal” bundle of the Grassmannian G(t, j) to V defines via apolarity a family of graded Artinian level quotients, A_{V}, over V with constant Hilbert function H. Since V is integral, it follows that the family (i.e. the morphism Spec(A_{V}) → V) is flat ([45], Lect. 6). Hence we have a morphism π : V → LevAlg(H) by the universal property of GradAlg(H). π(V) is irreducible and closed in LevAlg(H) (it is closed because an “inverse” (LevAlg(H))_{red} → L( H) on closed points exists, by [26], p. 249). So chains of closed irreducible subsets in L( H) and LevAlg(H) correspond, and the spaces have the same dimension. Since their tangent spaces are isomorphic, it follows that GradAlg^{H}(R) is smooth at (A) iff L( H) is smooth at (A). Then we conclude by Theorem 36 since the surjectivity of (17) in degree zero is equivalent to the exactness of the corresponding sequence in Remark 37.

As an application we consider certain type 2 level algebras studied by Iarrobino in [25], i.e. level algebras given by A = R/ann(F_{1}, F_{2}) where F_{1} and F_{2} are forms of the same degree j in the dual polynomial algebra of R, upon which R acts by differentiation (“without coefficients”). Let A_{i} := R/ann(F_{i}). Since we have I_{A} = I_{A_{1}} ∩ I_{A_{2}}, we get an exact sequence

0 → A → A_{1} ⊕ A_{2} → R/(ann(F_{1}) + ann(F_{2})) → 0.

In an extended draft of [25] the author determines the tangent space of LevAlg(H) at such an (A) and he gives it a manageable form in the case {F_{1}, F_{2}} is complementary, i.e. provided

H_{A}(i) = min{dim R_{i}, H_{A_{1}}(i) + H_{A_{2}}(i)} for any i .  (27)

where H_{A} = H. Our Theorem 36 gives us, not only a tangent space which coincides with his, but it provides us also with the following manageable form of the obstruction space.

Proposition 45. Let {F_{1}, F_{2}} be complementary forms of degree j, and let A = R/I_{A} be the Artinian level quotient with Hilbert function H given by I_{A} = ann(F_{1}, F_{2}). Let I_{A_{i}} = ann(F_{i}). Then (I_{A}/I_{A} · I_{A_{1}})_{j} ⊕ (I_{A}/I_{A} · I_{A_{2}})_{j} is the dual of the tangent space of GradAlg^{H}(R) at (A), and jH_{2}(R, A, A_{1}) ⊕_{j}H_{2}(R, A, A_{2}) is the dual of a space containing all obstructions of deforming A as a graded R-algebra. In particular if the sequences

I_{A} ⊗_{R} I_{A} ⟶^{λ} I_{A} ⊗_{R} I_{A_{i}} ↠ I_{A} · I_{A_{i}}

where λ(x ⊗ y) = x ⊗ y − y ⊗ x, are exact in degree j for i = 1, 2, then GradAlg(H) (and L( H)) is smooth at (A) and we have

dim_{(A)} GradAlg^{H}(R) = ∑_{i=1}^{2} dim(I_{A}/I_{A} · I_{A_{i}})_{j}

Remark 46. The map I_{A}⊗_{R}I_{A} ⟶^{λ′} I_{A}⊗_{R}I_{A}, defined by λ′(x⊗y) = x⊗y−y⊗x, obviously commutes with λ above. Since λ′ factors via the natural surjection I_{A} ⊗_{R} I_{A} ↠ Λ^{2}I_{A} (in char(k) ≠ 2), then λ also does. In char(k) ≠ 2 the exactness of the two sequences of Proposition 45 is therefore equivalent to the exactness of

Λ^{2}I_{A} → I_{A} ⊗_{R} I_{A_{i}} ↠ I_{A} · I_{A_{i}} ,  (28)

i = 1, 2, in degree j. Indeed, by Remark 37, _{j}H_{2}(R, A, A_{i}) is the homology of (28) in degree j. In particular if (I_{A} ⊗_{R} I_{A})_{j} ≃ (S_{2}I_{A})_{j} (e.g. (I_{A} ⊗_{R} I_{A})_{j} ≃ (I_{A}^{2})_{j}), then the exactness of the sequences of Proposition 45 is equivalent to (I_{A} ⊗_{R} I_{A_{i}})_{j} ≃ (I_{A} · I_{A_{i}})_{j}.

