ON THE CANONICAL MODULE OF A 0-DIMENSIONAL SCHEME

Martin Kreuzer

Abstract. The main topic of this paper is to give characterizations of geometric properties of 0-dimensional subschemes X ⊆ ℙ^{d} in terms of the algebraic structure of the canonical module of their projective coordinate ring. We characterize Cayley-Bacharach, (higher order) uniform position, linearly and higher order general position properties, and derive inequalities for the Hilbert functions of such schemes. Finally we relate the structure of the canonical module to properties of the minimal free resolution of X.

AMS Classification: 14C99, 13C05

Introduction. In the study of 0-dimensional schemes X embedded in some projective space ℙ^{d} over an algebraically closed field k one oftentimes considers the homogeneous ideal I_{X} and the projective coordinate ring R = k[X_{0}, …, X_{d}]/I_{X} of the embedding X ⊆ ℙ^{d}. Here we want to pursue a “dual” point of view and try to characterize geometric properties of that embedding in terms of algebraic properties of the canonical module ω_{R} of R.

The canonical module can be described by ω_{R} ≅ \underline{Hom}_{k[x_{0}]}(R, k[x_{0}])(−1), where \underline{Hom} means graded homomorphisms and x_{0} is the image in R of a linear form which does not pass through any point in the support of X. This module is a finitely generated graded R-module which starts in degree −σ_{X}, where σ_{X} = max{n ∈ ℤ : H_{X}(n) < deg X}. The multiplication maps of this module will be used to describe geometrical properties of X like the Cayley-Bacharach property, uniform position property, general position property, etc. The starting point of our investigations is the following theorem whose reduced version was shown in [GKR] and whose nonreduced version can be found in [K1].

Theorem 1. The scheme X is locally Gorenstein and a Cayley-Bacharach scheme if and only if there exists an element φ ∈ (ω_{R})_{−σ_{X}} which satisfies Ann_{R}(φ) = (0).

Here we say that X is a Cayley-Bacharach scheme, if every hypersurface of degree σ_{X} which contains a subscheme of degree deg X − 1 of X automatically contains X. In section 2 we shall give a new proof of Theorem 1 which is based on a detailed study of R and ω_{R} in section 1. It was already pointed out in [GKR] that structural results like Theorem 1 tend to have implications for the growth behaviour of the Hilbert function of X.

Corollary. If X is locally Gorenstein and a Cayley-Bacharach scheme, then H_{X}(n) + H_{X}(σ_{X} − n) ≤ deg X for all n ∈ ℤ.

If we drop the assumption “X is locally Gorenstein” in Theorem 1, we obtain the following characterization.

Theorem 2. The scheme X is a Cayley-Bacharach scheme if and only if the multiplication map R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is nondegenerate.

Equivalently, we could have defined Cayley-Bacharach schemes as the ones for which all subschemes Y ⊆ X of degree deg X − 1 have Hilbert function H_{Y} (n) = min{H_{X}(n), deg X − 1}. This approach is used in section 3 where we discuss higher uniformities. We say that X is n-uniform, if every subscheme Y ⊆ X with deg X − n ≤ deg Y ≤ deg X has the Hilbert function H_{Y} (n) = min{H_{X}(n), deg Y }. Then X is in uniform position if and only if X is (deg X − 1)-uniform. We prove the following characterizations.

Theorem 3. Let Δ_{X} := deg X − H_{X}(σ_{X}). Then X is Δ_{X}-uniform if and only if the multiplication map R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is biinjective.

Corollary. If X is Δ_{X}-uniform, then H_{X}(n) + H_{X}(σ_{X} − n) ≤ deg X − Δ_{X} + 1 for all n ∈ {0, …, σ_{X}}.

Theorem 4. The scheme X is in uniform position if and only if for every n ∈ {0, …, σ_{X}} the multiplication map R_{n} ⊗ (ω_{R})_{−n} ⟶ (ω_{R})_{0} is biinjective.

Here “biinjective” means that r ∈ R_{n}, φ ∈ (ω_{R})_{−n}, and r · φ = 0 imply r = 0 or φ = 0. Of course, many intermediate uniformities can be characterized in an analogous way.

Recently another kind of uniformity has received some attention. We say that X is in linearly general position, if deg(X ∩ L) ≤ 1 + dim L for every proper linear subspace L ⊂ ℙ^{d}. In section 4 we characterize schemes in linearly general position as follows.

Theorem 5. A nondegenerate subscheme X ⊆ ℙ^{d} is in linearly general position if and only if the multiplication map R_{1} ⊗ (ω_{R})_{−1} ⟶ (ω_{R})_{0} is biinjective.

Corollary. If X is nondegenerate and in linearly general position, then we have ΔH_{X}(n) ≥ d for all n ∈ {1, …, σ_{X}}.

In particular, H_{X}(n) ≥ min{1 + nd, deg X} for all n ≥ 0.

The second inequality of this corollary has been obtained in [EH] with a different method. Theorem 5 is also generalized for schemes in quadratically or higher order general position. We go on to describe the relation of those notions with the classical terminology “X imposes independent conditions on forms of degree n”.

Our last section deals with characterizations of properties of the minimal projective resolution of R as a k[X_{0}, …, X_{d}]-module using the multiplication maps of ω_{R}. For example, we show the following theorem.

Theorem 6. If the projective resolution of X is almost linear, then the multiplication map R_{σ_{X}−α_{X}+2} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−α_{X}+2} is injective.

Moreover, we give characterizations of schemes with almost linear or higher order resolutions in terms of the algebraic structure of ω_{R}.

1. The Projective Coordinate Ring and Its Canonical Module. This section consists of a series of definitions and easy lemmas which will prove useful later on. The canonical module is defined and some of its most elementary properties are noted.

First of all we want to fix the notations which will be used throughout the paper. We work over an algebraically closed field k of arbitrary characteristic. By ℙ^{d} we denote the d-dimensional projective space over k. We want to study 0-dimensional subschemes X of ℙ^{d}. Since we always consider X together with a fixed embedding X ⊆ ℙ^{d}, we shall say “X has property P ” when we really mean “the embedding X ⊆ ℙ^{d} has property P ”.

The coordinates {X_{0}, …, X_{d}} of ℙ^{d} are always chosen such that no point of the support of X lies on the hyperplane V(X_{0}). By I_{X} we denote the homogeneous saturated ideal of X in k[X_{0}, …, X_{d}]. The projective (or homogeneous) coordinate ring of X is then given by R := k[X_{0}, …, X_{d}]/I_{X}. It is a standard graded k-algebra R = ⨁_{n≥0} R_{n},

i.e. we have R_{0} = k, dim_{k} R_{1} is finite, and R = k[R_{1}]. We let 𝔪 := ⨁_{n>0} R_{n} be the

homogeneous maximal ideal of R. The image of X_{i} in R is denoted by x_{i} for i = 0, …, d. By the choice of coordinates, x_{0} ∈ R_{1} is not a zero divisor of R. Hence R is a 1dimensional Cohen-Macaulay ring and \overline{R} := R/(x_{0}) is a 0-dimensional ring.

The Hilbert function of X is denoted by H_{X} : ℤ ⟶ ℕ (n ⟼ dim_{k} R_{n}). We have H_{X}(n) = 0 for n < 0, H_{X}(0) = 1, and H_{X}(n) = deg X for n >> 0. The invariant

σ_{X} := max{n ∈ ℤ : H_{X}(n) < deg X}

will play an important rôle throughout this paper. By ΔH_{X} : ℤ ⟶ ℕ (n ⟼ H_{X}(n) − H_{X}(n−1)) we denote the first difference function of H_{X}. Because of the exact sequence of graded R-modules

0 ⟶ R(−1) ⟶^{x_{0}} R ⟶ \overline{R} ⟶ 0

we have ΔH_{X}(n) = dim_{k} \overline{R}_{n} for all n ∈ ℤ. Here R(−1) denotes the shift of R, i.e. the graded R-module with R(−1)_{n} = R_{n−1} for all n ∈ ℤ. Since \overline{R} is also a standard graded k-algebra, we have ΔH_{X}(n) ≠ 0 if and only if n ∈ {0, …, σ_{X} +1}, and therefore H_{X}(0) < … < H_{X}(σ_{X}+1) = deg X. The number Δ_{X} := H_{X}(σ_{X}+1)−H_{X}(σ_{X}) denotes the last nonzero difference of H_{X}. It is clear that R_{n} = (x_{0})_{n} for every n > σ_{X} + 1, hence R_{n} = x_{0}^{n−σ_{X}−1}R_{σ_{X}+1} for every n ≥ σ_{X} + 1.

Now we shall examine subschemes Y ⊆ X of degree deg X − 1. Let I_{Y/X} be the ideal of Y in R, and let α_{Y/X} := min{n ∈ ℕ : (I_{Y/X})_{n} ≠ 0} be its initial degree. Then I_{Y/X} is a saturated ideal of R and α_{Y/X} is well-defined, because dim_{k}(R/I_{Y/X})_{n} = H_{Y} (n) = deg Y < deg X = dim_{k} R_{n} for n >> 0.

The projective coordinate ring of Y is S = R/I_{Y/X}. As Y ⊆ X, the element x_{0} is not a zero divisor of S. Consequently, \overline{S} := S/(x_{0}) has Hilbert function dim_{k} \overline{S}_{n} = ΔH_{Y} (n) = H_{Y} (n) − H_{Y} (n − 1) for n ∈ ℤ. From Y ⊆ X and deg Y = deg X − 1 we get ΔH_{Y} (n) ≤ ΔH_{X}(n) for n ∈ ℤ and ∑_{n∈ℤ} ΔH_{Y} (n) = ∑_{n∈ℤ} ΔH_{X}(n) − 1. By definition of α_{Y/X}, we have ΔH_{Y} (α_{Y/X}) < ΔH_{X}(α_{Y/X}).

Altogether, the Hilbert function of Y is given by

H_{Y} (n) = \{ \begin{matrix}H_{X}(n) & for n < α_{Y/X}, \\ H_{X}(n) − 1 & for n ≥ α_{Y/X}. \end{matrix}

We shall call α_{Y/X} the degree of Y in X. From the above discussion it follows that α_{Y/X} ≤ σ_{X} + 1.

A nonzero element f_{Y}^{∗} ∈ (I_{Y/X})_{α_{Y /X}} is called a minimal separator of Y. Since x_{0} is not a zero divisor on R, the element x_{0}^{n}f_{Y}^{∗} is a k-basis of (I_{Y/X})_{α_{Y /X}+n} for each n ≥ 0. A nonzero element f_{Y} ∈ k · x_{0}^{σ_{X}+1−α_{Y /X}} f_{Y}^{∗} is called a separator of Y. For any r ∈ R_{n}, n ≥ 0, we get rf_{Y} ∈ (I_{Y/X})_{σ_{X}+1+n}, hence rf_{Y} = λx_{0}^{n}f_{Y} for some λ ∈ k.

