ON MAXIMAL CAYLEY-BACHARACH SCHEMES

Martin Kreuzer  Lorenzo Robbiano
Fakultät für Mathematik  Dipartimento di Matematica
Universität Regensburg  Università di Genova
Postfach 397  Via L.B. Alberti 4
D-93040 Regensburg, Germany  I-16132 Genova, Italy

1. Introduction

The notion of Cayley-Bacharach schemes (for short, CB-schemes) is very general and a common property of different classes of 0-dimensional subschemes of ℙ^{n}, such as 0-dimensional complete intersections, sets of points with the uniform position property, and generic sets of points. Now suppose we are dealing with a CB-scheme X. Is it possible to add one point to it and still get a CB-scheme? If not, we say that X is a maximal CB-scheme. A first motivation for investigating this kind of property is that it allows us to distinguish between the first and the last of the aforementioned classes. The reason is that generic sets of points are always CB-schemes, hence they cannot be maximal, while on the other hand we will see that complete intersections are (with a few minor exceptions, cf. 3.1) maximal CB-schemes.

As a source of inspiration, we mention the paper [4], where the corresponding property of maximality with respect to uniform position is introduced for plane 0-dimensional schemes. The main technical tools are those developed in [5] and [7] for many theoretical aspects, and in [9] for the production of suitable examples.

Now we give a more detailed description of the paper. Section 2 contains the fundamental definitions and the first results. The notion of maximal CBschemes is introduced, and some examples are given which show that this notion cannot be introduced in the context of reduced schemes. Instead, it requires the use of general 0-dimensional schemes. Then Proposition 2.4 shows that every non-maximal CB-scheme must have a minimal generator of its homogeneous ideal in high degrees. Thus CB-schemes with all of their minimal generators in low degrees (e.g. most complete intersections) are maximal. Next, Theorem 2.6 gives many conditions under which a CB-scheme is not maximal. As a special case, we recover the non-maximality of generic sets of points. Several examples motivate the assumptions in both theorems.

Section 3 begins with the description of precisely which 0-dimensional complete intersections are maximal CB-schemes. This is achieved in 3.1 as a consequence of Corollary 2.5 and Theorem 2.6. The remainder of the section is devoted to various generalizations of that result. In particular, Theorem 3.2 singles out maximal CB-schemes among 0-dimensional schemes whose Hilbert function has the tail of a complete intersection, while Theorem 3.6 detects maximal CB-schemes among 0-dimensional subschemes of ℙ^{n} whose homogeneous ideal is generated by the maximal minors of a 2 × (n+1)-matrix of homogeneous forms. Then Corollary 3.7 specializes to almost complete intersections in ℙ^{2}. Again many examples are provided at the end of the section.

The final section 4 deals with maximal CB-schemes from the point of view of liaison. Theorem 4.1 generalizes a result of [5] to not necessarily reduced schemes, and Corollary 4.2 yields a characterization of maximal CBschemes in terms of liaison. From this we deduce Corollary 4.6, which is a very useful tool for constructing maximal CB-schemes. Finally, Proposition 4.9 provides a partial converse to Corollary 4.6, and a couple of examples at the end show how to use it.

2. Definition and Basic Results

In what follows we work over an algebraically closed field k. By ℙ^{n} we denote the n-dimensional projective space over k. Our main objects of study are 0-dimensional subschemes X ⊆ ℙ^{n}. We let I(X) denote the saturated homogeneous ideal of X in k[X_{0}, …, X_{n}], and R(X) := k[X_{0}, …, X_{n}]/I(X) the projective coordinate ring of X. The coordinates {X_{0}, …, X_{n}} of ℙ^{n} are always chosen in such a manner that the residue class x_{0} := X_{0} + I(X) is not a zero divisor of R(X).

The Hilbert function of X is the function H_{X} : ℤ → ℕ (i ↦ dim_{k} R(X)_{i}), and its first difference function is ΔH_{X}(i) = h_{i} := H_{X}(i) − H_{X}(i − 1). We shall also have opportunity to use the numbers σ_{X} := max{i ∈ ℤ : H_{X}(i) < deg X} and Δ_{X} := ΔH_{X}(σ_{X} + 1) = h_{σ_{X}+1}.

Recall that X is called a Cayley-Bacharach scheme (for short, CBscheme), if every subscheme Y ⊆ X of degree deg Y = deg X − 1 has Hilbert function H_{Y} = min{H_{X}, deg Y } (cf. [7]). The purpose of this section is to study the following notion.

Definition. We say that X is a maximal CB-scheme, if X is a CB-scheme, and if every 0-dimensional scheme Y ⊇ X of degree deg Y = deg X + 1 is not a CB-scheme.

First of all, we give some examples which show that it is in general not sufficient in the definition to consider only 0-dimensional schemes Y of the form Y = X ∪ P with a reduced point P ∉ Supp X.

2.1. Example. Let X ⊆ ℙ^{2} be the 0-dimensional subscheme which is concentrated at the point P = (1 : 0 : 0) and satisfies 𝒪_{X,P} ≅ 𝒪_{ℙ^{2},P} /𝔪_{ℙ^{2},P}^{2} . We claim that X is not a maximal CB-scheme, while no scheme Y of the form Y = X ∪ Q with a reduced point Q ≠ P is a CB-scheme.

Let us first remark that X is a CB-scheme, because any scheme Z ⊆ X of degree deg Z = deg X − 1 = 2 has the same Hilbert function ΔH_{Z} : 1 1. The fact that X is a subscheme of the scheme Y which is concentrated at P and satisfies 𝒪_{Y,P} ≅ 𝒪_{ℙ^{2},P} /(x_{1}^{2}, x_{2}^{2}) and deg Y = 4 = deg X + 1 shows that X is not a maximal CB-scheme. On the other hand, let Q ≠ P be a reduced point in ℙ^{2}, and let Y := X ∪ Q. Then the line joining P and Q defines a subscheme X′ ⊆ X of degree 2, and the subscheme Y′ := X′ ∪ Q of Y has Hilbert function ΔH_{Y′} : 1 1 1. Thus Y is not a CB-scheme.

Our next examples demonstrate that even if we start with a reduced scheme X, we still have to allow nonreduced schemes Y in the above definition.

2.2. Example. Let char k = 0 or char k ≥ 13, and let X := {P_{1}, …, P_{15}} ⊆ ℙ^{2} be the reduced scheme with P_{1} = (1:−5:6), P_{2} = (1:−3:6), P_{3} = (1:0:6), P_{4} = (1:3:6), P_{5} = (1:−3:4), P_{6} = (1:0:5), P_{7} = (1:3:4), P_{8} = (1:−3:−4), P_{9} = (1:0:−5), P_{10} = (1:3:−4), P_{11} = (1:−5:−6), P_{12} = (1:−3:−6), P_{13} = (1:0:−6), P_{14} = (1:3:−6), and P_{15} = (1:−5:0).

Then X is a subscheme of the complete intersection scheme Z := 𝒵(f, g) with f = (X_{2}−6X_{0})(X_{2}+6X_{0})(X_{1}^{2}+X_{2}^{2}−25X_{0}^{2}) and g = X_{1}(X_{1}−3X_{0})(X_{1}+ 3X_{0})(X_{1} + 5X_{0}) which has degree 16. ¿From [7], 2.8, it follows that X is a CB-scheme. Since Z also is a CB-scheme, we see that X is not a maximal CB-scheme.

The Hilbert function of X is ΔH_{X} : 1 2 3 4 3 2. From the fact that I(X) is generated by f, g, and a form of degree 6 = σ_{X} +2 and our next proposition we conclude that any CB-scheme Y with X ⊆ Y and deg Y = 16 = deg X +1 has Hilbert function ΔH_{Y} : 1 2 3 4 3 2 1. Therefore Y has to equal 𝒵(f, g). In this case Y is not reduced. Its support agrees with the support of X, but dim_{k} 𝒪_{Y,P_{15}} = 2. Altogether, we have shown that schemes Y of the form Y = X ∪ P with a reduced point P ∉ Supp X are not CB-schemes.