Proof. Let s(I_{A}) be the minimal degree of a minimal generator of I_{A} and let \overline{A} = R/(ann(F_{1}) + ann(F_{2})) . Since \{F_{1}, F_{2}\} is complementary, we get (\overline{A})_{v} = 0, i.e. A_{v} ≃ (A_{1})_{v} ⊕(A_{2})_{v} for v ≥ s(I_{A}). It follows that

(K_{A_{1}})_{v} ⊕ (K_{A_{2}})_{v} ≃ (K_{A})_{v}

for v ≤ −s(I_{A}). Defining \overline{K} by the long exact sequence

0 → \overline{K} → K_{A_{1}} ⊕ K_{A_{2}} → K_{A} → 0 ,  (29)

we get (\overline{K})_{v} = 0 for v ≤ −s(I_{A}). By considering a minimal R-free resolution of I_{A}, it follows that

0Tor_{i}^{R}(I_{A}, \overline{K}) = 0 for i ≤ 0 .  (30)

Now applying I_{A} ⊗ (−) onto (29), or more precisely using the corresponding long exact sequence of algebra homology, we get

0H_{i}(R, A, K_{A_{1}} ⊕ K_{A_{2}}) ≃ _{0}H_{i}(R, A, K_{A})

for i = 1 and 2 because _{0}Tor_{1}^{R}(I_{A}, \overline{K}) ≃ _{0}Tor_{2}^{R}(A, \overline{K}) ↠ _{0}H_{2}(R, A, \overline{K}) is surjective (cf. (17)) and I_{A} ⊗_{R} \overline{K} ≃ H_{1}(R, A, \overline{K}), i.e. _{0}H_{i}(R, A, \overline{K}) vanishes for i = 1 and 2 by (30). Then we conclude easily by A_{i} ≃ K_{A_{i}}(−j), Theorem 36 and Remark 37. Indeed we have

(I_{A} ⊗_{R} K_{A_{i}})_{0} ≃ (I_{A} ⊗_{R} A_{i}(j))_{0} ≃ (I_{A} ⊗_{R} R/I_{A_{i}})_{j} ≃ (I_{A}/I_{A} · I_{A_{i}})_{j}

and we get (I_{A} ⊗_{R} K_{A})_{0} ≃ (I_{A} ⊗_{R} K_{A_{1}})_{0} ⊕ (I_{A} ⊗_{R} K_{A_{2}})_{0} as well as

0H_{2}(R, A, K_{A}) ≃ _{j}H_{2}(R, A, A_{1}) ⊕_{j}H_{2}(R, A, A_{2}) .

By the assumption of the exactness of the sequences and by Remark 37 (letting F = R and N = I_{A_{i}}), we get the vanishing of _{0}H_{2}(R, A, K_{A}) and we are done.

Remark 47. As Iarrobino points out in the draft of [25], Theorem 4.8A of [24] implies that if F_{1} is any form of degree j and F_{2} is a sum of length s of linear forms to the j-th power (i.e. the Hilbert function of A_{2} = R/ann(F_{2}) equals H(s, j, n) of Remark 26(v)), then {F_{1}, F_{2}} is complementary provided we choose the linear forms of F_{2} general enough. It follows that H_{A} is given by (27).

First we give an easy example which may also be treated by Proposition 42.