Next we want to derive a local description of separators which will be useful later on. The coordinate ring Γ of X in the affine space 𝔸^{d} ≅ D_{+}(X_{0}) is Γ = R/(x_{0} − 1). The canonical epimorphism R −↠ Γ is given by dehomogenization. If we equip Γ with the ascending filtration F induced by the degree filtration of k[X_{1}, …, X_{d}] via Γ ≅ k[X_{1}, …, X_{d}]/\overline{I}_{X}, where \overline{I}_{X} = I_{X}/I_{X} ∩ (X_{0} − 1), we can form the homogenization f^{∗} = x_{0}^{ord_{F} f}f(x_{1}/x_{0}, …, x_{d}/x_{0}) of any element f ∈ Γ. For details on those procedures the reader may consult [KK].

Lemma 1.1. If we restrict dehomogenization to elements of degree σ_{X} + 1, we obtain an isomorphism R_{σ_{X}+1} ⟶^{∼} Γ.

Proof: Because of dim_{k} R_{σ_{X}+1} = deg X = dim_{k} Γ, it suffices to show that every element of Γ is the dehomogenization of an element of R_{σ_{X}+1}. For f ∈ Γ we have ord_{F} f ≤ σ_{X} +1, because no element of gr_{F}(Γ) ≅ \overline{R} has a degree larger than that. Now x_{0}^{σ_{X}+1−ord_{F} f}f^{∗} ∈ R_{σ_{X}+1} has dehomogenization f. □

Combining the isomorphism of the lemma with the canonical isomorphism Γ ⟶^{∼} ∏_{P ∈X} O_{X,P}, we obtain an isomorphism ı : R_{σ_{X}+1} ⟶^{∼} ∏_{P ∈X} O_{X,P} which maps each homo-

geneous element r ∈ R_{σ_{X}+1} to the tupel (r_{P})_{P ∈X} of its germs at the points of X. In particular, we have ı(x_{0}^{σ_{X}+1}) = (1)_{P ∈X}.

The ideal of Y in ∏_{P ∈X} O_{X,P} is of the form k · (0, …, 0, s_{P} , 0, …, 0), where P ∈ X

and s_{P} ∈ 𝔖(O_{X,P}) is an element of the socle 𝔖(O_{X,P}) = {r_{P} ∈ O_{X,P} : 𝔪_{X,P} · r_{P} = 0} of O_{X,P}. Clearly, ı(f_{Y}) = (0, …, 0, s_{P} , 0, …, 0) for some separator f_{Y} ∈ R_{σ_{X}+1} of Y, and the image of any other separator of Y is a nonzero scalar multiple of this element. In particular, for two subschemes Y, Y′ ⊆ X with deg Y = deg Y′ = deg X −1 we obtain

f_{Y} · f_{Y′} ∈ \{ \begin{matrix}k · x_{0}^{σ_{X}+1}f_{Y} & if dim_{k} O_{X,P} = 1 and Y = Y′ = X \ \{P \}, \\ (0) & otherwise. \end{matrix}

When dealing with ideals of R defining subschemes of X, it is necessary to keep the following lemma in mind.

Lemma 1.2. If J ⊆ R is a homogeneous ideal, then its saturation J^{sat} := {r ∈ R : 𝔪^{n}r ⊆ J for some n ≥ 0} is already given by

J^{sat} = {r ∈ R : x_{0}^{n}r ∈ J for some n ≥ 0}.

In particular, J is saturated if and only if x_{0}r ∈ J implies r ∈ J.

Proof: As noted earlier, R_{n} = x_{0}^{n−σ_{X}−1}R_{σ_{X}+1} for n ≥ σ_{X} + 1. Thus x_{0}^{n}r ∈ J implies 𝔪^{σ_{X}+1+n}r = 𝔪^{σ_{X}+1}x_{0}^{n}r ⊆ J, and the claims of the lemma follow. □

In order to define the canonical module of R, we need some basic properties of the category of graded R-modules. Its homomorphisms are homogeneous R-linear maps φ : M ⟶ N. We also let \underline{Hom}_{R}(M, N) be the graded R-module whose homogeneous components are the sets of homogeneous R-linear maps φ : M ⟶ N(n) for n ∈ ℤ. The functor H_{𝔪}^{0}(M) := {x ∈ M : 𝔪^{n}x = 0 for some n ≥ 0} is a left-exact covariant functor on that category. We can form its right derived functors H_{𝔪}^{i}(–), i ≥ 0. The modules H_{𝔪}^{i}(M) are graded R-modules whose underlying R-modules agree with the usual local cohomology modules. We equip k with the trivial grading and let M^{∗} := \underline{Hom}_{k}(M, k) for every graded R-module M.

Definition. The graded R-module ω_{R} := H_{𝔪}^{1}(R)^{∗} is called the canonical module of R.

The following properties of ω_{R} are proven in [GW].

Lemma 1.3. a) The graded R-module ω_{R} is finitely generated.

b) There is a canonical isomorphism of graded R-modules

ω_{R} ≅ \underline{Hom}_{k[x_{0}]}(R, k[x_{0}])(−1).

In particular, x_{0} is not a zero divisor on ω_{R}.

c) There is an exact sequence of graded R-modules

0 ⟶ R ⟶ Γ_{∗}O_{X} ⟶ H_{𝔪}^{1}(R) ⟶ 0.

In particular, the Hilbert function of ω_{R} satisfies

H_{ω_{R}}(n) = deg X − H_{X}(−n)  for all n ∈ ℤ.

d) If Y ⊆ X is a subscheme and S = R/I_{Y/X} its projective coordinate ring, then there is a canonical isomorphism of graded R-modules

ω_{S} ≅ {φ ∈ ω_{R} : I_{Y/X} · φ = 0}.
Notice that 1.3.c implies −σ_{X} = min{n ∈ ℤ : (ω_{R})_{n} ≠ 0} and

H_{ω_{R}}(−σ_{X}) = Δ_{X} < … < H_{ω_{R}}(0) = deg X − 1 < H_{ω_{R}}(1) = deg X.

From now on, we shall always use 1.3.b to think of elements of ω_{R} as k[x_{0}]-linear forms on R, and we shall use 1.3.d to identify ω_{S} with a submodule of ω_{R}. Comparing Hilbert functions then yields σ_{Y} ≤ σ_{X} for every subscheme Y ⊆ X.

Our final three lemmas of this section will help us deal with those linear forms.

Lemma 1.4. For φ ∈ ω_{R}, the following conditions are equivalent.

a) φ = 0

b) φ |_{R_{σ_{X} +1}}: R_{σ_{X}+1} ⟶ k[x_{0}] is the zero linear transformation.

Proof: It suffices to show that b) implies a). If r ∈ R_{n}, n ≤ σ_{X}, then 0 = φ(x_{0}^{σ_{X}+1−n}r) = x_{0}^{σ_{X}+1−n}φ(r), so φ(r) = 0. And if r ∈ R_{n}, n ≥ σ_{X} + 1, then we can write r = x_{0}^{n−σ_{X}−1}r′ with r′ ∈ R_{σ_{X}+1}, and φ(r) = x_{0}^{n−σ_{X}−1}φ(r′) = 0 again. □

Lemma 1.5. There is a 1 - 1 correspondence between elements φ of (ω_{R})_{−σ_{X}} and k-linear maps \overline{φ} : R_{σ_{X}+1} ⟶ k with \overline{φ}(x_{0}R_{σ_{X}}) = 0.

Proof: If φ ∈ (ω_{R})_{−σ_{X}} ≅ Hom_{k[x_{0}]}(R, k[x_{0}])_{−σ_{X}−1}, then its restriction \overline{φ} := φ |_{R_{σ_{X} +1}}: R_{σ_{X}+1} ⟶ k vanishes on x_{0}R_{σ_{X}}, because we have φ(x_{0}R_{σ_{X}}) = x_{0}φ(R_{σ_{X}}) ⊆ x_{0}k[x_{0}]_{−1} = (0).

Conversely, if \overline{φ} : R_{σ_{X}+1} ⟶ k is a k-linear map, we define a homogeneous k-linear map φ : R ⟶ k[x_{0}] of degree −σ_{X} − 1 as follows:

1) For r ∈ R_{n}, n ≤ σ_{X}, we let φ(r) := 0.

2) For r ∈ R_{n}, n ≥ σ_{X} + 1, we write r = x_{0}^{n−σ_{X}−1}r′ with r′ ∈ R_{σ_{X}+1} and let

φ(r) := x_{0}^{n−σ_{X}−1}\overline{φ}(r′).

Now \overline{φ}(x_{0}R_{σ_{X}}) = 0 is exactly the right condition to make φ even k[x_{0}]-linear, so that φ defines an element of (ω_{R})_{−σ_{X}}. Obviously those two constructions are inverses of each other and define the desired bijection. □

Lemma 1.6. For φ ∈ ω_{R} and Y ⊆ X with deg Y = deg X − 1, the following conditions are equivalent.

a) f_{Y} · φ = 0

b) φ(f_{Y}) = 0

Proof: If φ(f_{Y}) = 0 and r ∈ R_{n}, n ≥ 0, then f_{Y} φ(r) = φ(rf_{Y}) ∈ k · x_{0}^{n}φ(f_{Y}) = (0), because rf_{Y} ∈ k · x_{0}^{n}f_{Y}, as noted earlier. □

2. Cayley-Bacharach Schemes. In this section we want to study the canonical module of 0-dimensional schemes having the Cayley-Bacharach property with respect to hypersurfaces of degree σ_{X}, the maximum possible degree (cf. [K2]). In particular, Theorems 1 and 2 of the introduction will follow from Theorems 2.4 and 2.6, respectively.

Definition. A 0-dimensional scheme X ⊆ ℙ^{d} is called a Cayley-Bacharach scheme, if every hypersurface of degree σ_{X} which contains a subscheme Y ⊆ X of degree deg Y = deg X − 1 automatically contains X.

Proposition 2.1. The following conditions are equivalent.

a) X is a Cayley-Bacharach scheme.

b) If Y ⊆ X and deg Y = deg X − 1, then α_{Y/X} = σ_{X} + 1.

c) Each subscheme Y ⊆ X of degree deg X − 1 has Hilbert function H_{Y} (n) = min{H_{X}(n), deg X − 1} for all n ∈ ℤ.

Proof: “a)⇒b)” By definition, α_{Y/X} is the least degree of a hypersurface containing Y, but not X. Our assumption implies α_{Y/X} ≥ σ_{X} + 1. The other inequality holds always.

“b)⇒c)” This follows from the description of H_{Y} in section 1.

“c)⇒a)” Since H_{X}(σ_{X}) ≤ deg X − 1, we have H_{Y} (σ_{X}) = H_{X}(σ_{X}), and this means that every hypersurface of degree σ_{X} which contains Y also contains X. □

In order to be able to characterize Cayley-Bacharach schemes in terms of their canonical modules, we need two lemmas.