Using the method of [9], we construct a similar example differently. 2.3. Example. Let char k ≠ 2, 7. Consider the subscheme X ⊆ 𝔸^{2} = D_{+}(X_{0}) ⊆ ℙ^{2} with affine ideal J(X) = (X_{1}(X_{1}^{14} − 1), X_{2} − X_{1}^{4}). Obviously X consists of the 14 reduced points (±ω^{i}, ω^{4i}), where ω is a primitive seventh root of unity in k and i = 1, …, 7, and of the reduced point (0, 0). One easily checks that I(X) = rad (J(X)^{hom}) = (X_{1}^{4} − X_{2}X_{0}^{3}, X_{2}^{4} − X_{1}^{2}X_{0}^{2}, X_{1}^{3}X_{2}^{3} − X_{1}X_{0}^{5}). It is also clear that X is contained in the nonreduced scheme Y ⊆ 𝔸^{2} defined by J(Y) = (X_{1}^{2}(X_{1}^{14} − 1), X_{2} − X_{1}^{4}), and that Y is the complete intersection in ℙ^{2} of the curves 𝒵(X_{1}^{4} − X_{2}X_{0}^{3}) and 𝒵(X_{2}^{4} − X_{1}^{2}X_{0}^{2}). As in Example 2.2, it follows that X is a CB-scheme, and the only CB-scheme of degree deg X + 1 containing X is the nonreduced scheme Y.

Our first proposition gives a simple criterion for detecting whether a given CB-scheme is maximal. Recall that a minimal generator of degree d of I(X) is nothing but an element of I(X)_{d} \ R(X)_{1}I(X)_{d−1}. If Y ⊆ X is a subscheme of degree deg X − 1, any nonzero element of the ideal I_{Y/X} of Y in R(X) is called a separator of Y in X. 2.4. Proposition. Let X ⊆ ℙ^{n} be an arbitrary 0-dimensional subscheme, and let Y ⊆ ℙ^{n} be a CB-scheme of degree deg Y = deg X + 1 containing X. Then one of the following two cases occurs.

a) We have σ_{Y} = σ_{X}, the first difference function of the Hilbert function of Y is ΔH_{Y} : 1 h_{1} h_{2} … h_{σ_{X}} Δ_{X} +1, and I(X) has a minimal generator in degree σ_{X} + 1.

b) We have σ_{Y} = σ_{X} + 1, the first difference function of the Hilbert function of Y is ΔH_{Y} : 1 h_{1} h_{2} … h_{σ_{X}} Δ_{X} 1, and I(X) has a minimal generator in degree σ_{X} + 2.

Proof: Since Y is a CB-scheme, we have H_{X} = min{H_{Y} , deg X}. In view of H_{X}(σ_{X}) = deg X−Δ_{X} < deg X, this implies H_{X}(σ_{X}) = H_{Y} (σ_{X}). Therefore σ_{Y} and ΔH_{Y} are as in a) or b).

In case a), we choose a separator F ∈ I(X)_{σ_{X}+1} of X in Y. Suppose F = X_{0}F_{0} + … + X_{n}F_{n} with F_{i} ∈ I(X)_{σ_{X}}. Then F_{i} ∉ I(Y) for some i, and F_{i} is a separator of X in Y of degree σ_{Y}, a contradiction. Thus F is a minimal generator of I(X).

In case b), a separator F ∈ I(X)_{σ_{X}+2} of X in Y turns out to be a minimal generator in the very same way. □

2.5. Corollary. If X ⊆ ℙ^{n} is a CB-scheme and I(X) is generated by elements of degree ≤ σ_{X}, then X is a maximal CB-scheme.

Next we want to give some instances in which we are able to show that certain CB-schemes are not maximal. For a homogeneous ideal I and n ≥ 0, we denote by ⟨I_{≤n}⟩ the ideal generated by the homogeneous elements of I of degree ≤ n.

2.6. Theorem. Let X ⊆ ℙ^{n} be a CB-scheme, and let J := ⟨I(X)_{≤σ_{X}} ⟩.

a) If J is a saturated ideal and σ_{X} + 1 is the largest degree of a minimal generator of I(X), then X is not a maximal CB-scheme.

b) If J is a radical ideal and I(X) has a minimal generator of degree σ_{X} + 1, then X is not a maximal CB-scheme.

c) If J is a prime ideal and X is locally Gorenstein, then X is not a maximal CB-scheme. More precisely, for a generic point P ∈ 𝒵(J), the scheme Y := X∪P is a CB-scheme of degree deg Y = deg X+1 containing X.

Proof: Let Z := 𝒵(J) ⊆ ℙ^{n} be the subscheme defined by J. Since J is saturated, it is the largest ideal defining Z.

“a)” By assumption, X is a proper subscheme of Z. There are two cases. Case 1: Suppose Supp X ⊊ Supp Z. Choose a reduced point P ∈

Supp Z \ Supp X, and let Y := X ∪ P. We claim that Y is a CB-scheme.

First of all, since there is an element of I(X)_{σ_{X}+1} which does not vanish at P, we have H_{X}(σ_{X} + 1) < H_{Y} (σ_{X} + 1), and therefore σ_{Y} = σ_{X} as well as ΔH_{Y} : 1 h_{1} h_{2} … h_{σ_{X}} Δ_{X} +1. There cannot be a separator of X in Y of degree σ_{Y}, because I(X)_{σ_{X}} = I(Y)_{σ_{X}}. If X′ ⊆ X is a subscheme of degree deg X′ = deg X − 1 and Y′ := X′ ∪ P, there cannot be a separator of Y′ in Y of degree σ_{Y}, since it would also be a separator of X′ in X.

Case 2: Suppose Supp X = Supp Z. Thus Z is a 0-dimensional scheme, too. Let P ∈ Supp Z be a point with dim_{k} 𝒪_{X,P} < dim_{k} 𝒪_{Z,P}. Choose a subscheme Y ⊆ Z such that X ⊆ Y, deg Y = deg X + 1, and 𝒪_{Y,Q} = 𝒪_{X,Q} for Q ∈ Supp X \ {P }. We shall show that Y is a CB-scheme.

Note that I(Y)_{σ_{X}+1} ⊊ I(X)_{σ_{X}+1}, because I(X) is generated by elements of degree ≤ σ_{X} + 1. This implies σ_{Y} = σ_{X} and ΔH_{Y} : 1 h_{1} h_{2} … h_{σ_{X}} Δ_{X} +1. In particular, by comparing ΔH_{X} and ΔH_{Y} we see that the degree of a minimal separator of X in Y is σ_{Y} + 1. If Y′ ⊆ Y is a subscheme of degree deg Y′ = deg Y − 1 which differs from Y at a point other than P, any separator f ∈ R(Y)_{σ_{Y}} = R(X)_{σ_{X}} of Y′ in Y would be a separator of the corresponding subscheme X′ ⊆ X, a contradiction to X being a CB-scheme.

Finally we have to consider subschemes Y′ ⊆ Y of degree deg Y′ = deg Y − 1 with dim_{k} 𝒪_{Y′,P} = dim_{k} 𝒪_{Y,P} − 1 and Y′ ≠ X. Let Y′ ⊆ Y be defined by a socle element s_{P} of 𝒪_{Y,P}. Then s_{P} is linearly independent from the socle element defining X ⊆ Y, and its image \overline{s}_{P} in 𝒪_{X,P} defines a subscheme X′ ⊆ X of degree deg X′ = deg X − 1. Suppose f ∈ R(Y)_{σ_{Y}} is a separator of Y′ in Y. Then x_{0}f maps to (0, …, 0, s_{P} , 0, …, 0) under the dehomogenization isomorphism R(Y)_{σ_{Y} +1} → ∏_{Q∈Y} 𝒪_{Y,Q} (cf. [7]). Using R(X)_{σ_{X}} = R(Y)_{σ_{Y}}, it follows that x_{0}f maps to (0, …, 0, \overline{s}_{P} , 0, …, 0) under the dehomogenization isomorphism R(X)_{σ_{X}+1} → ∏_{Q∈X} 𝒪_{X,Q}. Thus f ∈ R(X)_{σ_{X}} is a separator of X′ in X, a contradiction to X being a CB-scheme again.

“b)” Since Z is reduced, we have Supp X ⊊ Supp Z, and we can proceed as in the first case of the proof of a).

“c)” Because of b), we can assume that I(X) has no minimal generator in degree σ_{X} + 1. Since X is locally Gorenstein, there are only finitely many subschemes X_{i} ⊆ X of degree deg X_{i} = deg X − 1, namely the ones defined by the socles of the local rings of the points in the support of X. For each of them, let f_{i} ∈ R(X)_{σ_{X}+1} be a separator, and let F_{i} ∈ k[X_{0}, …, X_{n}]_{σ_{X}+1} be a preimage of f_{i}. Since J_{σ_{X}+1} = I(X)_{σ_{X}+1}, it follows that the hypersurface 𝒵(F_{i}) meets Z properly. Let P be a reduced point of Z not on any of those hypersurfaces. We claim that Y := X ∪ P is a CB-scheme.