Example 48. (a) Let H = (1, 3, 6, 10, 6, 2). By Remark 47 there are forms F_{1} and F_{2} where each F_{i} is a sum of length 5 of linear forms to the 5-th power (i.e. H_{A_{i}} = (1, 3, 5, 5, 3, 1)) and such that the Hilbert function of A = R/ann(F_{1}, F_{2}) is H. Then A is compressed. From the Hilbert functions we see that s(I_{A}) = 4 while s(I_{A_{i}}) = 2. Moreover, the socle degree of A and A_{i} are 5, and we get (I_{A} ⊗ I_{A_{i}})_{5} = 0 for i = 1 and 2. By Proposition 45 it follows that GradAlg(H) is unobstructed at (A) and we have

dim_{(A)} GradAlg(H) = 2 · dim(I_{A})_{5} = 38 .

(b) Let H = (1, 3, 6, 9, 6, 2), let F_{1} be as in (a), while we now let F_{2} be a sum of length 4 of general linear forms to the 5-th power. Hence H_{A_{2}} = (1, 3, 4, 4, 3, 1) and H = H_{A} where A = R/ann(F_{1}, F_{2}) by Remark 47. From the Hilbert functions we see that s(I_{A}) = 3 and s(I_{A_{i}}) = 2. Since we easily see that I_{A} ⊗ I_{A_{i}} ≃ I_{A} · I_{A_{i}} is an isomorphism in degree 5 for i = 1 and 2, we get by Proposition 45 that GradAlg(H) is unobstructed at (A) and that

dim_{(A)} GradAlg(H) = 2 · dim(I_{A})_{5} − dim(I_{A} · I_{A_{1}})_{5} − dim(I_{A} · I_{A_{2}})_{5} = 35 .

Loosely speaking it is, for i = 1, 2, the relations of I_{A} ·I_{A_{i}} in degree j, modulo those coming from the relations of I_{A} ⊗ I_{A_{i}} and the generators of ∧^{2}I_{A}, which contribute to _{0}H^{2}(R, A, A). Of course the vanishing of _{0}H^{2}(R, A, A) as well as the dimension of GradAlg^{H}(R) is usually straightforward to get from Proposition 45 provided s(I_{A}) + s(I_{A_{i}}) ≥ j for i = 1, 2, as in Example 48.

We finish this paper by proving a conjecture of Iarrobino, appearing in the draft of [25], namely that L( H) with H = (1, 3, 6, 10, 14, 10, 6, 2) contains at least two irreducible components, where one of the components contains Artinian level type 2 algebras given by 2 forms of Hilbert function H_{1} = (1, 3, 6, 9, 9, 6, 3, 1) and H_{2} = (1, 3, 4, 5, 5, 4, 3, 1), as in Remark 47, and the other contains level type 2 algebras constructed via 2 forms with Hilbert function H_{3} = (1, 3, 6, 10, 10, 6, 3, 1) and H_{4} = (1, 3, 4, 4, 4, 4, 3, 1). As pointed out in the Introduction, even though this conjecture was open until now, Iarrobino and Boij have in a joint work already constructed other examples of reducible L( H) whose general elements are level quotients of type 2, one with H = (1, 3, 6, 10, 14, 18, 20, 20, 12, 6, 2), and moreover got a doubly infinite series of such components [5].

Example 49. Let H = (1, 3, 6, 10, 14, 10, 6, 2). We claim that there are at least two components V_{1} and V_{2} of L( H) whose general elements are Artinian level type 2 algebras, that dim V_{1} = 46 and dim V_{2} = 47 and that both components are generically smooth.