Lemma 2.2. a) Let n ≥ 0 and r ∈ R_{n} \ {0}. Then there exists an element r′ ∈ R_{σ_{X}+1}, a subscheme Y ⊆ X with deg Y = deg X − 1, and a separator f_{Y} ∈ R_{σ_{X}+1} of Y such that rr′ = x_{0}^{n}f_{Y}.

b) If Y ⊆ X is a subscheme and deg Y ≤ n ≤ deg X, then there exists a subscheme Z of X such that deg Z = n and Y ⊆ Z ⊆ X.

Proof: “a)” Since r ≠ 0, there is a point P ∈ X such that (r_{Q})_{Q∈X} := ı(x_{0}^{σ_{X}+1−n}r) satisfies r_{P} ≠ 0. Then we can find an element r′_{P} ∈ O_{X,P} such that s_{P} := r_{P} r′_{P} ∈ 𝔖(O_{X,P}) is in the socle of O_{X,P}. Now we use r′ := ı^{−1}((0, …, 0, r′_{P} , 0, …, 0)) and f_{Y} := ı^{−1}((0, …, 0, s_{P} , 0 …, 0)) and obtain the desired equality rr′ = x_{0}^{n}f_{Y}.

“b)” By induction, it suffices to do the case n = deg X − 1. Let I_{Y/X} ⊆ R be the ideal of Y, and let r ∈ (I_{Y/X})_{m} \ {0} for some m ≥ 0. Using a) we find a subscheme Z ⊆ X with deg Z = deg X − 1 and a separator f_{Z} ∈ R_{σ_{X}+1} of Z such that rr′ = x_{0}^{m}f_{Z} for some r′ ∈ R_{σ_{X}+1}. Hence I_{Z/X} = (f_{Z})^{sat} ⊆ (r)^{sat} ⊆ I_{Y/X}, and thus Y ⊆ Z ⊆ X.

□

Lemma 2.3. A homogeneous element φ ∈ ω_{R} satisfies Ann_{R}(φ) = (0) if and only if φ(f_{Y}) ≠ 0 for any separator of a subscheme Y ⊆ X of degree deg Y = deg X − 1.

Proof: If Ann_{R}(φ) = (0), then f_{Y} ·φ ≠ 0, and thus φ(f_{Y}) ≠ 0 by 1.6. Conversely, if r · φ = 0 for some r ∈ R_{n} \ {0}, n ≥ 0, then we can use 2.2.a to conclude that x_{0}^{n}f_{Y} φ = 0 for some separator f_{Y} of a subscheme Y ⊆ X of degree deg X − 1. By 1.3, we then have f_{Y} · φ = 0 and φ(f_{Y}) = 0, a contradiction. □

Theorem 2.4. The following conditions are equivalent.

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

b) There exists an element φ ∈ (ω_{R})_{−σ_{X}} such that φ(f_{Y}) ≠ 0 for all separators f_{Y} ∈ R_{σ_{X}+1} of subschemes Y ⊆ X of degree deg Y = deg X − 1.

c) There exists an element φ ∈ (ω_{R})_{−σ_{X}} such that Ann_{R}(φ) = (0).

d) A generic element φ ∈ (ω_{R})_{−σ_{X}} satisfies Ann_{R}(φ) = (0).

Proof: “a)⇒d)” Since X is locally Gorenstein, there is for each point P ∈ X a unique subscheme Y ⊆ X with deg Y = deg X − 1 corresponding to a socle element of O_{X,P}. Since X is a Cayley-Bacharach scheme, a separator f_{Y} ∈ R_{σ_{X}+1} satisfies f_{Y} ∉ x_{0}R_{σ_{X}}. Thus a generic k-linear map \overline{φ} : R_{σ_{X}+1} ⟶ k with \overline{φ}(x_{0}R_{σ_{X}}) = 0 satisfies \overline{φ}(f_{Y}) ≠ 0 for those finitely many subschemes Y ⊆ X. Now 1.5 and 2.3 together imply the claim.

“d)⇒c)” is clear and “b)⇔c)” follows from 2.3.

“b)⇒a)” From R(σ_{X}) ≅ Rφ ⊆ ω_{R} and dim_{k} R_{n} = deg X = dim_{k}(ω_{R})_{n} for n >> 0 we conclude that ω_{X,P} ≅ O_{X,P} for all P ∈ X, i.e. that X is locally Gorenstein. Since φ(f_{Y}) ≠ 0 and φ(x_{0}R_{σ_{X}}) ⊆ k[x_{0}]_{−1} = (0), we must have f_{Y} ∉ x_{0}R_{σ_{X}}, and therefore α_{Y/X} = σ_{X} + 1, for all subschemes Y ⊆ X of degree deg X − 1. Hence X is a Cayley-Bacharach scheme by 2.1. □

The corollary of Theorem 1 stated in the introduction follows now by simply comparing Hilbert functions for the inclusion R(σ_{X}) ≅ Rφ ⊆ ω_{R}. If we have equality here, ω_{R} is a graded free R-module of rank one. It is well-known that this is the case if and only if R is a Gorenstein ring or, in other words, if X is arithmetically Gorenstein.

Corollary 2.5. a) If X is a Cayley-Bacharach scheme and Δ_{X} = 1, then X is

locally Gorenstein.

b) A 0-dimensional scheme X is arithmetically Gorenstein if and only if X is a Cayley-Bacharach scheme and H_{X}(n) + H_{X}(σ_{X} − n) = deg X for all n ∈ ℤ.

Proof: In view of what we explained above, it only remains to show “a)”. Suppose there is a point P ∈ X such that dim_{k} 𝔖(O_{X,P}) ≥ 2. Let f_{Y} , f_{Y′} ∈ R_{σ_{X}+1} be separators of subschemes Y, Y′ ⊆ X with deg Y = deg Y′ = deg X − 1 such that ı(f_{Y}) and ı(f_{Y′}) are k-linearly independent elements of 𝔖(O_{X,P}). Since Δ_{X} = dim_{k} \overline{R}_{σ_{X}+1} = 1, there are scalars λ, λ′ ∈ k such that λf_{Y} + λ′f_{Y′} ∈ x_{0}R_{σ_{X}}. But then the subscheme Y′′ = V(λf_{Y} + λ′f_{Y′}) ⊆ X satisfies deg Y′′ = deg X − 1 and α_{Y′′/X} ≤ σ_{X}, a contradiction.

□

Example. Let X ⊆ ℙ^{2} be concentrated at two points P_{1} and P_{2} with O_{X,P_{i}} ≅ O_{ℙ^{2},P_{i}}/𝔪_{ℙ^{2},P_{i}}^{2} for i = 1, 2. Then X is not locally Gorenstein, and from its Hilbert function H_{X} : 1 3 5 6 6 ⋯ we read off Δ_{X} = 1. Therefore we know immediately that X is not a Cayley-Bacharach scheme.

Our next theorem contains Theorem 2 of the introduction.

Theorem 2.6. The following conditions are equivalent.

a) X is a Cayley-Bacharach scheme.

b) The multiplication map R_{σ_{X}+1} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{1} is nondegenerate.

c) The multiplication map R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is nondegenerate.

d) For all m, n ≥ 0, the multiplication map R_{m} ⊗ (ω_{R})_{−σ_{X}+n} ⟶ (ω_{R})_{−σ_{X}+m+n} is nondegenerate.

Proof: If φ ∈ (ω_{R})_{−σ_{X}+n} and R_{m}φ = 0 for some m, n ≥ 0, then x_{0}^{m}φ = 0, and therefore φ = 0. Thus the multiplication maps appearing in the theorem are always nondegenerate in the second argument.

“a)⇒b)” Suppose that r ∈ R_{σ_{X}+1} \ {0} and r(ω_{R})_{−σ_{X}} = 0. Then 2.2.a shows that f_{Y} (ω_{R})_{−σ_{X}} = 0 for some separator f_{Y} ∈ R_{σ_{X}+1} of a subscheme Y ⊆ X of degree deg X −1. Let \overline{f}_{Y} be the image of f_{Y} in \overline{R}_{σ_{X}+1}. Since X is a Cayley-Bacharach scheme, we have \overline{f}_{Y} ≠ 0. Choose a complement V of k · \overline{f}_{Y} in \overline{R}_{σ_{X}+1}, and let π : \overline{R}_{σ_{X}+1} ⟶ k be the projection to \overline{f}_{Y} along V. Then π lifts to a k-linear map \overline{φ} : R_{σ_{X}+1} ⟶ k with \overline{φ}(x_{0}R_{σ_{X}}) = 0 and \overline{φ}(f_{Y}) = 1. Using 1.5 we obtain an element φ ∈ (ω_{R})_{−σ_{X}} such that φ(f_{Y}) = 1. In view of 1.6 this is a contradiction.

“b)⇒d)” Let r ∈ R_{m} such that r(ω_{R})_{−σ_{X}+n} = 0. In particular, this implies rx_{0}^{n}(ω_{R})_{−σ_{X}} = 0, and therefore r(ω_{R})_{−σ_{X}} = 0. If m ≤ σ_{X} + 1, we conclude from rx_{0}^{σ_{X}+1−m} ∈ R_{σ_{X}+1} and rx_{0}^{σ_{X}+1−m}(ω_{R})_{−σ_{X}} = 0 that r = 0. If m > σ_{X} + 1, we write r = x_{0}^{m−σ_{X}−1}r′ with r′ ∈ R_{σ_{X}+1} and conclude from r′(ω_{R})_{−σ_{X}} = 0 that r = 0.

Since “d)⇒c)” is clear, it remains to show “c)⇒a)”. Suppose that X is not a Cayley-Bacharach scheme, i.e. that there is a subscheme Y ⊆ X of degree deg X − 1 such that f_{Y} ∈ x_{0}R_{σ_{X}} for some separator f_{Y} of Y. Write f_{Y} = x_{0}g_{Y} with g_{Y} ∈ R_{σ_{X}}. For φ ∈ (ω_{R})_{−σ_{X}} we have φ(f_{Y}) = x_{0}φ(g_{Y}) ∈ x_{0}k[x_{0}]_{−1} = (0), hence f_{Y} φ = 0 by 1.6. Thus x_{0}g_{Y} (ω_{R})_{−σ_{X}} = f_{Y} (ω_{R})_{−σ_{X}} = 0, implying g_{Y} (ω_{R})_{−σ_{X}} = 0, a contradiction.

□

Corollary 2.7. If X is a Cayley-Bacharach scheme, then

H_{X}(n) ≤ Δ_{X}(deg X − H_{X}(σ_{X} − n))  for all n ≥ 0.

Proof: Since R_{n} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−σ_{X}+n} is nondegenerate for every n ≥ 0, we get injections R_{n} ⟶ Hom_{k}((ω_{R})_{−σ_{X}} , (ω_{R})_{−σ_{X}+n}). Now compare dimensions. □

Of course, if Δ_{X} = 1, the inequalities of Corollary 2.7 and the corollary of Theorem 1 are equivalent. We do not know whether the stronger inequalities of the latter corollary hold for arbitrary Cayley-Bacharach schemes. The following example shows that, at any rate, Theorems 2.4 and 2.6 are not equivalent, if Δ_{X} > 1.