By the assumptions, we have I(Y)_{σ_{X}+1} = I(X)_{σ_{X}+1}, and therefore σ_{Y} = σ_{X} + 1 as well as ΔH_{Y} : 1 h_{1} h_{2} … h_{σ_{X}} Δ_{X} 1. If f ∈ R(Y)_{σ_{Y}} = R(X)_{σ_{X}+1} is a separator of X in Y, this implies f ∈ I_{X/Y}, hence f = 0, a contradiction. Now let X_{i} ⊆ X be a subscheme of degree deg X_{i} = deg X −1, and let Y_{i} := X_{i} ∪ P. If f ∈ R(Y)_{σ_{Y}} = R(X)_{σ_{X}+1} is a separator of Y_{i} in Y, then f is also a separator of X_{i} in X. Therefore f is a scalar multiple of f_{i} and does not vanish at P, a contradiction again. □

Let us discuss the hypotheses of this theorem for a moment. Example 2.2 shows that 𝒵(J) need not be reduced in general. Our next pair of examples demonstrates that J is not even saturated in general, and that this assumption cannot be omitted from a).

2.7. Example. a) Let char k = 0 or char k ≥ 11. Consider the reduced scheme X ⊆ ℙ^{2} consisting of the following 45 points, sketched in the affine plane 𝔸^{2} = D_{+}(X_{0}) = {P = (p_{0}, p_{1}, p_{2}) ∈ ℙ^{2} : p_{0} ≠ 0}:

(0, 7) • • • • • • • • (7, 7)
• • • • • • • •
• • • • • • • •
•  • • •
•  • • •
(0, 2)•  • • • •
•  • • •
•  • • • (7, 0)

Obviously X is contained in the complete intersection defined by f = X_{1}(X_{1} − X_{0})(X_{1} − 2X_{0}) ⋯ (X_{1} − 7X_{0}) and g = X_{2}(X_{2} − X_{0})(X_{2} − 2X_{0}) ⋯ (X_{2} − 7X_{0}). The Hilbert function of X is given by ΔH_{X} : 1 2 3 4 5 6 7 8 6 3, so that σ_{X} = 8. Furthermore, the polynomial h = (X_{1} − 5X_{0})(X_{1} − 6X_{0})(X_{1}−7X_{0})(X_{2}−5X_{0})(X_{2} − 6X_{0})(X_{2} − 7X_{0})(X_{2}−X_{1}+2)(X_{2}−X_{1}−2) clearly defines the third minimal generator of I(X) in degree σ_{X}, and I(X) = (f, g, h)^{sat.}.

But when we compute a minimal system of generators of I(X), we find that there is another minimal generator in degree 9 = σ_{X} + 1. Hence J = (f, g, h) is not saturated. Using [5], 4.6, we see that X is a CB-scheme. In fact, X is even a maximal CB-scheme, since any CB-scheme Y of degree 46 containing X would have (f, g, h) ⊆ I(Y) by Proposition 2.4, in contradiction to I(X) = (f, g, h)^{sat.}.

b) Let char k = 0, and let X ⊆ 𝔸^{2} = D_{+}(X_{0}) ⊆ ℙ^{2} be the 0-dimensional subscheme with affine ideal (f, g), where f = X_{1}^{27} −X_{1} −1 and g = X_{2} −X_{1}^{21}. Since f is irreducible over ℚ, it follows from [9] that X is a reduced CBscheme. Using the computer algebra system CoCoA, it is easy to check that ΔH_{X} : 1 2 3 4 5 6 4 2, the ideal J = ⟨I(X)_{≤σ_{X}} ⟩ is not saturated, and we even have J^{sat.} ⊊ I(X).

The following example shows that the assumption “X is locally Gorenstein” cannot be dispensed with in part c) of the theorem.

2.8. Example. Let char k ≠ 2, and let X ⊆ ℙ^{2} be the subscheme concentrated at the points P_{1} = (1 : 5 : 0), P_{2} = (1 : 4 : 3), P_{3} = (1 : 3 : 4), and P_{4} = (1 : 0 : 5) which satisfies 𝒪_{X,P_{1}} = 𝒪_{ℙ^{2},P_{1}}/𝔪_{ℙ^{2},P_{1}}^{2} and in which P_{2}, P_{3}, and P_{4} are reduced points. Obviously X is not locally Gorenstein. Since the Hilbert function of X is ΔH_{X} : 1 2 3, it is also clear that X is a CB-scheme. Furthermore, σ_{X} = 1 and J = (0) is a prime ideal.

But there is no reduced point P_{5} ∈ ℙ^{2} \ Supp X such that Y := X ∪ P_{5} is a CB-scheme. Such a scheme Y has Hilbert function ΔH_{Y} : 1 2 3 1. There exists a conic Q through the reduced points P_{1}, …, P_{5}. Then there is a subscheme X′ ⊆ X of degree deg X′ = 5 = deg X−1 such that dim_{k} 𝒪_{X′,P_{1}} = 2 and X′ ⊆ Q. Now the subscheme Y′ := X′ ∪ P_{5} of Y has degree deg Y′ = 6 = deg Y − 1 and Hilbert function ΔH_{Y′} : 1 2 2 1. Hence Y is not a CB-scheme, as we had claimed.

Definitions. a) We say that X has generic Hilbert function, if H_{X} = min{H_{ℙ}, deg X}.

b) We call α_{X} := min{i ∈ ℕ : I(X)_{i} ≠ 0} the initial degree of X.

c) If Y ⊆ X is a proper subscheme and I_{Y/X} its ideal in R(X), we call α_{Y/X} := min{i ∈ ℕ : (I_{Y/X})_{i} ≠ 0} the initial degree of Y in X.

The reader may verify that X has a generic Hilbert function if and only if α_{X} > σ_{X}. An easy application of Theorem 2.6 provides us with a large class of nonmaximal CB-schemes.

2.9. Corollary. Let X be a CB-scheme with generic Hilbert function. Then X is not a maximal CB-scheme, unless deg X = H_{ℙ}(σ_{X} + 1) and X is not locally Gorenstein. More precisely, for a generic point P ∈ ℙ^{n}, the scheme Y := X ∪ P is a CB-scheme.

Proof: Since α_{X} > σ_{X}, we can apply Theorem 2.6.b,c with J = (0). □

In view of Corollary 2.5 and Theorem 2.6.a, it seems natural to classify CB-schemes X ⊆ ℙ^{n} according to m(X), the largest degree of a minimal generator of I(X). Corollary 2.5 says that CB-schemes X with m(X) ≤ σ_{X} are maximal, while Theorem 2.6.a states that most CB-schemes with m(X) = σ_{X} + 1 are not maximal. It is well known that m(X) ≤ σ_{X} + 2 holds all the time. Therefore we are left with considering CB-schemes X such that m(X) = σ_{X} + 2. ¿From the subsequent four examples we can conclude that in this case X can be a maximal CB-scheme or not, independent of whether there are minimal generators of degree σ_{X} + 1 of I(X) or not.

2.10. Example. A set of three non-collinear points in ℙ^{2} has minimal generators in degree 2 = σ_{X} + 2 only, and is not a maximal CB-scheme by Corollary 2.9.

2.11. Example. Let char k ≠ 2, 3, and let X ⊆ ℙ^{2} be the 0-dimensional subscheme consisting of the following 22 reduced points:

(0, 4) • • •
• • •  • (4, 3)
• • • • •
• • • • •
(0, 0) • • • • • (4, 0)

The Hilbert function of X is ΔH_{X} : 1 2 3 4 5 4 3, and we have σ_{X} = 5. Using [5], 4.6, we see that X is a CB-scheme. The homogeneous ideal of X has two minimal generators in degree 5 = σ_{X}, and two in degree 7 = σ_{X} + 2.

Any CB-scheme Y of degree 23 containing X has ΔH_{Y} : 1 2 3 4 5 4 3 1 by Proposition 2.4, hence it is contained in the 0-dimensional reduced complete intersection Z defined by the two forms X_{1}(X_{1} − X_{0})(X_{1} − 2X_{0})(X_{1} − 3X_{0})(X_{1} − 4X_{0}) and X_{2}(X_{2} − X_{0})(X_{2} − 2X_{0})(X_{2} − 3X_{0})(X_{2} − 4X_{0}) which generate I(X)_{4}. Using [5], 4.6, again, we see that no point of Z \ X can be added to X to make up a CB-scheme. Thus X is a maximal CB-scheme.

2.12. Example. A set of five reduced points in ℙ^{2}, no four of which are on a line, has minimal generators in degrees 2 = σ_{X} + 1 and 3 = σ_{X} + 2, and is not a maximal CB-scheme by Corollary 2.9.