To get the component V_{1} of dimension 46, take F_{1} to be a sum of length 4 of general linear forms to the 7-th power and take F_{2} to be a general polynomial of degree 7. If A_{i} = R/ann(F_{i}) and A = R/ann(F_{1}, F_{2}), then H_{A_{2}} = (1, 3, 6, 10, 10, 6, 3, 1), H_{A_{1}} = (1, 3, 4, 4, 4, 4, 3, 1) and H_{A} = H. It suffices to show that A is unobstructed and that dim_{(A)} GradAlg(H) = 46. To do so we use Proposition 45. Indeed from the Hilbert functions we see that s(I_{A}) = s(I_{A_{2}}) = 4 and s(I_{A_{1}}) = 2. Hence (I_{A} ⊗ I_{A_{2}})_{7} = 0. Moreover since I_{A} has one generator of degree 4 and 8 generators of degree 5 and A_{1} is a complete intersection of type (2, 2, 6), it follows that all relations of I_{A} · I_{A_{1}} must be of degree greater or equal to 8. We get that (I_{A} ⊗ I_{A_{1}})_{7} ≃ (I_{A} · I_{A_{1}})_{7} is an isomorphism of vector spaces of dimension 2 · (3 + 8) = 22. Hence Proposition 45 applies and we get the unobstructedness of A and

dim_{(A)} GradAlg(H) = 2 · dim(I_{A})_{7} − dim(I_{A} · I_{A_{1}})_{7} = 46 .

To get the other component, let now F_{1} be a sum of length 9 of general linear forms to the 7-th power (i.e. H_{A_{1}} = (1, 3, 6, 9, 9, 6, 3, 1)), let F_{2} be, say F_{2} = x^{6}y+xy^{6}+z^{7} and let A = R/ann(F_{1}, F_{2}). Then the the Hilbert function of A is H by Remark 47. We claim that A is licci! Indeed it is easily checked by Macaulay 2 that A above admits the following CI-linkages to a CI of type (1, 1, 3). We start with a general CI of type (4, 5, 7) whose generators are contained in I_{A} and follow up by general CI-linkages of type (4, 5, 6), (4, 4, 6), (4, 4, 5), (3, 3, 5), (3, 3, 4), (2, 2, 4) and (2, 2, 3), in this order. Then A is unobstructed and dim_{(A)} GradAlg(H) = 47 by Remark 26(iii) or Theorem 24 and we are done (of course, once using Macaulay 2 it is easier to see that the tangent space is 47-dimensional by computing _{0}ext^{1}(I_{A}, I_{A}). The unobstructedness of A is, however, not at all easy to see by Macaulay 2 computations because _{0}Ext_{R}^{1}(I_{A}, A) ≃ _{0}Ext^{2}(I_{A}, I_{A}) is 1-dimensional and so is _{0}H^{2}(R, A, A) by Proposition 45 and Remark 46. Hence we really need to use that the unobstructedness property is preserved under CI-linkages, which is true by Theorem 24).

Macaulay 2 also provides us with the following minimal resolution

0 → R(−10)^{2} → R(−8)^{2} ⊕ R(−7) ⊕ R(−6)^{8} → R(−7) ⊕ R(−5)^{8} ⊕ R(−4) → R → A → 0 .

Thanks to Theorem 24 we claim this is the minimal resolution of the general element of V_{2}! Indeed by Theorem 24 we know that general CI-linkages take open sets onto open sets. Hence if we start with a general CI of type (1, 1, 3) and reverse all the general CI-linkages above, we get a general element of V_{2} which one may check (Macaulay 2) has the minimal resolution described above. Note the ghost-term R(−7). This term is not present in the minimal resolution of the algebra we described in V_{1} which, by Macaulay 2, has another ghost-term, namely R(−6). Still we claim that the general elements of V_{1} and V_{2} have different but comparable sets of graded Betti numbers. Indeed in Iarrobino’s draft [25] he also mention that there is another algebra A′ = R/ann(F_{1}, F_{2}) with H_{A′} = (1, 3, 6, 10, 14, 10, 6, 2) obtained by taking two general enough forms {F_{1}, F_{2}} with A_{i} = R/ann(F_{i}) and H_{A_{i}} = (1, 3, 5, 7, 7, 5, 3, 1) for i = 1, 2. We have used Macaulay 2 to see that if each F_{i} for i = 1, 2 is of the form F_{i} = (l1)^{5} ∗(l2)∗(l3)+(l1)∗(l2)^{5} ∗(l3) where lj are general linear forms, then A′ has a minimal resolution as above without ghost-terms. Hence the set of graded Betti number of the general elements of V_{1} and V_{2} have a common minimum and the claim is proved.