Example. Let X ⊆ ℙ^{2} be the subscheme defined by I_{X} = (X_{1}^{2}, X_{1}X_{2}, X_{2}^{2}). It is concentrated at the point P = (1 : 0 : 0) and has Hilbert function H_{X} : 1 3 3 ⋯ and σ_{X} = 0. Since O_{X,P} ≅ O_{ℙ^{2},P} /𝔪_{ℙ^{2},P}^{2} , we see that X is not locally Gorenstein. In particular, X has infinitely many subschemes of degree two. But each subscheme Y ⊆ X of degree two has Hilbert function H_{Y} : 1 2 2 ⋯, so X is a Cayley-Bacharach scheme.

Let us also check what happens in the canonical module in this example. For an element φ ∈ (ω_{R})_{0}, let a := φ(x_{1}) and b := φ(x_{2}). We claim that (bx_{1} − ax_{2})φ = 0. This follows from Lemma 1.4, since ((bx_{1} − ax_{2})φ)(x_{0}) = x_{0}bφ(x_{1}) − x_{0}aφ(x_{2}) = 0, and ((bx_{1} − ax_{2})φ)(x_{i}) = bφ(x_{1}x_{i}) − aφ(x_{2}x_{i}) = (b − a)φ(0) = 0 for i = 1, 2. Hence no element φ ∈ (ω_{R})_{0} has Ann_{R}(φ) = 0, while the multiplication map R_{0}⊗(ω_{R})_{0} ⟶ (ω_{R})_{0} is clearly nondegenerate.

The final proposition of this section applies Theorem 2.6 to give a generalization of the well-known fact that if one removes a point from a 0-dimensional reduced complete intersection, one is still left with a Cayley-Bacharach scheme.

Proposition 2.8. Suppose that X is a Cayley-Bacharach scheme with Δ_{X} = 1, and Y ⊆ X is a subscheme of degree deg Y = deg X −1. Then Y is a Cayley-Bacharach scheme.

In particular, if X is arithmetically Gorenstein, every subscheme of degree deg X−1 is a Cayley-Bacharach scheme.

Proof: Let f_{Y} ∈ R_{σ_{X}+1} be a separator of Y. As explained in section 1, the projective coordinate ring of Y is S = R/(f_{Y}), and its canonical module is given by ω_{S} ≅ {ψ ∈ ω_{R} : f_{Y} ψ = 0}. Since Δ_{X} = 1, the Hilbert function of Y is H_{Y} (n) = H_{X}(n) for n ≤ σ_{X} and H_{Y} (n) = deg Y for n ≥ σ_{X}. Thus σ_{Y} = σ_{X} − 1 and S_{σ_{Y}} ≅ R_{σ_{X}−1}.

Suppose that r ∈ R_{σ_{X}−1} annihilates (ω_{S})_{−σ_{X}+1}. By 2.5.a and 2.4, there is an element φ ∈ (ω_{R})_{−σ_{X}} such that Ann_{R}(φ) = (0). So, (ω_{S})_{−σ_{X}+1} contains {ℓφ : ℓ ∈ R_{1}, f_{Y} ℓφ = 0}, and therefore rℓ = 0 for all ℓ ∈ R_{1} such that f_{Y} ℓ = 0.

We let ı(x_{0}^{2}r) =: (r_{Q})_{Q∈X}, ı(x_{0}^{σ_{X}} ℓ) =: (ℓ_{Q})_{Q∈X}, and ı(f_{Y}) =: (0, …, 0, s_{P} , 0, …, 0) with P ∈ X, s_{P} ∈ 𝔖(O_{X,P}). If ℓ ∈ R_{1} defines a hyperplane V(ℓ) which passes through P, but through no other point of X, we have f_{Y} ℓ = ı^{−1}((0, …, 0, s_{P} ℓ_{P} , 0, …, 0)) = 0. Thus rℓ = 0 and r_{P} ℓ_{P} = 0 for such ℓ. Since the various germs ℓ_{P} generate the maximal ideal of O_{X,P}, it follows that r_{P} is an element of the socle of O_{X,P}. Then the hypothesis that X is locally Gorenstein implies that r_{P} is a multiple of s_{P}.

On the other hand, for Q ∈ X, Q ≠ P, the germ ℓ_{Q} is a unit of O_{X,Q}, and rℓ = 0 implies r_{Q} = 0. Altogether this shows that x_{0}^{2}r is a scalar multiple of f_{Y}. But f_{Y} ∉ x_{0}R_{σ_{X}}, because X is a Cayley-Bacharach scheme, so the only possibility left is r = 0. Hence the multiplication S_{σ_{Y}} ⊗ (ω_{S})_{−σ_{Y}} ⟶ (ω_{S})_{0} is nondegenerate, and therefore Y is a Cayley-Bacharach scheme by 2.6. □

Example. If X ⊆ ℙ^{2} is reduced and concentrated at the points (1 : 0 : 0), (1 : 1 : 0), (1 : 2 : 0), (1 : 0 : 1), (1 : 1 : 1), (1 : 0 : 2), (1 : 2 : 2), then its Hilbert function is H_{X} : 1 3 6 7 7 ⋯, and X is a Cayley-Bacharach scheme because of [GKR], 4.9. From the proposition it follows that any subset of six points of X is a Cayley-Bacharach scheme, too.

3. Higher Uniformities. In Proposition 2.1 we have characterized Cayley-Bacharach schemes X as those 0-dimensional schemes for which every subscheme Y of degree deg X − 1 has the same Hilbert function H_{Y} (n) = min{H_{X}(n), deg X − 1} for all n ∈ ℤ. This can be interpreted as a weak uniformity of X and invites the following generalization.

Definition. Let n ≥ 1. We say that X is n-uniform, if every subscheme Y ⊆ X of degree deg X − n ≤ deg Y ≤ deg X has Hilbert function H_{Y} (m) = min{H_{X}(m), deg Y } for all m ∈ ℤ.

We say that X is in uniform position, if X is (deg X − 1)-uniform.

Examples. a) By definition, X is 1-uniform if and only if X is a Cayley-Bacharach

scheme.

b) Proposition 2.8 says that every Cayley-Bacharach scheme X with Δ_{X} = 1 is 2-uniform. In particular, every arithmetically Gorenstein scheme is 2-uniform.

c) J. Harris has shown in [H] that if char k = 0 and X is the general hyperplane section of an integral curve C ⊆ ℙ^{d+1}, then X is in uniform position. For most cases, this has been extended to char k > 0 by J. Rathmann in [R].

d) The 0-dimensional reduced complete intersection scheme X = {(1 : 0 : 0), (1 : 1 : 0), (1 : 2 : 0), (1 : 0 : 1), (1 : 1 : 1), (1 : 2 : 1)} ⊆ ℙ^{2} is 2-uniform by 2.8, but not 3-uniform, because Y := {(1 : 0 : 0), (1 : 1 : 0), (1 : 2 : 0)} has H_{Y} (1) = 2 < 3 = min{H_{X}(1), deg Y }. Therefore Proposition 2.8 cannot be improved for complete intersections.

The next theorem contains Theorem 3 of the introduction. In the sequel we shall say that a k-linear map μ : U ⊗ V ⟶ W of finite dimensional k-vector spaces is biinjective, if μ(u ⊗ v) = 0 implies u = 0 or v = 0 for all u ∈ U, v ∈ V.

Theorem 3.1. The following conditions are equivalent.

a) X is Δ_{X}-uniform.

b) The multiplication map R_{σ_{X}} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is biinjective.

c) For each n ∈ {0, …, σ_{X}}, the multiplication map R_{n}⊗(ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−σ_{X}+n} is biinjective.

Proof: “a)⇒b)” Let r ∈ R_{σ_{X}} \ {0}, and let Y := V(r) ⊆ X. Then r ∈ (I_{Y/X})_{σ_{X}} implies H_{Y} (σ_{X}) < H_{X}(σ_{X}). Since X is Δ_{X}-uniform, this yields deg Y < H_{X}(σ_{X}) = deg X − Δ_{X}. By applying Lemma 2.2.b, we find a subscheme Y′ ⊆ X such that deg Y′ = deg X−Δ_{X} and Y ⊆ Y′. By assumption, σ_{Y} ≤ σ_{Y′} = σ_{X} −1. If S := R/I_{Y/X}, it follows that no nonzero element of (ω_{R})_{−σ_{X}} lies in ω_{S} = {φ ∈ ω_{R} : I_{Y/X} · φ = 0}. By 1.2, this means that no element of (ω_{R})_{−σ_{X}} \ {0} is annihilated by r.

“b)⇒c)” Let φ ∈ (ω_{R})_{−σ_{X}} and r ∈ R_{n}, n ≤ σ_{X}, such that rφ = 0. Then rx_{0}^{σ_{X}−n} ∈ R_{σ_{X}} and rx_{0}^{σ_{X}−n}φ = 0 imply rx_{0}^{σ_{X}−n} = 0 or φ = 0. Hence we have r = 0 or φ = 0.

“c)⇒a)” Let Y ⊆ X be a subscheme of degree deg X − Δ_{X} ≤ deg Y ≤ deg X. It suffices to show H_{Y} (σ_{X}) = H_{X}(σ_{X}), because H_{Y} (n) = deg Y for n ≥ σ_{X} + 1 follows already from σ_{Y} ≤ σ_{X}. Suppose that H_{Y} (σ_{X}) < H_{X}(σ_{X}) ≤ deg Y. Then there exists a nonzero element r ∈ (I_{Y/X})_{σ_{X}}, and we can also conclude that σ_{Y} = σ_{X}. Therefore there is a nonzero element φ ∈ (ω_{R})_{−σ_{X}} which is annihilated by I_{Y/X}. In particular, rφ = 0, a contradiction. □

Remark. It is useful to be able to check computationally whether the multiplication map R_{σ_{X}} ⊗(ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} is biinjective. Otherwise we would have to compute the Hilbert functions of all subschemes Y ⊆ X of degree deg Y ≥ deg X − Δ_{X} in order to check Δ_{X}-uniformity. We start by computing a minimal homogeneous presentation

⨁_{i=1}^{β_{d−1}} R(α_{d−1 i} − d − 1) ⟶^{δ} ⨁_{i=1}^{β_{d}} R(α_{di} − d − 1) ⟶^{ϵ} ω_{R} ⟶ 0

as explained in section 5. Let e_{1}, …, e_{Δ} ∈ ⨁_{i=1}^{β_{d}} R(α_{di} − d − 1) be those standard basis

vectors which have degree −σ_{X}. Their images φ_{i} := ϵ(e_{i}) form a k-basis of (ω_{R})_{−σ_{X}}. Compute monomials (or polynomials) m_{1}, …, m_{s−Δ} ∈ k[X_{0}, …, X_{d}]_{σ_{X}} whose images in R_{σ_{X}} form a k-basis of that vector space (s = deg X).