2.13. Example. Let char k = 0 or char k ≥ 11. Consider the reduced subscheme X ⊆ 𝔸^{2} = D_{+}(X_{0}) ⊆ ℙ^{2} consisting of the following 75 points:

(0, 9) • • • • • • • • • • (9, 9)
• • • • • • • • • •
• • • • • • • • • •
• • • • • • • • • •
•  • • • •
•  • • • •
•  • • • • •
(0, 2) •  • • • • • •
• •  • • • •
• •  • • • • (9, 0)

Obviously X is contained in the complete intersection defined by f = X_{1}(X_{1} − X_{0})(X_{1} − 2X_{0}) ⋯ (X_{1} − 9X_{0}) and g = X_{2}(X_{2} − X_{0})(X_{2} − 2X_{0}) ⋯ (X_{2} − 9X_{0}). The Hilbert function of X is given by ΔH_{X} : 1 2 3 ⋯ 10 9 7 4, so that σ_{X} = 11. Furthermore, the polynomial h = (X_{1} − 6X_{0})(X_{1} − 7X_{0})(X_{1}−8X_{0})X_{1}−9X_{0})(X_{2} − 6X_{0})(X_{2} − 7X_{0})(X_{2}−8X_{0})(X_{2}−9X_{0})(X_{2}− X_{1}−2X_{0})(X_{2}−X_{1}+2X_{0})(X_{2}−X_{1}+3X_{0}) clearly defines a minimal generator of I(X) of degree 11, and I(X) = (f, g, h)^{sat.}.

But when we compute a minimal system of generators of I(X), we find that there are two more minimal generators of degrees 12 and 13. Thus J = ⟨I(X)_{≤σ_{X}} ⟩ = (f, g, h) is not saturated. Using [5], 4.6, we see that X is a CB-scheme. In fact, X is even a maximal CB-scheme, since any CB-scheme Y of degree 76 containing X would have (f, g, h) ⊆ I(Y) by Proposition 2.4, in contradiction to I(X) = (f, g, h)^{sat.}.

3. Complete and Almost Complete Intersections

Using the notations and results of the previous section, it is not difficult to describe precisely which 0-dimensional complete intersections in ℙ^{n} are maximal CB-schemes.

3.1. Remark. Let X ⊆ ℙ^{n} be the complete intersection of n hypersurfaces of degrees d_{1} ≤ … ≤ d_{n}.

a) If n = 2 and d_{1} ≥ 3, or if n ≥ 3 and d_{1} ≥ 2, then X is a maximal CB-scheme by Corollary 2.5, because I(X) is generated in degrees ≤ d_{n}, and d_{n} ≤ σ_{X} = d_{1} + … + d_{n} − n − 1.

b) If X ⊆ ℙ^{2} is the complete intersection of a curve of degree two and a curve of degree d ≥ 2, then X is not a maximal CB-scheme by Theorem 2.6.a, since the largest degree of a minimal generator of I(X) is d = (2+d−3)+1 = σ_{X} + 1, and ⟨I(X)_{≤σ_{X}} ⟩ is saturated.

Our next theorem provides a generalization of Remark 3.1. We say that a 0-dimensional subscheme X ⊆ ℙ^{n} has the tail of a complete intersection, if h_{σ_{X}} = n and h_{σ_{X}+1} = Δ_{X} = 1. Recall that the canonical module of R(X) is the graded R(X)-module ω_{R(X)} ≅ \underline{Hom}_{k[x_{0}]}(R(X), k[x_{0}])(−1) (cf. [7]).

3.2. Theorem. Suppose X ⊆ ℙ^{n} is a 0-dimensional CB-scheme which is not contained in a quadric hypersurface and has the tail of a complete intersection. Then X is a maximal CB-scheme.

Proof: By assumption, ΔH_{X} is of the form ΔH_{X} : 1 n \binom{n+1}{2} … n 1. Suppose Y is a CB-scheme of degree deg Y = deg X +1 containing X. According to Proposition 2.4, we distinguish two cases.

Case 1: Suppose ΔH_{Y} is of the form ΔH_{Y} : 1 n \binom{n+1}{2} … n 2. Using ω_{R(X)} ≅ {φ ∈ ω_{R(Y)} : I_{X/Y} ·φ = 0} and dim_{k}(ω_{R(X)})_{−σ_{X}} = Δ_{X} = 1, we find an element φ_{1} ∈ (ω_{R(Y)})_{−σ_{Y}} which is a k-basis of the image of (ω_{R(X)})_{−σ_{X}}. Since X is a CB-scheme, Theorem 2.6 of [7] shows that Ann_{R(Y)}(φ_{1}) = I_{X/Y}. In particular, this means that rφ_{1} ≠ 0 for each r ∈ R(Y)_{2}, since I(X)_{2} = (0) by assumption. Now choose φ_{2} ∈ (ω_{R(Y)})_{−σ_{Y}} such that {φ_{1}, φ_{2}} is a k-basis of (ω_{R(Y)})_{−σ_{Y}}.

Next we consider the ring \overline{R} := R(Y)/x_{0}R(Y), where x_{0} ∈ R(Y)_{1} is not a zerodivisor. We have ω\overline{R} ≅ (ω_{R(Y)}/x_{0}ω_{R(Y)})(1). Let ϕ_{1} resp. ϕ_{2} denote the image of φ_{1} resp. φ_{2} in ω\overline{R}. Again we have rϕ_{1} ≠ 0 for each r ∈ \overline{R}_{2}. Thus dim_{k}(ω\overline{R})_{−σ_{Y}} = n implies (ω\overline{R})_{−σ_{Y}} = \overline{R}_{1}ϕ_{1}. For i = 1, …, n we denote by x_{i} the image of X_{i} in \overline{R}, and we write x_{i}ϕ_{2} = ℓ_{i}ϕ_{1} with ℓ_{i} ∈ \overline{R}_{1}. Then (x_{i}ℓ_{j} − x_{j}ℓ_{i})ϕ_{1} = x_{i}x_{j}ϕ_{2} − x_{j}x_{i}ϕ_{2} = 0 implies x_{i}ℓ_{j} − x_{j}ℓ_{i} = 0 for i, j = 1, …, n. Since \overline{R}_{2} ≅ k[X_{1}, …, X_{n}]_{2}, this yields ℓ_{i} = λx_{i} for i = 1, …, n and some λ ∈ k^{×}. Hence x_{i}(ϕ_{2} − λϕ_{1}) = 0 for i = 1, …, n, contradicting the fact that the socle of ω\overline{R} is exactly (ω\overline{R})_{0}.

Case 2: Suppose ΔH_{Y} is of the form ΔH_{Y} : 1 n \binom{n+1}{2} … n 1 1. In this case, any element φ ∈ (ω_{R(Y)})_{−σ_{Y}} \ {0} is annihilated by n − 1 k-linearly independent elements of R(Y)_{1}. Since Y is a CB-scheme with Δ_{Y} = 1, it is locally Gorenstein by [7], 2.5.a. But then Ann_{R(Y)}(φ) ≠ (0) contradicts [7], 2.4. □

Clearly, 0-dimensional complete intersections with d_{1} ≥ 3 in Remark 3.1.a satisfy the assumptions of this theorem. More generally, we have the following corollary.

3.3. Corollary. Suppose X ⊆ ℙ^{n} is a 0-dimensional, arithmetically Gorenstein scheme which is not contained in a quadric hypersurface. Then X is a maximal CB-scheme.

Proof: It suffices to note that X is a CB-scheme with symmetric Hilbert function (cf. [8], 1.1) and to apply the theorem. □

The following pair of examples shows that Theorem 3.2 can also be applied to CB-schemes X which are not arithmetically Gorenstein. Notice that we construct two very similar sets of points using the very different, contrasting methods of section 4 and [9].

3.4. Example. a) Let char k ≠ 2, 3, and let X ⊆ ℙ^{2} be the following 0-dimensional reduced subscheme of degree 13:

(0, 3) •  • • (3, 3)
• •  •
• • •
(0, 0) • • • • (3, 0)

Then ΔH_{X} is given by ΔH_{X} : 1 2 3 4 2 1, and X is a CB-scheme because of [5], Theorem 4.6. The theorem shows that X actually is a maximal CBscheme.

b) Let char k = 0, and let X ⊆ 𝔸^{2} = D_{+}(X_{0}) ⊆ ℙ^{2} be the 0-dimensional subscheme with affine ideal (f, g), where f = X_{1}^{13} − 1 and g = X_{2} − X_{1}^{4}. It follows from [9] that X is a reduced CB-scheme. Again ΔH_{X} is given by ΔH_{X} : 1 2 3 4 2 1, and X is a maximal CB-scheme by the theorem.

