Short running title: On 0-dimensional complete intersections

On 0-dimensional Complete Intersections

by Martin Kreuzer

Fakultät für Mathematik, Universität Regensburg, D-8400 Regensburg, FRG

1. Introduction

Given a 0-dimensional subscheme Y of ℙ^{d}, how can we use invariants and geometric properties of the embedding Y ⊆ ℙ^{d} to characterize whether Y is a complete intersection ? In the case Y ⊆ ℙ^{2} there is a well-known answer to this question (cf. [DM], § 4):

The scheme Y is a complete intersection of two curves if and only if the Hilbert function of Y is symmetrical and Y has the Cayley-Bacharach property with respect to O_{ℙ}(a_{Y}).

Here a_{Y} is the maximal degree for which the Hilbert function H_{Y} of Y is below deg Y, the Hilbert function is called symmetrical if H_{Y} (n) + H_{Y} (a_{Y} − n) = deg Y for all n ≥ 0, and the Cayley-Bacharach property means that every curve of degree a_{Y} passing through a subscheme of colength one of Y has to contain Y.

This is no longer true for Y ⊆ ℙ^{d}, d ≥ 3. Instead, we generalize the above result as follows.

Theorem 1.1. A 0-dimensional subscheme Y ⊆ ℙ^{d}, d ≥ 2 is arithmetically Gorenstein (i.e. the projective coordinate ring of Y is a Gorensteinring) if and only if the Hilbert function of Y is symmetrical and Y has the Cayley-Bacharach property with respect to O_{ℙ}(a_{Y}).

For reduced schemes Y this has been shown in [DGO]. In the nonreduced case we give a proof which is based on a detailed study of the Cayley-Bacharach property using traces.

In view of this theorem we want to find further necessary conditions for the characterization of 0-dimensional complete intersections in ℙ^{3}. Our approach is to divide this problem into two steps:

1) Give conditions on Y to be the zeroscheme of a regular section of a suitable vector bundle of rank 3 on ℙ^{3}.

2) Give additional conditions which force the vector bundle to split into a direct sum of line bundles. Step 1 is carried out in section 3. We utilize the study of suitable rank 3 vector bundles,

called vector bundles with good resp. very good sections, which was started in [K1]. We characterize the 0-dimensional zeroschemes of regular sections of vector bundles with very good sections in the following way.

Theorem 1.2. A 0-dimensional subscheme Y of ℙ^{3} is the zeroscheme of a regular section of a rank 3 vector bundle with very good sections if and only if there is a locally complete intersection (l.c.i.) curve C such that Y is the associated subscheme of an effective Cartier divisor on C and O_{C}(Y) is subcanonical and globally generated.

Examples show that this representation is in general not unique. The theorem can also be used to construct “good” vector bundles on ℙ^{3} with given invariants. In section 4 we use a modification of Horrocks’ splitting criterion for vector bundles to achieve step 2. We obtain a characterization of 0-dimensional complete intersections in ℙ^{3}.

Theorem 1.3. A 0-dimensional subscheme Y ⊆ ℙ^{3} is a complete intersection if and only if Y is arithmetically Gorenstein and there exists a projectively normal, l.c.i. curve C such that Y is the associated subscheme of an effective Cartier divisor on C and O_{C}(Y) ≅ ω_{C}(−a_{Y}) is globally generated.

Finally this theorem is reformulated in a more computational way and illustrated by examples.

2. On 0-dimensional Arithmetically Gorenstein Schemes

Let k be an algebraically closed field, V a projective scheme over k, L a locally free O_{V} -module of rank one, and Y a 0-dimensional closed subscheme of V with ideal sheaf J_{Y}. The following definition generalizes the one given in [GH], p. 470.

Definition 2.1. We say that Y has the Cayley-Bacharach property (CBP) with respect to L, if for every subscheme Z ⊆ Y with deg Z = deg Y −1 and for every section s ∈ H^{o}(V, L) with Z ⊆ (s)_{o} we have Y ⊆ (s)_{o}.

Here (s)_{o} denotes the zeroscheme of s, whose ideal sheaf is the image of the dual morphism sˇ : Lˇ ⟶ O_{V}.