Using elimination, we can find the variety V ⊆ 𝔸_{k}^{s} of all solutions (a_{1}, …, a_{Δ}, b_{1}, …, b_{s−Δ}) ∈ k^{s} of

(a_{1}e_{1} + … + a_{Δ}e_{Δ})(b_{1}m_{1} + … + b_{s−Δ}m_{s−Δ}) ∈ im δ.

Let L ⊆ 𝔸_{k}^{s−Δ} be the linear subspace of all solutions (b_{1}, …, b_{s−Δ}) ∈ k^{s−Δ} of b_{1}m_{1} + … + b_{s−Δ}m_{s−Δ} ∈ (I_{X})_{σ_{X}}. Now check whether V equals {0} × 𝔸_{k}^{s−Δ} ∪ 𝔸_{k}^{Δ} × L in 𝔸_{k}^{s} ≅ 𝔸_{k}^{Δ} × 𝔸_{k}^{s−Δ}.

More generally, it is clear that the described method can be applied to check biinjectivity of any bilinear map of vector spaces whose matrix we know. Another method can be derived from [E], 1.1.

Like Theorem 2.4, Theorem 3.1 also has implications for the growth behaviour of H_{X}. The key here is the following lemma.

Biinjective Map Lemma. Let μ : U ⊗ V ⟶ W be a k-linear, biinjective map of finite dimensional k-vector spaces. Then

dim_{k} W ≥ dim_{k} U + dim_{k} V − 1.

For a nice, elementary proof of this lemma see [G]. In the situation of 3.1 we can apply it and obtain the corollary stated in the introduction. This corollary was obtained earlier in [HE] under the stronger hypothesis that X is reduced and in uniform position.

Next we want to characterize 0-dimensional schemes with even higher uniformities.

Theorem 3.2. Let i ∈ {0, …, σ_{X}}. The following conditions are equivalent.

a) X is (deg X − H_{X}(i))-uniform.

b) For each n ∈ {i, …, σ_{X}}, the multiplication map R_{n} ⊗ (ω_{R})_{−n} ⟶ (ω_{R})_{0} is biinjective.

c) If n ∈ {0, …, σ_{X} − i} and m ∈ {0, …, σ_{X} − n}, the multiplication map R_{m} ⊗ (ω_{R})_{−σ_{X}+n} ⟶ (ω_{R})_{−σ_{X}+m+n} is biinjective.

Proof: “a)⇒b)” We proceed by downward induction on i. The case i = σ_{X} is handled by 3.1. By induction hypothesis, we only have to show that R_{i} ⊗ (ω_{R})_{−i} ⟶ (ω_{R})_{0} is biinjective. Suppose r ∈ R_{i} \ {0} and φ ∈ (ω_{R})_{−i} are such that rφ = 0. Let Y := V(r) ⊆ X. Then the ideal I_{Y/X} of Y in R satisfies r ∈ (I_{Y/X})_{i} ≠ (0). Since X is (deg X − H_{X}(i))-uniform, this implies deg Y < H_{X}(i). Using 2.2.b we find a subscheme Y′ ⊆ X such that deg Y′ = H_{X}(i) and Y ⊆ Y′. By assumption, σ_{Y} ≤ σ_{Y′} = i − 1. Let S := R/I_{Y/X}. From rφ = 0 and 1.2 we conclude I_{Y/X} · φ = (r)^{sat}φ = 0. Hence φ ∈ (ω_{S})_{−i}. Now −i < −σ_{Y} yields φ = 0, as was to be shown.

“b)⇒c)” Suppose that r ∈ R_{m} and φ ∈ (ω_{R})_{−σ_{X}+n} satisfy rφ = 0. Then rx_{0}^{σ_{X}−n−m} ∈ R_{σ_{X}−n} and rx_{0}^{σ_{X}−n−m}φ = 0 imply rx_{0}^{σ_{X}−n−m} = 0 or φ = 0 by b). Hence r = 0 or φ = 0.

Since “c)⇒b)” is clear, it remains to show “b)⇒a)”. Again we proceed by downward induction on i, the case i = σ_{X} being the corresponding statement in 3.1. Suppose that Y ⊆ X is a subscheme of degree H_{X}(i) ≤ deg Y ≤ deg X. If deg Y ≥ H_{X}(i + 1), we get the claim from the induction hypothesis. Thus we can assume that H_{X}(i) ≤ deg Y < H_{X}(i + 1). We find a subscheme Y′ ⊆ X such that deg Y′ = H_{X}(i + 1) and Y ⊆ Y′. From the induction hypothesis we know H_{Y′}. Hence σ_{Y} ≤ σ_{Y′} = i, and therefore we have H_{Y} (n) = deg Y for n ≥ i + 1. It remains to show H_{Y} (i) = H_{X}(i). Suppose H_{Y} (i) < H_{X}(i). Then there is an element r ∈ (I_{Y/X})_{i} \{0}, and H_{Y} (i) < deg Y implies σ_{Y} ≤ i. Thus we also find an element φ ∈ (ω_{S})_{−i} \ {0}. Now rφ = 0 contradicts b). □

Notice that in order to obtain Theorem 4 of the introduction, we only have to apply Theorem 3.2 with i = 0.

Example. Consider the 0-dimensional reduced subscheme X ⊆ ℙ^{2} concentrated at the eleven points P_{1} = (1 : 2 : 1), P_{2} = (1 : 1 : 2), P_{3} = (1 : 2 : −1), P_{4} = (1 : 1 : −2), P_{5} = (1 : −2 : 1), P_{6} = (1 : −2 : −1), P_{7} = (1 : 3 : 2), P_{8} = (1 : 3 : 3), P_{9} = (1 : 4 : 0), P_{10} = (1 : 5 : 0), and P_{11} = (1 : 0 : 6). Its Hilbert function is H_{X} : 1 3 6 10 11 11 ⋯, so Δ_{X} = 1, and using the procedure given in [GKR] it is easy to check that X is a Cayley-Bacharach scheme.

Since Y_{1} := {P_{1}, …, P_{6}} has Hilbert function H_{Y_{1}} : 1 3 5 6 6 ⋯ and Y_{2} := {P_{1}, P_{2}, P_{8}, …, P_{11}} has Hilbert function H_{Y_{2}} : 1 3 6 6 ⋯, we see that X is not 5uniform, where 5 = deg X − H_{X}(σ_{X} − 1). Therefore there exist nonzero elements ψ ∈ (ω_{R})_{−2} and r ∈ R_{2} such that rψ = 0. After calculating a presentation of ω_{R} as in section 5, it is a straightforward exercise to compute those elements explicitly.

Corollary 3.3. If X is locally Gorenstein and in uniform position, the multiplication maps R_{m} ⊗ R_{n} ⟶ R_{m+n} are biinjective for all m, n ≥ 0 such that m + n ≤ σ_{X}.

Proof: Since X is a locally Gorenstein Cayley-Bacharach scheme, we find an element φ ∈ (ω_{R})_{−σ_{X}} such that Ann_{R}(φ) = 0. Now restrict the multiplication maps from part c) of the theorem to Rφ ⊆ ω_{R} and identify R(σ_{X}) ≅ Rφ. □

Of course, unlike Theorem 4, the above corollary is not a characterization of 0dimensional schemes in uniform position, since the inclusion Rφ ⊆ ω_{R} is a strict one as long as X is not arithmetically Gorenstein.

Finally, we also want to give a characterization of n-uniform schemes for those values of n which lie between two consecutive values of deg X−H_{X}(i). Here we restrict ourselves to the case of reduced schemes. Also, we only formulate the case n ∈ {1, …, Δ_{X}} explicitly, and leave appropriate generalizations to the reader.

In the case of reduced 0-dimensional subschemes X ⊆ ℙ^{d} we shall use the following notations. We let s := deg X and write X = {P_{1}, …, P_{s}}. For each i ∈ {1, …, s}, we let f_{i} ∈ R_{σ_{X}+1} be a separator corresponding to the subscheme X \{P_{i}} ⊆ X. As shown in [GKR], the set \{f_{1}, …, f_{s}\} is a k-basis of R_{σ_{X}+1}. Therefore the images \{\overline{f}_{1}, …, \overline{f}_{s}\} in \overline{R}_{σ_{X}+1} are a set of generators of that vector space. The relations among those images determine the structure of the multiplication map R_{σ_{X}} ⊗(ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{0} in a natural way.

Proposition 3.4. Let n ∈ {1, …, Δ_{X}}. The following conditions are equivalent.

a) X is n-uniform.

b) Every subset of n elements from \{\overline{f}_{1}, …, \overline{f}_{s}\} is linearly independent in \overline{R}_{σ_{X}+1}.

Proof: “a)⇒b)” Let ν_{1}, …, ν_{n} ∈ {1, …, s} be pairwise distinct elements, and let Y := X \ {P_{ν_{1}}, …, P_{ν_{n}}}. Then H_{Y} (i) = H_{X}(i) for i ≤ σ_{X} and H_{Y} (i) = s − n for i ≥ σ_{X} + 1. Also note that (f_{ν_{1}}, …, f_{ν_{n}}) ⊆ I_{Y/X}. If we compare Hilbert functions, we see that in fact we must have I_{Y/X} = (f_{ν_{1}}, …, f_{ν_{n}}). Thus (f_{ν_{1}}, …, f_{ν_{n}}) is a saturated ideal of R, i.e. λ_{1}f_{ν_{1}} + … + λ_{n}f_{ν_{n}} ∈ x_{0}R_{σ_{X}} with λ_{i} ∈ k implies λ_{1} = … = λ_{n} = 0. This is clearly equivalent to what we claimed in b).

“b)⇒a)” Let m ≤ n, let ν_{1}, …, ν_{m} ∈ {1, …, s} be pairwise distinct, and let Y := X \ {P_{ν_{1}}, …, P_{ν_{m}}}. We have to show H_{Y} (i) = min{H_{X}(i), s − m} for all i ∈ ℤ. Since the ideal (f_{ν_{1}}, …, f_{ν_{m}}) defines Y in X scheme-theoretically, and since R/(f_{ν_{1}}, …, f_{ν_{m}}) has the correct Hilbert function, the claim is equivalent to showing that the ideal (f_{ν_{1}}, …, f_{ν_{m}}) is saturated.

Let i ≥ 0 and r ∈ R_{i} such that x_{0}r ∈ (f_{ν_{1}}, …, f_{ν_{m}}). We consider three cases.

1) If i ≥ σ_{X} +1, we can write x_{0}r = λ_{1}x_{0}^{i−σ_{X}} f_{ν_{1}} +…+λ_{m}x_{0}^{i−σ_{X}} f_{ν_{m}} with λ_{j} ∈ k,

and we get r = λ_{1}x_{0}^{i−σ_{X}−1}f_{ν_{1}} + … + λ_{m}x_{0}^{i−σ_{X}−1}f_{ν_{m}} ∈ (f_{ν_{1}}, …, f_{ν_{m}}).