After examining the complete intersection case, we now turn to 0-dimen-sional subschemes of ℙ^{n} which are defined by the 2×2-minors of a 2×(n+1)- matrix of homogeneous polynomials. As a special case we shall show that almost complete intersections in ℙ^{2} are usually maximal CB-schemes.

3.5. Lemma. Let A be a ring and {f_{1}, …, f_{n+1}} a regular sequence in A. We shall write f⃗^{t} for the row vector (f_{1}, …, f_{n+1}) and f⃗ for the corresponding column vector.

a) If g⃗ ∈ A^{n+1} satisfies g^{t} · f⃗ = 0, then there exists an antisymmetric matrix 𝒜 = (a_{ij}) ∈ Mat_{n+1}(A) (i.e. a_{ii} = 0 and a_{ij} = −a_{ji} for 1 ≤ i < j ≤ n+1) such that g⃗ = 𝒜 · f⃗.

b) If 𝒜 ∈ Mat_{n+1}(A) is an antisymmetric matrix and g⃗ ∈ A^{n+1}, then g^{t} · 𝒜 · g⃗ = 0.

Proof: This easy induction on n is left to the reader. □

3.6. Theorem. Let a_{1}, …, a_{n+1}, b ≥ 0, let f_{i} ∈ k[X_{0}, …, X_{n}]_{a_{i}} for i = 1, …, n + 1, and let g_{i} ∈ k[X_{0}, …, X_{n}]_{a_{i}+b} for i = 1, …, n + 1. Suppose that the ideal generated by the 2 × 2-minors of the matrix ℳ = \binom{f_{1} f_{2} ... f_{n+1}}{g_{1} g_{2} ... g_{n+1}} defines a 0-dimensional subscheme X ⊆ ℙ^{n}, and that {f_{1}, …, f_{n+1}} is a regular sequence. Then we have the following statements.

a) X is a locally Gorenstein CB-scheme.

b) If ∑_{ν≠i,j} a_{ν} + (n − 1)b ≥ n + 1 for 1 ≤ i < j ≤ n + 1, then X is a maximal CB-scheme.

Proof: For 1 ≤ i < j ≤ n + 1 let Δ_{ij} := g_{i}f_{j} − f_{i}g_{j}. Then dim X = 0 implies that the ideal (Δ_{ij})_{1≤i<j≤n+1} has height n. From [1], 2.7 we conclude that this ideal is saturated, and therefore it is exactly I(X). Since {f_{1}, …, f_{n+1}} is a regular sequence, we have a_{i} > 0 for i = 1, …, n + 1.

A minimal free resolution of the A := k[X_{0}, …, X_{n}]-module R(X) = A/I(X) is given by the Eagon-Northcott resolution (cf. [1], 2.16)

0 ⟶ (∧^{0}G ⊗ S_{n−1}(F))ˇ ⟶^{∂}ˇ (∧^{1}G ⊗ S_{n−2}(F))ˇ ⟶^{∂}ˇ ⋯

⋯ ⟶^{∂}ˇ (∧^{n−1}G ⊗ S_{0}(F))ˇ ⟶^{ν_{0}} ∧^{0}G ⊗ S_{0}(F) ⟶ R(X) ⟶ 0

where ℳ defines an A-linear map g : G = ⨁_{i=1}^{n+1} A(−a_{i}) ⟶ F = A ⊕ A(b) and

ˇ means \underline{Hom}_{A}(—, A). Notice that the maps ∂ˇ are homogeneous of degree zero in our case, while ν_{0} is homogeneous of degree −a_{1} − ⋯ − a_{n+1} − b.

By dualizing and twisting this sequence, we obtain a minimal graded free resolution of the canonical module ω_{R(X)} of R(X). In particular, this module has a minimal graded presentation of the form

⨁_{i=1}^{n+1} ⨁_{j=0}^{n−2} A(jb−a_{i}) ⟶^{𝒜} ⨁_{j=0}^{n−1} A(jb) ⟶^{ℬ} ω_{R(X)}(−a_{1}−⋯−a_{n+1}−b+n+1) → 0.

Let e_{ij} and ϵ_{j} denote the standard bases of the first and second module of this sequence, respectively. By construction of the Eagon-Northcott resolution, the map 𝒜 is given by 𝒜(e_{ij}) = f_{i}ϵ_{j} + g_{i}ϵ_{j+1}. Furthermore, let φ_{j} := ℬ(ϵ_{j}) for j = 0, …, n − 1. We conclude that σ_{X} = a_{1} + ⋯ + a_{n+1} + nb − n − 1 and that deg φ_{j} = −σ_{X} + (n − 1 − j)b for j = 0, …, n − 1.

By [7], 2.4, we need to show Ann_{R(X)}(φ_{n−1}) = (0) in order to show a). Suppose r ∈ R(X) is a homogeneous element such that rφ_{n−1} = 0. We choose a homogeneous polynomial p ∈ A such that r = p + I(X). Then p · ϵ_{n−1} ∈ ker ℬ means that there are elements c_{ij} ∈ A such that p · ϵ_{n−1} = ∑_{i=1}^{n+1} ∑_{j=0}^{n−2} c_{ij}(f_{i}ϵ_{j} + g_{i}ϵ_{j+1}). This yields the following system of equations:

p = c_{1 n−2}g_{1} + ⋯ + c_{n+1 n−2}g_{n+1}
0 = c_{1 n−3}g_{1} + ⋯ + c_{n+1 n−3}g_{n+1} + c_{1 n−2}f_{1} + ⋯ + c_{n+1 n−2}f_{n+1}
⋮
0 = c_{1 0}g_{1} + ⋯ + c_{n+1 0}g_{n+1} + c_{1 1}f_{1} + ⋯ + c_{n+1 1}f_{n+1}
0 = c_{1 0}f_{1} + ⋯ + c_{n+1 0}f_{n+1}

Let us write c_{i}^{t} := (c_{1 i}, …, c_{n+1 i}), f⃗^{t} := (f_{1}, …, f_{n+1}), and g^{t} := (g_{1}, …, g_{n+1}). Then, by the lemma, c_{0}^{t} · f⃗ = 0 implies that there is an antisymmetric matrix 𝒜_{0} ∈ Mat_{n+1}(A) such that c⃗_{0} = 𝒜_{0} · f⃗. Hence 0 = g^{t} · c⃗_{0} + c_{1}^{t} · f⃗ = (g^{t} · 𝒜_{0} + c_{1}^{t}) · f⃗ = (c⃗_{1} − 𝒜_{0} · g⃗)^{t} · f⃗, and another application of the lemma yields c⃗_{1} − 𝒜_{0} · g⃗ = 𝒜_{1} · f⃗ for some antisymmetric matrix 𝒜_{1} ∈ Mat_{n+1}(A). Next we obtain

0 = g^{t}·c⃗_{1}+c_{2}^{t}·f⃗ = g^{t}·𝒜_{0}·g⃗+g^{t}·𝒜_{1}·f⃗+c_{2}^{t}·f⃗ = (g^{t}·𝒜_{1}+c_{2}^{t})·f⃗ = (c⃗_{2}−𝒜_{1}·g⃗)^{t}·f⃗,

and therefore c⃗_{2} − 𝒜_{1} · g⃗ = 𝒜_{2} · f⃗ with an antisymmetric matrix 𝒜_{2} ∈ Mat_{n+1}(A).

Continuing this way, we finally find c⃗_{n−2} = 𝒜_{n−3} · g⃗ + 𝒜_{n−2} · f⃗ and p = g^{t} · c⃗_{n−2} = g^{t} · 𝒜_{n−2} · f⃗ with an antisymmetric matrix 𝒜_{n−2} = (a_{ij}) ∈ Mat_{n+1}(A). This implies p = ∑_{i<j} a_{ij}(g_{i}f_{j} − g_{j}f_{i}) ∈ (Δ_{ij}) = I(X). Thus we have r = p + I(X) = 0, as we wanted to show.

It remains to prove b). Because of the hypothesis we know that σ_{X} = a_{1} + … + a_{n+1} + nb − n − 1 ≥ a_{i} + a_{j} + b = deg Δ_{ij} for 1 ≤ i < j ≤ n + 1. An application of Corollary 2.5 now finishes the proof. □

Recall that a 0-dimensional subscheme of ℙ^{2} is an almost complete intersection, if its homogeneous ideal has a minimal system of generators consisting of three homogeneous polynomials. From the Hilbert-Burch Theorem we then know that I(X) is in fact the ideal generated by the 2 × 2-minors of some 2 × 3-matrix of homogeneous polynomials. The following corollary is an easy consequence of the theorem.