Remarks. Let α_{Y} : H^{o}(V, L) ⟶ H^{o}(Y, L ⊗ O_{Y}) be the restriction homomorphism. By applying the snake lemma to the diagram

0 ⟶ H^{o}(V, J_{Y} · L) ⟶ H^{o}(V, L) ⟶^{α_{Y}} H^{o}(Y, L ⊗ O_{Y}) ⟶ coker α_{Y} ⟶ 0
 ; ↓  ∥  ↓ ; ↓  ↓ ; ↓
0 ⟶ H^{o}(V, J_{Z} · L) ⟶ H^{o}(V, L) ⟶^{α_{Z}} H^{o}(Z, L ⊗ O_{Z}) ⟶ coker α_{Z} ⟶ 0

we see that Y has the CBP with respect to L if and only if

dim_{k}(coker α_{Z}) = dim_{k}(coker α_{Y}) − 1

for every subscheme Z ⊆ Y with deg Z = deg Y − 1.

A subscheme Z ⊆ Y with deg Z = deg Y − 1 corresponds to an ideal of H^{o}(Y, O_{Y}) ≅ ∏_{P ∈Y} O_{Y,P} which is 1-dimensional over k, i.e. to an element of the form (0, …, 0, s_{P} , 0, …, 0)

where s_{P} is in the socle of O_{Y,P}. If we denote by 𝔖_{P} (L) the image of {x ∈ L_{P} ⊗ O_{Y,P} : 𝔪_{Y,P} · x = 0} in H^{o}(Y, L ⊗ O_{Y}) ≅ ∏_{P ∈Y} L_{P} ⊗ O_{Y,P}, then another application of the snake

lemma shows that Y has the CBP with respect to L if and only if

im α_{Y} ∩ 𝔖_{P} (L) = 0  for all P ∈ Y.

The following proposition provides a useful reformulation of the CBP in terms of traces. Here a trace of a module M over a finite algebra A/k means an A-module basis of Hom_{k}(M, k), where the A-module structure of Hom_{k}(M, k) is defined as usual by

\begin{matrix}A × Hom_{k}(M, k) & ⟶ & Hom_{k}(M, k) \\ (a, l) & ⟼ & (m ↦ l(a · m)). \end{matrix}

Proposition 2.2. The following conditions are equivalent:

(a) Y is a Gorensteinscheme (i.e. the local rings O_{Y,P} are Gorensteinrings) and has the CBP with respect to L.

(b) There is a trace σ : H^{o}(Y, L⊗O_{Y}) ⟶ k of the H^{o}(Y, O_{Y})-module H^{o}(Y, L⊗O_{Y}) such that im α_{Y} ⊆ ker σ.

Proof: “(a)⇒(b)” According to the preceding remark, 𝔖_{P} (L) has for every point P ∈ Y a 1-dimensional image in H^{o}(Y, L ⊗ O_{Y})/im α_{Y}. Therefore there is a 1-codimensional subvectorspace \overline{W} ⊆ H^{o}(Y, L ⊗ O_{Y})/im α_{Y} which meets these images only trivially. The preimage W of \overline{W} in H^{o}(Y, L ⊗ O_{Y}) is also 1-codimensional and intersects 𝔖_{P} (L) only trivially.

Let x ∈ H^{o}(Y, L ⊗ O_{Y}) \ W and let σ be the projection to k · x along W. Every nonzero element of H^{o}(Y, L⊗O_{Y}) has a nonzero multiple in one of the local socles 𝔖_{P} (L). Therefore σ is linearly independent.

We use the identification H^{o}(Y, L ⊗ O_{Y}) ≅ ∏_{P ∈Y} L_{P} ⊗ O_{Y,P} and denote by σ_{P} the

restriction of σ to L_{P} ⊗ O_{Y,P}. If σ_{P} ∈ 𝔪_{Y,P} · Hom_{k}(L_{P} ⊗ O_{Y,P} , k), then σ_{P} (y) = 0 for all y ∈ L_{P} ⊗ O_{Y,P} such that 𝔪_{Y,P} · y = 0. Thus σ(𝔖_{P} (L)) = 0, in contradiction to the choice of W = ker σ. Therefore σ_{P} ∉ 𝔪_{Y,P} · Hom_{k}(L_{P} ⊗ O_{Y,P} , k) for all P ∈ Y.