2) If i = σ_{X}, we can write x_{0}r = λ_{1}f_{ν_{1}} + … + λ_{m}f_{ν_{m}} with λ_{j} ∈ k, and using b)

we get λ_{1} = … = λ_{m} = 0, and thus r = 0.
3) If i < σ_{X}, we have x_{0}r ∈ (f_{ν_{1}}, …, f_{ν_{m}})_{i+1} = (0), hence r = 0.

In any case, x_{0}r ∈ (f_{ν_{1}}, …, f_{ν_{m}}) implies r ∈ (f_{ν_{1}}, …, f_{ν_{m}}). By 1.2, this yields the desired conclusion. □

Example. Suppose char k = p > 0 and X ⊆ ℙ^{d} is a reduced 0-dimensional subscheme consisting of s := deg X ≥ p + 2 𝔽_{p}-rational points and having Δ_{X} = 2. Then X is not 2-uniform.

Suppose that X was 2-uniform. Since X is defined over 𝔽_{P}, its separators f_{i} and their residue classes \overline{f}_{i} are elements of 𝔽_{p}[x_{0}, …, x_{d}] resp. 𝔽_{P} [x_{1}, …, x_{d}] for i = 1, …, s. By the proposition, we have \overline{f}_{i} ≠ 0 and \overline{f}_{i} ∉ k · \overline{f}_{j} for all i, j = 1, …, s such that i ≠ j. W.l.o.g. let \{\overline{f}_{1}, \overline{f}_{2}\} be a k-basis of \overline{R}_{σ_{X}+1}. For i = 1, …, s write \overline{f}_{i} = λ_{i}\overline{f}_{1} +μ_{i}\overline{f}_{2} with λ_{i}, μ_{i} ∈ 𝔽_{p}. Then {(λ_{1} : μ_{1}), …, (λ_{s} : μ_{s})} is a set of s ≥ p + 2 distinct points in ℙ_{𝔽_{p}}^{1}, a contradiction.

4. Schemes in General Position. In this section let X again be an arbitrary 0-dimensional subscheme of ℙ^{d}. We want to study 0-dimensional schemes which exhibit the following kind of uniformity.

Definition. We say that X is in linearly general position, if deg(L∩X) ≤ 1+dim L for every proper linear subspace L ⊊ ℙ^{d}.

It is useful to rephrase this condition in terms of Hilbert functions of various subschemes of X.

Proposition 4.1. The following conditions are equivalent.

a) X is in linearly general position.

b) If Y ⊆ X is a subscheme of degree deg Y ≤ d + 1, then H_{Y} (n) = deg Y for all n ≥ 1.

c) Each subscheme Y ⊆ X of degree deg Y ≤ d + 1 satisfies σ_{Y} ≤ 0.

d) If deg X ≥ d + 1, then H_{Y} (1) = d + 1 for every subscheme Y ⊆ X of degree deg Y = d + 1, and if 1 ≤ deg X ≤ d, then H_{X}(1) = deg X.

Proof: “a)⇒b)” Let ⟨X⟩ be the linear span of X, i.e. the linear subspace ⟨X⟩ := V((I_{X})_{1}) ⊆ ℙ^{d}. Then we have dim⟨X⟩ = d − dim_{k}(I_{X})_{1} = H_{X}(1) − 1. If X does not span ℙ^{d}, i.e. if dim⟨X⟩ < d, we can use L = ⟨X⟩ in the definition, and we obtain deg X ≤ H_{X}(1) ≤ d. Therefore H_{X}(n) = deg X for all n ≥ 1. Clearly, this must then also be true for all subschemes of X.

In case dim⟨X⟩ ≥ d, we have H_{X}(1) = d + 1 ≤ deg X. Choose a subscheme Z ⊆ X such that deg Z = d + 1 and Y ⊆ Z. It suffices to show H_{Z}(1) = d + 1. Then σ_{Y} ≤ σ_{Z} yields the claim. Suppose there is a hyperplane L ⊆ ℙ^{d} such that Z ⊆ L. Then X ⊈ L implies L ∩ X ⊊ X. Thus d + 1 = deg Z ≤ deg(L ∩ X) ≤ 1 + dim L = d, a contradiction. Therefore Z is not contained in any hyperplane, i.e. H_{Z}(1) = d + 1.

“b)⇔c)” is clear by definition of σ_{Y}, and “c)⇒d)” is also clear.

“d)⇒a)” First we consider the case deg X ≥ d + 1. Let Z = L ∩ X for some proper linear subspace L ⊊ ℙ^{d}. If deg Z ≥ d + 1, we find a subscheme Y ⊆ Z of degree deg Y = d + 1. Then H_{Y} (1) = d + 1 follows from our assumption, but contradicts Y ⊆ Z ⊆ X. Hence we must have deg Z ≤ d. Then we find a subscheme Y ⊆ X such that deg Y = d + 1 and Z ⊆ Y. By assumption we have σ_{Y} = 0, hence σ_{Z} ≤ 0. Altogether we obtain

deg(L ∩ X) = deg Z = H_{Z}(1) ≤ H_{L}(1) = 1 + dim L.

Finally, we consider the case 1 ≤ deg X ≤ d. By assumption we have σ_{X} ≤ 0. Therefore σ_{L∩X} ≤ 0 for every proper linear subspace L ⊊ ℙ^{d}. Thus we find deg(L∩X) = H_{L∩X}(1) ≤ H_{L}(1) = 1 + dim L, as desired. □

Example. Schemes in linearly general position are, for instance, obtained naturally by taking general hyperplane sections of nondegenerate integral curves C ⊆ ℙ^{d+1} which are not strange. This is the content of the General Position Lemma shown in [R].

A 0-dimensional scheme X ⊆ ℙ^{d} is called nondegenerate, if it is not contained in any hyperplane. Our next theorem characterizes nondegenerate schemes in linearly general position and contains Theorem 5 of the introduction. The easy task of formulating a similar theorem also in the degenerate case is left to the reader.

Notice that if X is in linearly general position and deg X ≥ d + 1, then X is automatically nondegenerate. Conversely, if X is nondegenerate, we obviously must have deg X ≥ d + 1.

Theorem 4.2. Let X ⊆ ℙ^{d} be a nondegenerate 0-dimensional subscheme. The following conditions are equivalent.

a) X is in linearly general position.

b) The multiplication map R_{1} ⊗ (ω_{R})_{−1} ⟶ (ω_{R})_{0} is biinjective.

c) For each n ∈ {1, …, σ_{X}}, the multiplication map R_{1} ⊗ (ω_{R})_{−n} ⟶ (ω_{R})_{−n+1} is biinjective.

Proof: “a)⇒b)” Let ℓ ∈ R_{1} \ {0} and φ ∈ (ω_{R})_{−1} such that ℓφ = 0. Since X is in linearly general position, we have deg V(ℓ) ≤ d. Using 2.2.b we find a subscheme Y ⊆ X such that deg Y = d + 1 and V(ℓ) ⊆ Y. Then I_{Y/X}, the ideal of Y in R, is contained in (ℓ)^{sat}, the ideal of V(ℓ) in R. By 1.2, this implies that for every r ∈ I_{Y/X} there exists a number n ≥ 0 such that x_{0}^{n}r ∈ (ℓ). In particular, we have x_{0}^{n}rφ = 0, and therefore rφ = 0. Let S := R/I_{Y/X}. By assumption and 4.1, we have H_{Y} (1) = d + 1. Thus H_{ω_{S}}(−1) = deg Y − H_{Y} (1) = 0. It follows that φ ∈ {ψ ∈ (ω_{R})_{−1} : I_{Y/X} · ψ = 0} = (ω_{S})_{−1} = (0), hence φ = 0, as was to be shown.

“b)⇒c)” If ℓ ∈ R_{1} \ {0} and φ ∈ (ω_{R})_{−n} are such that ℓφ = 0, then ℓx_{0}^{n−1}φ = 0 implies x_{0}^{n−1}φ = 0 by b), and hence φ = 0.

“c)⇒a)” Suppose that X is not in linearly general position. By 4.1, there is a subscheme Y ⊆ X such that deg Y = d + 1 and H_{Y} (1) ≤ d. Since X is nondegenerate, we find an element ℓ ∈ R_{1}\{0} such that ℓ ∈ I_{Y/X}. Let S := R/I_{Y/X}. Since H_{ω_{S}}(−1) = deg Y − H_{Y} (1) ≥ 1, we find an element φ ∈ (ω_{S})_{−1} \ {0}. But then I_{Y/X} · φ = 0 implies ℓφ = 0, a contradiction. □

As usual, also this theorem has implications for the Hilbert function of schemes in linearly general position. In fact, an application of the Biinjective Map Lemma to b) yields the corollary stated in the introduction. The second statement of that corollary follows by simply adding up the inequalities obtained before.

Example. If X is a nondegenerate 0-dimensional subscheme of ℙ^{3} and ΔH_{X}(σ_{X}) = 2, i.e. if H_{X} is of the form H_{X} : 1 3 ⋯ s−Δ_{X}−2 s−Δ_{X} s s ⋯, where s := deg X, then X is not in linearly general position. This follows from the aforementioned corollary, since ΔH_{X}(σ_{X}) = 2 < 3 = ΔH_{X}(1).

Our next corollary is an immediate consequence of Theorem 3.2 and Theorem 4.2.

Corollary 4.3. If X is nondegenerate and in uniform position, then X is in linearly general position.

In view of Proposition 4.1.b, we find it natural to extend the concept of linearly general position as follows.

Definition. Let i ≥ 1. We say that X is in i^{th}-order general position, if every subscheme Y ⊆ X of degree deg Y ≤ H_{ℙ^{d}}(i) satisfies H_{Y} (n) = min{H_{ℙ^{d}}(n), deg Y } for all n ≥ 0.

In other words, the Hilbert function of each subscheme Y ⊆ X of degree deg Y ≤ H_{ℙ^{d}}(i) agrees with H_{ℙ^{d}} as long as possible, and then immediately attains its maximum value. By 4.1.b, X is in 1^{st}-order general position if and only if X is in linearly general position. Also, X is in (σ_{X} + 1)^{th}-order general position if and only if X is in uniform position and in generic position (i.e. H_{X}(n) = min{H_{ℙ^{d}}(n), deg X} for all n ≥ 0).

By now, it should be clear to the reader how i^{th}-order general position is reflected by the structure of the canonical module. Again we restrict ourselves to the case of sufficiently many points and leave the degenerate cases as an exercise.