3.7. Corollary. Suppose that X ⊆ ℙ^{2} is a 0-dimensional almost complete intersection whose homogeneous ideal is of the form I(X) = (f_{1}g_{2} − f_{2}g_{1}, f_{1}g_{3} − f_{3}g_{1}, f_{2}g_{3} − f_{3}g_{2}) with polynomials f_{i} ∈ k[X_{0}, X_{1}, X_{2}]_{a_{i}} and g_{i} ∈ k[X_{0}, X_{1}, X_{2}]_{a_{i}+b} for i = 1, 2, 3. Furthermore, assume that {f_{1}, f_{2}, f_{3}} is a regular sequence. Then we have the following statements.

a) X is a locally Gorenstein Cayley-Bacharach scheme.

b) If a_{i} + b ≥ 3 for i = 1, 2, 3, then X is a maximal CB-scheme.

c) If a_{i} + b = 2 for at least two indices i ∈ {1, 2, 3} and a_{i} + b ≥ 3 for the remaining ones, then X is not a maximal CB-scheme.

Proof: Since parts a) and b) follow immediately from the theorem, it remains to show c). In that case at most one generator of I(X) has degree ≤ σ_{X}, and the remaining ones have degree σ_{X} +1. Thus X is not a maximal CB-scheme by Theorem 2.6.a. □

Let us illustrate both the theorem and the corollary with some examples.

3.8. Example. Let X ⊆ ℙ^{2} be the subscheme defined by the 2×2-minors of the matrix ℳ = \binom{X_{0} X_{1} X_{2}}{X_{2}^{3} X_{0}^{3} X_{1}^{3}}. If we dehomogenize those minors with respect to X_{0}, we get the ideal (X_{2}^{4}−X_{1}^{3}, X_{1}X_{2}^{3}−1, X_{1}^{4}−X_{2}) = (X_{1}^{13}−1, X_{2}−X_{1}^{4}) of k[X_{1}, X_{2}], i.e. X agrees with the scheme considered in Example 3.4.b. Since {X_{1}^{13} − 1, X_{2} − X_{1}^{4}} is a regular sequence and X ∩ 𝒵(X_{0}) = ∅, it follows that X is a 0-dimensional subscheme of ℙ^{2}. Now the corollary yields another proof that X is a locally Gorenstein, maximal CB-scheme.

3.9. Example. Let X ⊆ ℙ^{2} be the subscheme defined by the 2×2-minors of the matrix ℳ = \binom{X_{0}^{3} X_{1}^{4} X_{2}^{5}}{X_{2}^{3} X_{0}^{4} X_{1}^{5}}. If we dehomogenize those minors with respect to X_{0}, we get the ideal (X_{1}^{47}−1, X_{2}−X_{1}^{30}) of k[X_{1}, X_{2}]. Since X∩𝒵(X_{0}) = ∅, this implies that X is a 0-dimensional subscheme of ℙ^{2}. By the corollary, X is a locally Gorenstein, maximal CB-scheme. It is easy to check that ΔH_{X} : 1 2 3 4 5 6 7 7 6 4 2, σ_{X} = 9, and X is reduced as long as char k ≠ 47.

3.10. Example. Let X ⊆ ℙ^{3} be the subscheme defined by the 2 × 2minors of the matrix ℳ = \binom{X_{0} X_{1} X_{2} X_{3}}{X_{1}^{2} X_{2}^{2} X_{3}^{2} X_{0}^{2}}. If we dehomogenize those minors with respect to X_{0}, we get the ideal (X_{3}^{15} − 1, X_{1} − X_{3}^{7}, X_{2} − X_{3}^{3}). Since X ∩ 𝒵(X_{0}) = ∅, this implies that X is a 0-dimensional subscheme of ℙ^{3}. By the theorem, X is a locally Gorenstein, maximal CB-scheme. It is easy to check that ΔH_{X} : 1 3 6 4 1, σ_{X} = 3, and X is reduced as long as char k ≠ 3, 5.

We continue with two examples of almost complete intersections in ℙ^{2} which are CB-schemes, but not maximal.

3.11. Example. Let X ⊆ ℙ^{2} be the subscheme defined by the 2 × 2minors of the matrix ℳ = \binom{X_{0}^{2} X_{1}^{2} X_{2}^{3}}{X_{1}^{2} X_{2}^{2} X_{0}^{3}}. As above, we can show that X is a 0-dimensional subscheme of ℙ^{2}. From parts a) and c) of the corollary it follows that X is a locally Gorenstain CB-scheme which is not a maximal CB-scheme. It is easy to check that ΔH_{X} : 1 2 3 4 4 2, σ_{X} = 4, and X is reduced as long as char k ≠ 2.

3.12. Example. Let ℳ = \binom{X_{0}^{2} X_{1}^{3} X_{2}^{4}}{X_{2}^{2} X_{0}^{3} X_{1}^{4}}, and let X ⊆ ℙ^{2} be the subscheme defined by the 2 × 2-minors of ℳ. Dehomogenization of those minors with respect to X_{0} yields the ideal (X_{1}^{13} − 1, X_{2}^{2} − X_{1}^{10}) of k[X_{1}, X_{2}]. Since X ∩ 𝒵(X_{0}) = ∅, this implies that X is a 0-dimensional subscheme of ℙ^{2}. By the corollary, X is a locally Gorenstein CB-scheme. It is also easy to check that ΔH_{X} : 1 2 3 4 5 5 4 2, σ_{X} = 6, and X is reduced as long as char k ≠ 13.

Notice that we cannot use the corollary to decide whether X is a maximal CB-scheme, since a_{1} + b = 2 and a_{2} + b ≥ 3, a_{3} + b ≥ 3. In fact, the ideal J = ⟨I(X)_{≤σ_{X}} ⟩ = (X_{0}^{5} − X_{1}^{3}X_{2}^{2}, X_{2}^{6} − X_{0}^{2}X_{1}^{4}) is a complete intersection, hence saturated, and thus X is not a maximal CB-scheme by Theorem 2.6.a. More concretely, the reader may check that the scheme Y = X ∪ P with P = (0 : 1 : 0) is a CB-scheme with ΔH_{Y} : 1 2 3 4 5 5 4 3, σ_{Y} = 6, and I(Y) = (X_{0}^{5} − X_{1}^{3}X_{2}^{2}, X_{2}^{6} − X_{0}^{2}X_{1}^{4}, X_{0}(X_{1}^{7} − X_{0}^{3}X_{2}^{4}), X_{2}(X_{1}^{7} − X_{0}^{3}X_{2}^{4})).

Finally, we give an example of a 0-dimensional almost complete intersection X ⊆ ℙ^{2} which is not even a CB-scheme. Notice that sequence {f_{1}, f_{2}, f_{3}} is not regular.

3.13. Example. Let X ⊆ ℙ^{2} be the reduced subscheme of degree four which is concentrated at the points P_{1} = (1 : 0 : 0), P_{2} = (1 : 1 : 0), P_{3} = (1 : 2 : 0), and P_{4} = (1 : 0 : 1). The homogeneous ideal of X is the ideal of 2 × 2-minors of the matrix ℳ = \binom{X_{1} 0 X_{2}−X_{0}}{0 X_{2} (X_{1}−X_{0})(X_{1}−2X_{0})}, but X is not even a CB-scheme, since ΔH_{X} : 1 2 1 and ΔH_{Y} : 1 1 1 for Y = {P_{1}, P_{2}, P_{3}} ⊆ X.

4. Maximal Cayley-Bacharach Schemes and Liaison

In this section we want to characterize maximal CB-schemes X which are contained in a 0-dimensional arithmetically Gorenstein scheme Z by properties of their residual scheme Y = Z \ X. For reduced subschemes X of a 0-dimensional complete intersection Z, the CB-property of X has been characterized in [5], 4.6. Here we need a more general version of that theorem.

4.1. Theorem. Suppose Z ⊆ ℙ^{n} is a 0-dimensional, arithmetically Gorenstein scheme, X ⊆ Z is a subscheme, and Y ⊆ Z is the residual scheme of X in Z, i.e. the subscheme defined by I_{Y/Z} = Ann_{R(Z)}(I_{X/Z}). Then the following conditions are equivalent.

a) X is a CB-scheme.

b) For each subscheme Y′ ⊆ Z with Y ⊆ Y′ and deg Y′ = deg Y + 1 we have (I_{Y′/Z})_{α_{Y /Z}} ⊊ (I_{Y/Z})_{α_{Y /Z}}.