Remark 50. (a) We have tried to look for other examples of several “level type 2 components” of smaller socle degree, but have not yet fully succeeded. A promising candidate is H = (1, 3, 6, 9, 9, 6, 2) where we get a level type 2 algebra A by starting with a CI of type (2, 2, 3) and linking in one step via a CI of type (4, 4, 3). By Remark 26(iii) we have _{0}hom_{R}(I_{A}, A) = 33. Moreover we have an A′ = R/ann(G_{1}, G_{2}) with _{0}hom_{R}(I_{A′}, A′) = 35 (checked by Macaulay 2) by taking G_{1} (resp. G_{2}) to be a sum of length 3 (resp. 6) of general linear forms to the 6-th power. It follows that A belongs to a 33-dimensional generically smooth component while, due to the size of the tangent spaces, there are only two possibilities for A′. It is either obstructed, or it is unobstructed in which case it belongs to an irreducible component different from the “licci” component. We have not yet been able to decide which of the possibilities that occur.

(b) One may construct other examples of several “level type 2 components” of larger socle type by taking the two components of Example 49 and performing a biliaison, starting with general CI’s of type (5, 5, b) containing the general elements of the components and follow up by general CI-linkages of type (b, b, b), b ≥ 7. Using Theorem 24, we get two irreducible components of GradAlg(H′) whose general elements are level type 2 quotients of socle degree 3b − 8 (H′ may be computed from H = (1, 3, 6, 10, 14, 10, 6, 2)).

Remark 51. If we want to compare the parameter space of type 2 codimension 3 level algebras to the corresponding space of Gorenstein algebras, we see many differences. In the level type 2 case,

(i) the parameter space may be reducible (Example 49 and Remark 50(b)),

(ii) _{0}H^{2}(R, A, A) may be non-vanishing (e.g. Example 49, there are many more). In the Gorenstein case (i) and (ii) are false. We have, however, not yet been able to find two irreducible “level type 2 codimension 3 components” with a type 2 level algebra in the intersection, nor have we been able to find an obstructed type 2 codimension 3 level algebra.

Since the general elements of the components of Example 49 have different sets of graded Betti numbers, one may look for multiple components in LevAlg(H) (e.g. of “type 2”) or in GradAlg(H) whose general elements have the same sets of graded Betti numbers. We have in [32], Ex. 26 and Rem. 27 described several such examples, the simplest one consists of “level type 3 codimension 3 components” (resp. “Gorenstein codimension 4 components”) whose general algebras are level (resp. Gorenstein) of dimension 2 (resp. dimension one). We may truncate the algebras (Proposition 33), or better, divide by some twist of the canonical module (Theorem 27) to get many examples of e.g. multiple “Artinian Gorenstein codimension 5 components” whose general elements have the same sets of graded Betti numbers. We are, however, not aware of examples of multiple components in the same Betti stratum in the level (resp. Gorenstein) Artinian codimension 3 (resp. 4) case.

References

[1] M. André. Méthode Simpliciale en Algèbre Homologique et Algèbre Commutative. no 32 Springer-Verlag, Berlin-New York, 1967.

[2] M. André. Homologie des algèbres commutatives. Die Grundlehren der mathematischen Wissenschaften, Band 206. Springer-Verlag, Berlin-New York, 1974.

[3] M. Boij. Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function. Pacific J. of Math. 187 no. 1 (1999), 1–11.

[4] M. Boij. Gorenstein Artin algebras and points in projective space. Bull. London Math. Soc. 31, no. 1 (1999), 11–16.