Theorem 4.4. Let i ≥ 0 and let α_{X} := min{n ∈ ℕ : (I_{X})_{n} ≠ 0} ≥ i + 1. The following conditions are equivalent.

a) X is in i^{th}-order general position.

b) For each n ∈ {1, …, i}, the multiplication map R_{n} ⊗ (ω_{R})_{−n} ⟶ (ω_{R})_{0} is biinjective.

c) For all m ∈ {1, …, i} and n ∈ {0, …, σ_{X} − m}, the multiplication map R_{m} ⊗ (ω_{R})_{−σ_{X}+n} ⟶ (ω_{R})_{−σ_{X}+m+n} is biinjective.

Proof: “a)⇒b)” We proceed by induction on i. The case i = 1 is contained in Theorem 4.2. Using the induction hypothesis, we only have to show that R_{i}⊗(ω_{R})_{−i} ⟶ (ω_{R})_{0} is biinjective. Let r ∈ R_{i} \ {0} and φ ∈ (ω_{R})_{−i} such that rφ = 0. Consider the subscheme Z := V(r) ⊆ X. Since r ∈ (I_{Z/X})_{i} and X is in i^{th}-order general position, we have deg Z < H_{ℙ^{d}}(i). Choose a subscheme Y ⊆ X such that deg Y = H_{ℙ^{d}}(i) and Z ⊆ Y. Then I_{Y/X} is contained in I_{Z/X} = (r)^{sat}. Thus, for every s ∈ I_{Y/X}, there exists n ≥ 0 such that x_{0}^{n}s ∈ (r). This implies I_{Y/X} · φ = 0. Let S = R/I_{Y/X}. From H_{ω_{S}}(−i) = 0 and φ ∈ (ω_{S})_{−i} we conclude φ = 0.

“b)⇒c)” is standard by now, and “c)⇒b)” is clear, so we still have to prove “b)⇒a)”. Again we proceed by induction on i, the case i = 1 being provided by 4.2. By induction hypothesis, each subscheme Y ⊆ X of degree deg Y ≤ H_{ℙ^{d}}(i − 1) has the desired Hilbert function. So, let Y ⊆ X be a subscheme of degree H_{ℙ^{d}}(i − 1) < deg Y ≤ H_{ℙ^{d}}(i). By choosing a subscheme of degree H_{ℙ^{d}}(i − 1) of Y and applying the induction hypothesis, we see that H_{Y} (i − 1) = H_{ℙ^{d}}(i − 1). Therefore it only remains to show σ_{Y} = i − 1.

Find a subscheme Z ⊆ X such that deg Z = H_{ℙ^{d}}(i) and Y ⊆ Z. Since deg Y > H_{ℙ^{d}}(i − 1) implies i − 1 ≤ σ_{Y} ≤ σ_{Z}, it suffices to show H_{Z}(i) = H_{ℙ^{d}}(i). Suppose there is a hypersurface of degree i containing Z. The image r ∈ R_{i} of its equation in R does not vanish because of α_{X} ≥ i + 1. Let S := R/I_{Z/X}. Since H_{ω_{S}}(−i) ≥ 1, we find a nonzero element φ ∈ (ω_{S})_{−i}. Then I_{Z/X} · φ = 0 implies rφ = 0, a contradiction. □

Remark. More generally, if we drop the assumption α_{X} ≥ i + 1 in 4.4, condition 4.4.b is equivalent to the statement “every subscheme Y ⊆ X of degree deg Y ≤ H_{X}(i) has Hilbert function H_{Y} (n) = min{H_{X}(n), deg Y } for all n ≥ 0”. This can be shown in a completely analogous manner and is left to the reader.

Let us return for a moment to the example of eleven points in ℙ^{2} considered in section 3.

Example. Let X = {P_{1}, …, P_{11}} ⊆ ℙ^{2} be the scheme defined in the example after Theorem 3.2. We have seen that the multiplication map R_{2} ⊗ (ω_{R})_{−2} ⟶ (ω_{R})_{0} is not biinjective. Since no three points of X are on a line, X is in linearly general position. Therefore the multiplication map R_{1} ⊗ (ω_{R})_{−1} ⟶ (ω_{R})_{0} is biinjective. Using the method described after Theorem 3.1, it is posiible to check this directly.

Altogether we conclude that X is in linearly general position, but not in 2^{nd}-order general position. The latter statement corresponds geometrically to the fact that the six points {P_{1}, …, P_{6}} of X are contained in a conic.

Theoren 4.4 has the following consequences for Hilbert functions of schemes in i^{th}-order general position.

Corollary 4.5. Let i ≥ 0, let X be in i^{th}-order general position, let α_{X} ≥ i + 1, and let m ∈ {1, …, i}. Then the sum of m consecutive terms of the sequence \{ΔH_{X}(1), …, ΔH_{X}(σ_{X})\} is at least \binom{m+d}{d} − 1.

The proof of this corollary is obtained by applying the Biinjective Map Lemma to 4.4.c.

Example. If X ⊆ ℙ^{3} is a 0-dimensional subscheme of degree deg X ≥ 10, and if no subscheme of degree 10 of X is contained in a quadric surface, then the Hilbert function of X cannot satisfy ΔH_{X}(σ_{X} − 1) = 4 and ΔH_{X}(σ_{X}) ≤ 4. This follows from the corollary, because X is in 2^{nd}-order general position and ΔH_{X}(σ_{X} − 1) + ΔH_{X}(σ_{X}) ≤ 8 < 9 = \binom{5}{3} − 1.

Finally, we want to explain the connection of our notion of “i^{th}-order general position” with the notion “imposes independent conditions on forms of degree i”. The following definition is adapted from [EK].

Definition. Let i ≥ 1. We say that X imposes independent conditions on forms of degree i, if deg X ≤ H_{ℙ^{d}}(i) implies H_{X}(i) = deg X, and if deg X ≥ H_{ℙ^{d}}(i) implies H_{Y} (i) = H_{ℙ^{d}}(i) for all subschemes Y ⊆ X of degree deg Y = H_{ℙ^{d}}(i).

Proposition 4.6. Let i ≥ 0. The following conditions are equivalent.

a) X is in i^{th}-order general position.

b) X imposes independent conditions on forms of degree j for every j ∈ {1, …, i}.

Proof: “a)⇒b)” Let j ∈ {1, …, i}. We consider two cases.

1) If deg X ≤ H_{ℙ^{d}}(j), we choose Y = X in the definition of i^{th}-order general position and get H_{X}(n) = min{H_{ℙ^{d}}(n), deg X} for n ≥ 0. In particular, H_{X}(j) = deg X.

2) If deg X ≥ H_{ℙ^{d}}(j), we choose a subscheme Y ⊆ X of degree H_{ℙ^{d}}(j) in the definition of i^{th}-order general position. We get H_{Y} (j) = min{H_{ℙ^{d}}(j), deg Y } = H_{ℙ^{d}}(j).

Consequently, X imposes independent conditions on forms of degree j.

“b)⇒a)” Let Y ⊆ X be a subscheme of degree deg Y ≤ H_{ℙ^{d}}(i). Since we can exclude the trivial case deg Y = 1, we can find j ∈ {1, …, i} such that H_{ℙ^{d}}(j − 1) < deg Y ≤ H_{ℙ^{d}}(j). Choose subschemes Z, Z′ ⊆ X such that deg Z = H_{ℙ^{d}}(j − 1), deg Z′ = H_{ℙ^{d}}(j), and Z ⊆ Y ⊆ Z′ ⊆ X.

Since X imposes independent conditions on forms of degree j, we have H_{Z′}(j) = H_{ℙ^{d}}(j) = deg Z′. Hence σ_{Y} ≤ σ_{Z′} = j − 1, and henceforth H_{Y} (n) = deg Y = min{H_{ℙ^{d}}(n), deg Y } for n ≥ j. Since X imposes independent conditions on forms of degree j − 1, we have H_{Z}(j − 1) = H_{ℙ^{d}}(j − 1). Thus also H_{Y} (j − 1) = H_{ℙ^{d}}(j − 1), implying H_{Y} (n) = H_{ℙ^{d}}(n) = min{H_{ℙ^{d}}(n), deg Y } for all n ∈ {0, …, j − 1}.

Altogether we see that X is in i^{th}-order general position. □

5. The Projective Resolution. The last topic of this paper is to exhibit some connections between the canonical module ω_{R} and the minimal graded free resolution of R. Here we consider R as a module over the polynomial ring A := k[X_{0}, …, X_{d}]. Since R is a 1-dimensional Cohen-Macaulay ring, its resolution is of the form

0 → ⨁_{i=1}^{β_{d}} A(−α_{di}) ⟶^{Φ_{d}} ⨁_{i=1}^{β_{d−1}} A(−α_{d−1 i}) → ⋯ → ⨁_{i=1}^{β_{1}} A(−α_{1i}) ⟶^{Φ_{1}} A → R → 0

where α_{ij} ∈ ℕ and β_{1}, …, β_{d} ∈ ℕ are the Betti numbers of X.

W.l.o.g. we can assume that α_{i1} ≤ … ≤ α_{i β_{i}} for i = 1, …, d. Let 𝔄_{i} be the matrix of Φ_{i} for i = 1, …, d. As the above resolution is minimal, no entry of any of the matrices 𝔄_{i} is a nonzero element of k. Hence α_{11} < … < α_{d1}. Also notice that α_{11} = α_{X}, where α_{X} := min{n ∈ ℕ : (I_{X})_{n} ≠ 0} denotes the least degree of a hypersurface containing X.

Now we dualize the above resolution and observe that Ext_{A}^{i}(R, A) = 0 for i = 0, …, d − 1 and Ext_{A}^{d}(R, A) ≅ ω_{R}(d + 1), cf. [GW]. We obtain a homogeneous exact sequence

0 → A ⟶^{Φ_{1}} ˇ ⨁_{i=1}^{β_{1}} A(α_{1i}) → ⋯ → ⨁_{i=1}^{β_{d−1}} A(α_{d−1 i}) ⟶^{Φ_{d}} ˇ ⨁_{i=1}^{β_{d}} A(α_{di}) → ω_{R}(d + 1) → 0

Since also this resolution is minimal, we conclude that α_{1 β_{1}} < … < α_{d β_{d}}. Notice that σ_{X} = − min{n ∈ ℤ : (ω_{R})_{n} ≠ 0} = α_{d β_{d}} − d − 1.

Definition. Let n ≥ 1.

a) We say that X has a resolution of order n, if α_{i β_{i}} ≤ α_{X} + i + n − 2 for i = 1, …, d.

b) We say that X has a resolution almost of order n, if α_{i β_{i}} ≤ α_{X} + i + n − 2 for i = 1, …, d − 1.

In particular, if n = 1 and a) (resp. b)) is satisfied, we say that X has a linear (resp. almost linear) resolution.

Notice that if X has a resolution of order n (resp. almost of order n), then for i = 2, …, d (resp. for i = 2, …, d − 1) each matrix 𝔄_{i} contains only homogeneous polynomials of degree at most n.

The following proposition generalizes the analogous statement for linear resolutions in [S] and follows also from [L].