Proof: Condition a) is equivalent to H_{X′}(σ_{X}) = H_{X}(σ_{X}) for each subscheme X′ ⊆ X of degree deg X′ = deg X − 1. Using H_{ω_{R(X)}}(n) = deg X − H_{X}(−n) for n ∈ ℤ, this translates to H_{ω_{R(X′)}}(−σ_{X}) = H_{ω_{R(X)}}(−σ_{X}) − 1 for each subscheme X′ ⊆ X of degree deg X′ = deg X − 1. Now we use σ_{Z} = σ_{X} + α_{Y/Z} (cf. [3]) and ω_{R(X)}(−σ_{X}) ≅ \underline{Hom}_{R(Z)}(I_{X/Z}, ω_{R(Z)})(−σ_{X}) ≅ Ann_{R(Z)}(I_{X/Z})(σ_{Z} − σ_{X}) ≅ (I_{Y/Z})(α_{Y/Z}). We obtain that a) is equivalent to dim_{k}(I_{Y′/Z})_{α_{Y /Z}} = dim_{k}(I_{Y/Z})_{α_{Y /Z}} − 1 for each subscheme Y′ ⊆ Z residual to a subscheme X′ ⊂ X of degree deg X′ = deg X −1. This is clearly equivalent to condition b). □

From this theorem we get the desired characterization of maximal CBschemes via liaison.

4.2. Corollary. Suppose that X, Y, and Z are as in the theorem. Furthermore, assume that X is a CB-scheme and I(Z) is generated in degrees ≤ σ_{X}. Then the following conditions are equivalent.

a) X is a maximal CB-scheme.

b) For each subscheme Y′ ⊆ Y of degree deg Y′ = deg Y − 1 there exists a subscheme Y′′ ⊆ Z with Y′ ⊆ Y′′ and deg Y′′ = deg Y such that (I_{Y′′/Z})_{α_{Y ′/Z}} = (I_{Y′/Z})_{α_{Y ′/Z}}.

Proof: If X is a maximal CB-scheme, then no scheme X′ ⊆ Z with X ⊆ X′ and deg X′ = deg X + 1 is a CB-scheme. Using the theorem, this yields exactly statement b).

Conversely, if b) holds, then no scheme X′ ⊆ Z with X ⊆ X′ and deg X′ = deg X + 1 is a CB-scheme by the theorem. From Proposition 2.4 we conclude that any CB-scheme X′ ⊇ X of degree deg X′ = deg X +1 would have to satisfy I(Z) = ⟨I(Z)_{≤σ_{X}} ⟩ ⊆ ⟨I(X)_{≤σ_{X}} ⟩ = ⟨I(X′)_{≤σ_{X}} ⟩ ⊆ I(X′), and thus would be contained in Z. Since such schemes don’t exist, X is a maximal CB-scheme. □

The following example shows that our assumption on the degrees of the generators of I(Z) is essential.

4.3. Example. Let char k ≠ 2, and let X = {P_{1}, P_{2}, P_{3}} ⊆ ℙ^{2} be the reduced subscheme with P_{1} = (1 : 0 : 0), P_{2} = (1 : 1 : 0), and P_{3} = (1 : 2 : 0). Then X is contained in the 0-dimensional reduced complete intersection Z = X ∪ {Q_{1}, Q_{2}, Q_{3}} with Q_{1} = (1 : 0 : 1), Q_{2} = (1 : 1 : 1), and Q_{3} = (1 : 2 : 1). Clearly, X is a CB-scheme and Y = Z \ X satisfies condition b) of Corollary 4.2, but X is not a maximal CB-scheme.

Sometimes it is advantageous to use arithmetically Gorenstein schemes Z which are not complete intersections. For instance, consider the following case.

4.4. Example. Let Q = 𝒵(X_{0}X_{3} − X_{1}X_{2}) ⊆ ℙ^{3} be the nonsingular quadric surface, and let L ⊆ Q be a line of type (1, 0) on Q. Fix three points P_{1}, P_{2}, P_{3} on L. Then there exist irreducible curves C_{1} of type (3, 4) and C_{2} of type (4, 3) through Y := {P_{1}, P_{2}, P_{3}}. Let Z ⊆ Q be the intersection scheme of C_{1} and C_{2}. By [6], 3.2 we know the minimal free resolution of the ideal sheaf 𝒥_{Z} as an 𝒪_{ℙ^{3}}-module:

0 → 𝒪_{ℙ^{3}}(−9) → 𝒪_{ℙ^{3}}(−7) ⊕ 𝒪_{ℙ^{3}}(−5)^{4} → 𝒪_{ℙ^{3}}(−4)^{4} ⊕ 𝒪_{ℙ^{3}}(−2) → 𝒥_{Z} → 0

Thus Z is a 0-dimensional arithmetically Gorenstein scheme with deg Z = 25, with ΔH_{Z} : 1 3 5 7 5 3 1, and with σ_{Z} = 5. Clearly, Z is not a complete intersection in ℙ^{3}.

Let us use Corollary 4.2 to show that the scheme X := Z\Y is a maximal CB-scheme. The Hilbert function of X is given by ΔH_{X} : 1 3 5 7 4 2, so σ_{X} = 4 and I(Z) is generated by elements of degree ≤ σ_{X}. Now observe that the hyperplanes through two of the points of Y cut out on C_{1} (and hence Z) precisely Y. Then the claim follows immediately from 4.2.

Our next example shows that it is not always possible in 4.2.b) to choose Y′′ = Y.

4.5. Example. Let us reconsider the example of 22 points X ⊆ ℙ^{2} given in 2.11. Clearly, X is contained in the 0-dimensional complete intersection Z = 𝒵(f, g) with f = X_{1}(X_{1} − X_{0}) ⋯ (X_{1} − 4X_{0}) and g = X_{2}(X_{2} − X_{0}) ⋯ (X_{2} − 4X_{0}). Note that I(Z) is generated in degree 5 = σ_{X}. The residual scheme of X in Z is Y = {Q_{1}, Q_{2}, Q_{3}}, where Q_{1} = (1:3:3), Q_{2} = (1:3:4), and Q_{3} = (1:4:4). Since α_{Y/Z} = 2, and since the forms of degree two through Y do not have a fixed point in X, it follows from Theorem 4.1 that X is a CB-scheme.

Now let us use Corollary 4.2 to show that X is a maximal CB-scheme. For Y′ = {Q_{1}, Q_{2}} we can take Y′′ = Y′ ∪ {(1:3:2)}, for Y′ = {Q_{1}, Q_{3}} we can take Y′′ = Y′ ∪ {(1:2:2)}, and for Y′ = {Q_{2}, Q_{3}} we can take Y′′ = Y′ ∪ {(1:2:4)} in order to satisfy condition 4.2.b. Thus X is a maximal CBscheme. Notice that we could not have used Y′′ = Y in either of the three cases.

Of course, we would like to have conditions which are easier to check than 4.2.b) and which nevertheless suffice to show via liaison that certain CB-schemes are maximal. Our next corollary comes in handy here.

Recall that we say that a 0-dimensional subscheme X ⊆ ℙ^{n} has the Cayley-Bacharach property of degree n ≥ 0 (in short, X has CBP (n)), if every subscheme Y ⊆ X of degree deg Y = deg X − 1 satisfies H_{Y} (n) = H_{X}(n). In this terminology, X is a CB-scheme if and only if X has CBP (σ_{X}). Moreover, if X has CBP (n), then X has CBP (n − 1), and every scheme of degree deg X ≥ 2 has CBP (0).

4.6. Corollary. Suppose that X, Y, and Z are as in Corollary 4.2. If Y has CBP (α_{Y/Z}), then X is a maximal CB-scheme.

Proof: In Corollary 4.2, we have I(Y′)_{n} = I(Y)_{n} for each subscheme Y′ ⊆ Y of degree deg Y′ = deg Y − 1 and each n ≤ α_{Y/Z}. Thus we can take Y′′ := Y each time in order to satisfy condition 4.2.b. □

It follows from Example 4.5 that the converse of this corollary is not true. Nevertheless, it turns out to be very efficient in practice. For instance, several examples in section 2 were constructed this way.