[5] M. Boij and A. Iarrobino. Families of level algebras. Talk by Iarrobino at the Magic05 conference, Notre Dame University, Oct. 2005. Paper to appear.

[6] G. Bolondi, J.O. Kleppe and R.M. Miro-Roig. Maximal rank curves and singular points of the Hilbert scheme Compositio Math. 77 (1991), 269-291.

[7] R. Buchweitz. “Contributions a la theorie des singularites,” Thesis l’Université Paris VII (1981).

[8] R. Buchweitz and B. Ulrich. Homological properties which are invariant under linkage. Preprint 1983.

[9] J.V. Chipalkatti, A.V. Geramita. On parameter spaces for Artin level Algebras, Michigan Math. J. 51 (2003), 187–207.

[10] S.J. Diesel Some irreducibility and dimension theorems for families of height 3 Gorenstein algebras. Pacific J. Math. 172 (1996) 365–397.

[11] D. Eisenbud. Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Math., volume 150, Springer–Verlag, New York, 1995.

[12] G. Ellingsrud. Sur le schéma de Hilbert des variétés de codimension 2 dans ℙ^{e} a cône de Cohen-Macaulay Ann. Scient. Éc. Norm. Sup. 8 (1975), 423-432.

[13] A.V. Geramita, P. Maroscia and L.G. Roberts. The Hilbert Function of a Reduced k-algebra, J. London Math. Soc. 28 (1993), 443–452.

[14] A.V. Geramita, T. Harima, J. Migliore and Y.S. Shin. The Hilbert function of a level algebra, to appear in Mem. Amer. Math. Soc..

[15] G. Gotzmann. A Stratification of the Hilbert Scheme of Points in the Projective Plane. Math. Z. 199 no. 4 (1988) 539–547.

[16] D. Grayson and M. Stillman. Macaulay 2-a software system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2/ .

[17] A. Grothendieck. Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les schemas des Hilbert. Séminaire Bourbaki 13, (1960/61), no. 221.

[18] A. Grothendieck. Eléments de la géométrie algebriques IV. Etude locale des schémas et des morphismes de schémas Publ. Math. I.H.E.S. 32 (1967).

[19] R. Hartshorne. Local Cohomology. Lectures Notes in Math. Vol. 41 Springer–Verlag, New York, 1967.

[20] R. Hartshorne. Connectedness of the Hilbert Scheme. Publ. Math. I.H.E.S. 29 (1966), 5–48.

[21] J. Herzog. Deformationen von Cohen-Macaulay Algebren. J. reine angew. Math. 318 (1980), 83–105.

[22] C. Huneke. Numerical invariants of liaison classes. Invent. Math. 75 (1984), 301-325.

[23] A. Iarrobino. Punctual Hilbert Schemes. Mem. Amer. Math. Soc. Vol. 10, no 188, (1977).

[24] A. Iarrobino. Compressed algebras: Artin algebras having given socle degrees and maximal length. Trans. Amer. Math. Soc. Vol. 285, no 1, (1984), 337-378.

[25] A. Iarrobino. Hilbert functions of Gorenstein algebras associated to a pencil of forms, in “Projective Varieties and Unexpected Properties, A Volume in Memory of Giuseppe Veronese, Proceedings (Siena, 2004)”, Walter de Gruyter, Berlin-New York, 2005.

[26] A. Iarrobino and V. Kanev. Power sums, Gorenstein Algebras and Determinantal Loci. Lectures Notes in Math. Vol. 1721 Springer–Verlag, New York, 1999.

[27] A. Iarrobino and H. Srinivasan. Some Gorenstein Artin algebras of embedding dimension four,I: Components of PGor(H) for H = (1, 4, 7, ..., 1). J. Pure Appl. Algebra 201 no1-3 (2005), 62–96.