Proposition 5.1. Let n ≥ 1.

a) X has a resolution of order n if and only if σ_{X} ≤ α_{X} + n − 3.

b) X has a resolution almost of order n if and only if α_{d−1 β_{d−1}} ≤ α_{X} + n + d − 3.

Proof: “a)” In view of the definition, it suffices to show “⇐”. From α_{d β_{d}} = σ_{X} + d + 1 ≤ α_{X} + d + n − 2 and α_{d β_{d}} ≥ α_{d−1 β_{d−1}} + 1 ≥ … ≥ α_{1β_{1}} + d − 1 we obtain the desired inequalities.

“b)” follows from α_{1 β_{1}} < … < α_{d−1 β_{d−1}} ≤ α_{X} + n + d − 3. □

Corollary 5.2. The following conditions are equivalent.

a) X has a linear resolution.

b) σ_{X} = α_{X} − 2

c) H_{X}(n) = min\{\binom{n+d}{d}, \binom{α_{X}+d−1}{d}\} for all n ∈ ℕ.

In particular, in this case we have deg X = \binom{α_{X}+d−1}{d} = \binom{σ_{X}+d+1}{d}.

Proof: “a)⇔b)” Because of the proposition we only have to show σ_{X} ≥ α_{X} − 2. This follows from σ_{X} = α_{d β_{d}} −d−1 ≥ α_{d1} −d−1 ≥ α_{d−1 1} −d ≥ … ≥ α_{11} −2 = α_{X} −2.

“a)⇒c)” From the presentation A(−α_{X})^{β_{1}} ⟶ A ⟶ R ⟶ 0 we obtain that H_{X}(n) = \binom{n+d}{d} for 0 ≤ n ≤ α_{X} − 1. Since deg X = H_{X}(σ_{X} + 1) = H_{X}(α_{X} − 1) = \binom{α_{X}+d−1}{d}, the conclusion follows.

“c)⇒b)” is clear from the definition of σ_{X}. □

Our next result contains Theorem 6 of the introduction. We characterize schemes with almost linear resolutions using the algebraic structure of their canonical module. The reader may consult [EG] and [L] for related results.

Theorem 5.3. Let {φ_{1}, …, φ_{β_{d}}} with φ_{i} ∈ (ω_{R})_{−α_{di}+d+1} be a minimal homogeneous system of generators of ω_{R}. The following conditions are equivalent.

a) X has an almost linear resolution.

b) If for i = 1, …, β_{d} there are elements r_{i} ∈ R_{α_{di}−α_{X}−d+1} such that r_{1}φ_{1} +…+ r_{β_{d}}φ_{β_{d}} = 0, then r_{1} = … = r_{β_{d}} = 0.

In particular, if X has an almost linear resolution, then the multiplication maps R_{n} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−σ_{X}+n} are injective for n = 0, …, σ_{X} − α_{X} + 2.

Proof: “a)⇒b)” Consider the minimal homogeneous presentation of ω_{R} induced

by ϵ : ⨁_{i=1}^{β_{d}} A(α_{di} − d − 1) −↠ ω_{R} with ϵ(e_{i}) = φ_{i} for i = 1, …, β_{d}. Because of what we

know about the graded Betti numbers of X it has the shape

⨁_{i=1}^{β_{d−1}} A(α_{X} − 3) ⟶ ⨁_{i=1}^{β_{d}} A(α_{di} − d − 1) ⟶^{ϵ} ω_{R} ⟶ 0.

Let K be the kernel of ϵ. Then K_{−α_{X}+2} = 0 implies that any relation r_{1}φ_{1} + … + r_{β_{d}}φ_{β_{d}} = 0 of degree −α_{X} + 2 (i.e. with r_{i} ∈ R_{−α_{X}+2−deg φ_{i}} = R_{α_{di}−α_{X}−d+1}) is trivial.

“b)⇒a)” Consider the minimal homogeneous presentation

⨁_{i=1}^{β_{d−1}} A(α_{d−1 i} − d − 1) ⟶ ⨁_{i=1}^{β_{d}} A(α_{di} − d − 1) ⟶^{ϵ} ω_{R} ⟶ 0

induced by ϵ(e_{i}) = φ_{i} for i = 1, …, β_{d}. Let K be the kernel of ϵ. The hypothesis implies that K_{−α_{X}+2} = 0. Hence we have −α_{d−1 i} +d+1 > −α_{X} +2 for i = 1, …, β_{d−1}. In particular, we have α_{d−1 β_{d−1}} ≤ α_{X} + d − 2, so that Proposition 5.1 shows that X has an almost linear resolution.

Finally we prove the additional claim. Let Δ := Δ_{X} = dim_{k} (ω_{R})_{−σ_{X}}. We conclude from b) that r_{1}φ_{β_{d}−Δ+1} + … + r_{Δ}φ_{β_{d}} = 0 with r_{1}, …, r_{Δ} ∈ R_{σ_{X}−α_{X}+2} implies r_{1} = … = r_{Δ} = 0. Since {φ_{β_{d}−Δ+1}, …, φ_{β_{d}}} is a k-basis of (ω_{R})_{−σ_{X}}, this means that R_{σ_{X}−α_{X}+2} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−α_{X}+2} is injective. The injectivity of the other multiplication maps follows now easily from the fact that x_{0} is not a zero divisor of R. □

Example. Let us return to the example of eleven points X = {P_{1}, …, P_{11}} in ℙ^{2} given after Theorem 3.2 for one last time. The projective resolution of X is

0 ⟶ A^{2}(−5) ⊕ A(−6) ⟶ A^{4}(−4) ⟶ A ⟶ R ⟶ 0.

Therefore X has an almost linear resolution, and Theorem 5.3 shows that the multiplication map (ω_{R})_{−3} ⊗ R_{1} ⟶ (ω_{R})_{−2} is injective. Of course, in the present example this follows also from the fact that X is a Cayley-Bacharach scheme with Δ_{X} = 1.

By combining the various informations which we obtained in the last three sections, we have now a clear picture of the multiplication maps of ω_{R} in degrees ≤ 0:

1) For i = 1, 2, 3 the multiplication maps (ω_{R})_{−3} ⊗R_{i} ⟶ (ω_{R})_{−3+i} are injective.

2) For i = 1, 2 the multiplications maps (ω_{R})_{−i}⊗R_{1} ⟶ (ω_{R})_{−i+1} are biinjective. They cannot be injective because of dimension reasons.

3) The map (ω_{R})_{−2} ⊗ R_{2} ⟶ (ω_{R})_{0} is neither injective nor biinjective.

Of course, also the previous theorem admits a generalization for schemes with almost quadratical or higher order resolutions. Since the proof of our final theorem is completely analogous to the one given above, we leave it to the reader.

Theorem 5.4. Let n ∈ {1, …, σ_{X} − α_{X} + 3}, and let {φ_{1}, …, φ_{β_{d}}} be a minimal homogeneous system of generators of ω_{R}, where φ_{i} ∈ (ω_{R})_{−α_{di}+d+1} for i = 1, …, β_{d}. Choose ν ∈ {1, …, β_{d} − Δ_{X} + 1} such that {φ_{ν}, …, φ_{β_{d}}} are precisely those elements in {φ_{1}, …, φ_{β_{d}}} which have degree at most −α_{X} − n + 3. (This is possible because of n ≤ σ_{X} − α_{X} + 3.) The following conditions are equivalent.

a) X has a resolution almost of order n.

b) If for i = 1, …, β_{d} − ν + 1 we have elements r_{i} ∈ R_{α_{di}−α_{X}−d−n+2} such that r_{1}φ_{v} + … + r_{β_{d}−ν+1}φ_{β_{d}} = 0, then r_{1} = … = r_{β_{d}−ν+1} = 0.

In particular, if X has a resolution almost of order n, then the multiplication maps R_{m} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−σ_{X}+m} are injective for m = 1, …, σ_{X} − α_{X} − n + 3.

The injectivity claim in Theorem 5.4 implies strong inequalities for the Hilbert function of X.

Corollary 5.5. Let n ∈ {1, …, σ_{X} −α_{X} +3}, and suppose that X has a resolution almost of order n. Then we have

Δ_{X} · H_{X}(m) + H_{X}(σ_{X} − m) ≤ deg X

for every m ∈ {1, …, σ_{X} − α_{X} − n + 3}.

Clearly every 0-dimensional scheme X ⊆ ℙ^{d} has a resolution almost of order σ_{X} − α_{X} + 3. The next lower case is somewhat more interesting.

Corollary 5.6. The following conditions are equivalent.

a) X has a resolution almost of order σ_{X} − α_{X} + 2.

b) α_{d−1 β_{d−1}} ≤ σ_{X} + d − 1

c) The multiplication map R_{1} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−σ_{X}+1} is injective.

Proof: In view of 5.1 and 5.4 we only have to show “c)⇒a)”. Choose ν ∈ {1, …, β_{d}} as in 5.4, and let r_{i} ∈ R_{α_{di}−σ_{X}−d} be such that r_{1}φ_{ν} + … + r_{β_{d}−ν+1}φ_{β_{d}} = 0. For i = 1, …, Δ_{X} we have r_{i} ∈ R_{0} = k. Since {φ_{1}, …, φ_{β_{d}}} is minimal, this implies r_{1} = … = r_{Δ} = 0. For i = Δ_{X} + 1, …, β_{d} − ν + 1 we have r_{i} ∈ R_{1}. Since R_{1} ⊗ (ω_{R})_{−σ_{X}} ⟶ (ω_{R})_{−σ_{X}+1} is injective, the relation r_{Δ_{X}+1}φ_{β_{d}−Δ_{X}+1} + … + r_{β_{d}−ν+1}φ_{β_{d}} = 0 implies r_{Δ+1} = … = r_{β_{d}−ν+1} = 0. An application of the theorem now finishes the proof. □

The following corollary is a special case of Corollary 5.5.

Corollary 5.7. If X ⊆ ℙ^{d} is nondegenerate and has a resolution almost of order σ_{X} − α_{X} + 2, then ΔH_{X}(σ_{X}) ≥ d · Δ_{X}.

Remark. For subschemes X ⊆ ℙ^{2} the projective resolution is particularly short. In this case “X has a resolution almost of order σ_{X} − α_{X} + 2” easily translates into “I_{X} is generated by elements of degree ≤ σ_{X} + 1”. Notice that I_{X} is always generated by elements of degree ≤ σ_{X} + 2 because of what we explained at the beginning of this section. If I_{X} is generated by elements of degree ≤ σ_{X} + 1, Corollary 5.7 yields that the Hilbert function of X satisfies ΔH_{X}(σ_{X}) ≥ 2Δ_{X}.

Acknowledgements. Part of this work was done while the author was visiting Queen’s University, Kingston, and was supported by NSERC. He is deeply indebted to A.V. Geramita, E. Kunz, and to the participants of seminars at both Queen’s University and Universität Regensburg, for valuable discussions about preliminary versions of this paper.

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