4.7. Example. Let us show that the scheme X of Example 2.7 is a maximal CB-scheme by using 4.1 and 4.6. The residual scheme Y of X in the complete intersection Z = 𝒵(f, g) satisfies ΔH_{Y} : 1 2 3 4 5 3 1, σ_{Y} = 5, and α_{Y/Z} = σ_{Z} − σ_{X} = 5. Sketch of Y:

• •  • •
•  • • •
• • •
• • •  •
• •  • •

Clearly, (I_{Y/Z})_{α_{Y /Z}} is generated by \overline{F}, \overline{G}, and \overline{H}, where F = X_{1}(X_{1} − X_{0}) ⋯ (X_{1} − 4X_{0}), G = X_{2}(X_{2} − X_{0}) ⋯ (X_{2} − 4X_{0}), H = (X_{2} − X_{1})(X_{2} − X_{1} − X_{0})(X_{2} − X_{1} + X_{0})H′, and H′ defines the unique conic through Q_{1} = (1:0:3), Q_{2} = (1:0:4), Q_{3} = (1:1:4), Q_{4} = (1:3:0), Q_{5} = (1:4:0), and Q_{6} = (1:4:1). Thus (I_{Y/Z})_{α_{Y /Z}} has no fixed points in X, and X is a CB-scheme by 4.1.

By Corollary 4.6, it suffices now to show that Y is a CB-scheme. For this we observe that Y is contained in the complete intersection W = 𝒵(F, G) and apply 4.1 once again. The unique curve of minimal degree through W \ Y = {(1:0:2), (1:1:3), (1:2:0), (1:2:4), (1:3:1), (1:4:2)} is 𝒵((X_{2} − X_{1} − 2X_{0})(X_{2} − X_{1} + 2X_{0})) and does not contain any point of Y. Thus Y is a CB-scheme, and X is a maximal CB-scheme.

Asking Y to have CBP (α_{Y/Z}) is usually a much weaker condition than asking Y to be a CB-scheme. Here is an example where Y indeed has CBP (α_{Y/Z}), but not CBP (σ_{Y}).

4.8. Example. Let char k = 0 or char k ≥ 11. Consider the reduced subscheme X ⊆ D_{+}(X_{0}) ⊆ ℙ^{2} consisting of the following 47 points:

(0, 7) •  • • • • • • (7, 7)
•  • • • • •
•  •  • • • •
• •  •  • • •
• • •  •  • •
• • •  •  •
•  • • •  •
(0, 0) • •  • • •  • (7, 0)

Obviously X is contained in the complete intersection scheme Z = 𝒵(f, g) with f = X_{1}(X_{1} − X_{0}) ⋯ (X_{1} − 7X_{0}) and g = X_{2}(X_{2} − X_{0}) ⋯ (X_{2} − 7X_{0}). Let Y = Z \ X be the residual scheme of X in Z. Since α_{Y/Z} = 3 and (I_{Y/Z})_{3} = k · \overline{h} with h = (X_{1} + X_{2} − 2X_{0})(X_{1} + X_{2} − 6X_{0})(X_{1} + X_{2} − 8X_{0}), it follows from 4.1 that X is a CB-scheme. In fact, Y has CBP (3), so X is a maximal CB-scheme by Corollary 4.6. But from ΔH_{Y} : 1 2 3 3 3 2 2 1 we conclude that Y is not a CB-scheme (cf. [2], 2.5).

The next proposition provides a partial converse to Corollary 4.6. We obtain an explicit way to check whether Y has CBP (α_{Y/Z}) and to show that certain CB-schemes are not maximal.

4.9. Proposition. Suppose that X, Y, and Z are as in Theorem 4.1. Then Y does not have CBP (α_{Y/Z}) if and only if one of the following two cases occurs.

a) There exists a subscheme Y′ ⊆ Y of degree deg Y′ = deg Y − 1 such that α_{Y′/Z} = α_{Y/Z} − 1 and dim_{k}(I_{Y′/Z})_{α_{Y ′/Z}} = 1.

b) There exists a subscheme Y′ ⊆ Y of degree deg Y′ = deg Y − 1 such that α_{Y′/Z} = α_{Y/Z} and dim_{k}(I_{Y′/Z})_{α_{Y /Z}} = dim_{k}(I_{Y/Z})_{α_{Y /Z}} + 1.

Moreover, if X is a CB-scheme, Z is reduced, and Y′ as in a) or b) has the property that the forms in (I_{Y′/Z})_{α_{Y ′/Z}} do not vanish simultaneously at a point of Z \ Y′, then X is not a maximal CB-scheme.

Proof: It is clear that Y does not have CBP (α_{Y/Z}) if a) or b) occurs. Conversely, suppose that Y does not have CBP (α_{Y/Z}), i.e. there exists a subscheme Y′ ⊆ Y of degree deg Y′ = deg Y − 1 such that I(Y′)_{α_{Y /Z}} ⊋ I(Y)_{α_{Y /Z}}. It follows that H_{Y′}(α_{Y/Z}) = H_{Y} (α_{Y/Z}) − 1. Since α_{Y′/Z} ≤ α_{Y/Z}, it remains to show α_{Y′/Z} = α_{Y/Z} − 1. For this we observe that if f ∈ (I_{Y′/Z})_{α_{Y ′/Z}} and ℓ ∈ R(Z)_{1} is the image of the equation of a line through the point P ∈ ℙ^{n} with 𝒪_{Y′,P} ≠ 𝒪_{Y,P}, then fℓ ∈ I_{Y/Z}. Finally, the additional claim follows by applying Theorem 4.1 to the present situation. □

We end this section with a couple of examples which show how one can apply Proposition 4.9 in concrete situations.

4.10. Example Let char k = 0 or char k > 5, and let X ⊆ ℙ^{2} be given by

• • • •
(0, 2) • •  • • • (5, 2)
• • • •  •
(0, 0) • • • • • • (5, 0)

Clearly, X is contained in a 0-dimensional reduced complete intersection Z of degree 24. The residual scheme Y of X in Z satisfies α_{Y/Z} = 2, and the subscheme Y′ = Y \ {(1 : 5 : 3)} of Y has α_{Y′/Z} = 1. Thus we are in case a) of the proposition. Since the line through Y′ does not contain any point of Z \ Y′, it follows that X is not a maximal CB-scheme.

4.11. Example. Let char k = 0 or char k > 5, and let X ⊆ ℙ^{2} be given by

• • • •  • (6, 3)
(0, 2) • •  • • • •
• • • •  • • (6, 1)
(0, 0) •  • • • •

Clearly, X is contained in a 0-dimensional reduced complete intersection Z of degree 28. The residual scheme Y of X in Z has α_{Y/Z} = 2, Hilbert function ΔH_{Y} : 1 2 2 1, and σ_{Y} = 2. It is not a CB-scheme, i.e. it does not have CBP (2), because it is not a complete intersection (cf. [8], 1.1).

The subscheme Y′ = Y \ {(1 : 5 : 3)} of Y also has α_{Y′/Z} = 2, but (I_{Y′/Z})_{2} ⊋ (I_{Y/Z})_{2}. Thus we are in case b) of the proposition. More precisely, the curves of degree two passing through Y′ decompose into the line through the diagonal and an arbitrary line through (1 : 1 : 0). Since they do not simultaneously contain a point of Z \ Y′, it follows that X is not a maximal CB-scheme.

ACKNOWLEDGEMENTS

During the preparation of this paper, the authors had numerous opportunities to use the computer algebra systems CoCoA (developed at Universitá di Genova) and COP (developed at Universität Regensburg). They are also indebted to S. Giuffrida and A. Ragusa (Catania) for helping them with Example 4.4.

REFERENCES

[1] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math. 1327, Springer, Heidelberg 1988.

[2] E.D. Davis, 0-dimensional subschemes of ℙ^{2}: New applications of Castelnuovo’s function, Ann. Univ. Ferrara, Sez. VII 32 (1986), 93– 107.

[3] E.D. Davis, A.V. Geramita, and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), 593–597.

[4] J. Elias and R.M. Miró-Roig, A strange property of the sets of points in uniform position, Comm. Algebra 21 (1993), 1577–1585.

[5] A.V. Geramita, M. Kreuzer, and L. Robbiano, Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339 (1993), 163–189.

[6] S. Giuffrida, R. Maggioni, and A. Ragusa, Resolutions of 0-dimen-sional subschemes of a smooth quadric, in: F. Orecchia and L. Chiantini (eds.), Zero-Dimensional Schemes, Proc. Conf. Ravello / Italy 1992, de Gruyter, Berlin 1994, pp. 191–204.

[7] M. Kreuzer, On the canonical module of a 0-dimensional scheme, Can. J. Math. 46 (1994), 357–379.

[8] M. Kreuzer, On 0-dimensional complete intersections, Math. Ann. 292 (1992), 43–58.

[9] G. Niesi and L. Robbiano, Disproving Hibi’s conjecture with CoCoA or projective curves with bad Hilbert functions, in: F. Eyssette and A. Galligo (eds.), Proc. Conf. Computational Algebraic Geometry MEGA 92, Birkhäuser, Basel 1993, pp. 195–201.