[28] J. O. Kleppe. Deformations of graded algebras. Math. Scand. 45 (1979), 205–231.

[29] J. O. Kleppe. Liaison of families of subschemes in ℙ^{n}, in “Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988)” Lectures Notes in Math. Vol. 1389 Springer–Verlag, New York, 1989.

[30] J. O. Kleppe. The smoothness and the dimension of PGor(H) and of other strata of the punctual Hilbert scheme. J. Algebra, 200 no. 2 (1998), 606–628.

[31] J. O. Kleppe and R. Miro-Roig. The dimension of the Hilbert scheme of Gorenstein codimension 3 subschemes. J. Pure Appl. Algebra, 127 no. 1 (1998), 73–82.

[32] J. O. Kleppe. Maximal families of Gorenstein algebras. Trans Amer. Math. Soc., 358 no. 7 (2006), 3133–3167.

[33] J. O. Kleppe and R. Miro-Roig. Dimension of families of determinantal schemes. Trans Amer. Math. Soc, 357 no. 7 (2005), 2871–2907.

[34] J. O. Kleppe. Unobstructedness and dimension of families of Gorenstein algebras. Preprint August 2004.

[35] J. O. Kleppe. The Hilbert Scheme of Space Curves of Small Diameter. Annales de l’institut Fourier, 56 no. 5 (2006), p. 1297-1335

[36] J.O. Kleppe, J. Migliore, R. Miro-Roig, U. Nagel and C. Peterson. Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness. Mem. Amer. Math. Soc. Vol. 154, no 732, (2001).

[37] Johannes Kleppe. Additive splittings of Homogeneous Polynomials. Thesis, June 2005, Univ. of Oslo.

[38] A. Laudal. Formal Moduli of Algebraic Structures. Lectures Notes in Math., Vol. 754, Springer–Verlag, New York, 1979.

[39] S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism. Trans Amer. Math. Soc, 128 (1967), 41–70.

[40] D. Mall. Connectedness of the Hilbert function strata and other connectedness results. J. Pure Appl. Algebra, 150 (2000), 175–205.

[41] M. Martin-Deschamps and D. Perrin. Sur la classification des courbes gauches, I. Astérisque, 184–185 (1990).

[42] M. Martin-Deschamps and R. Piene. Arithmetically Cohen-Macaulay curves in ℙ^{4} of degree 4 and genus 0. Manuscripta Math. 93 (1997), 391–408.

[43] J. Migliore. Families of reduced zero-dimensional schemes. Collect. Math. 57, 2 (2006), 173-192.

[44] J. Migliore. Introduction to liaison theory and deficiency modules. Progress in Math., Vol. 165, Birkhäuser Boston, Inc., Boston, MA, 1998.

[45] D. Mumford. Lectures on Curves on an Algebraic Surface. Annals of Math. Studies, Vol. 59, Princeton Univ. Press, 1966.

[46] R. Notari and M.L. Spreafico, A Stratification of Hilbert Scheme by Initial Ideals and Applications. Manuscripta Math. 101 (2000),429-448.

[47] K. Pardue. Deformation Classes of Graded Modules and Maximal Betti Numbers Illinois J. Math. 40, no 4 (1996), 564–585.

[48] R. Piene and M. Schlessinger. On the Hilbert scheme compactification of the space of twisted cubics. I. Amer. J. Math. 107 (1985), 761–774.

[49] Ragusa and Zappala. On the reducibility of the postulation Hilbert scheme. Rend. Circ. Mat. Palermo, Serie II, no LIII (2004), 401-406.

[50] C. Walter. Some examples of obstructed curves in ℙ^{3} In: Complex Projective Geometry. London Math. Soc. Lecture Note Ser. 179, Cambridge University Press (1992).

Oslo University College, Faculty of Engineering, Pb. 4 St. Olavs plass, N-0130 Oslo, Norway. E-mail address: JanOddvar.Kleppe@iu.hio.no