Since L_{P} ⊗O_{Y,P} ≅ O_{Y,P} and O_{Y,P} is a Gorensteinring, σ_{P} generates the O_{Y,P} -module Hom_{k}(L_{P} ⊗ O_{Y,P} , k) by the lemma of Nakayama. Using the H^{o}(Y, O_{Y})-isomorphism

\begin{matrix}Hom_{k}(H^{o}(Y, L ⊗ O_{Y}), k) & ⟶^{∼} & ∏_{P ∈Y} Hom_{k}(L_{P} ⊗ O_{Y,P} , k) \\ σ & ⟼ & (σ_{P})_{P ∈Y} \end{matrix}

we find that σ generates the H^{o}(Y, O_{Y})-module Hom_{k}(H^{o}(Y, L ⊗ O_{Y}), k).

Altogether we have proved that σ is a trace of H^{o}(Y, L ⊗ O_{Y}). By construction, im α_{Y} ⊆ W = ker σ.

“(b)⇒(a)” Since H^{o}(Y, L ⊗ O_{Y}) ≅ H^{o}(Y, O_{Y}), there is also a trace of H^{o}(Y, O_{Y}), i.e. Y is a Gorensteinscheme. From the linear independence of σ we conclude σ(𝔖_{P} (L)) ≠ 0 for all P ∈ Y. Therefore im α_{Y} ∩ 𝔖_{P} (L) = 0 for all P ∈ Y, and Y has the CBP with respect to L by the above remark. q.e.d.

Now let us specialize this proposition to the case V = ℙ^{d} and L = O_{ℙ}(a_{Y}), where a_{Y} is defined as the last degree for which the Hilbert function H_{Y} of Y has not yet reached its maximal value deg Y, i.e.

a_{Y} = max{n ∈ ℤ : H_{Y} (n) < deg Y }
= max{n ∈ ℤ : H^{1}(ℙ, J_{Y} (a_{Y})) ≠ 0}.

Let I_{Y} := Γ_{∗}(J_{Y}) denote the saturated homogeneous ideal and S^{∗} := k[X_{o}, …, X_{d}]/I_{Y} the projective coordinate ring of Y. W.l.o.g. we can assume that x_{o} := X_{o} + I_{Y} is not a zero divisor of S^{∗}. The affine coordinate ring S of Y in the affine space D_{+}(X_{o}) is described by

S ≅ H^{o}(Y, O_{Y}) ≅ S^{∗}/(x_{o} − 1) ≅ k[X_{1}, …, X_{d}]/J_{Y} ,

where J_{Y} is the dehomogenization of I_{Y} with respect to X_{o}. Also, let F = (F_{i})_{i∈ℤ} denote the (ascending) filtration of S/k induced by the degree filtration of k[X_{1}, …, X_{d}], i.e.

F_{i} = {s ∈ S : ∃f ∈ k[X_{1}, …, X_{d}] : deg f ≤ i, s = f + J_{Y} }.

Then S^{∗} is the Rees algebra of the filtered algebra (S/k, F) (cf. [KK], § 3).

Remark. Let β_{Y} be the isomorphism

β_{Y} : S ≅ H^{o}(Y, O_{Y}) ⟶^{∼} H^{o}(Y, O_{Y} (a_{Y}))

induced by multiplication with x_{o}^{a_{Y}}. Then β_{Y} (F_{a_{Y}}) = im α_{Y}, because the composition β_{Y}^{−1} ∘ α_{Y} : H^{o}(ℙ, O_{ℙ}(a_{Y})) ⟶ S sends a homogeneous polynomial F of degree a_{Y} to the residue class of its dehomogenization F (1, X_{1}, …, X_{d}) in S.

Proposition 2.3. The following conditions are equivalent:

(a) Y is a Gorensteinscheme and has the CBP with respect to O_{ℙ}(a_{Y}).

(b) There is a trace σ : S ⟶ k with F_{a_{Y}} ⊆ ker σ.

(c) There is a homogeneous k[x_{o}]-linear map σ^{∗} : S^{∗} ⟶ k[x_{o}] of degree −a_{Y} − 1 such that Ann_{S^{∗}}(σ^{∗}) = 0 as an element of the S^{∗}-module Hom_{k[x_{o}]}(S^{∗}, k[x_{o}]).

Proof: “(a)⇔(b)” This follows from (2.2) and the preceding remark.

“(b)⇒(c)” Every element of S^{∗} is of the form x_{o}^{n} · f^{∗}, where f^{∗}(x_{o}, …, x_{d}) := x_{o}^{ord_{F} f} · f(\frac{x_{1}}{x_{o}} , …, \frac{x_{d}}{x_{o}}) is the homogenization of an element f ∈ S (cf. [KK], (1.1)). Therefore we can define a k[x_{o}]-linear map σ^{∗} : S^{∗} ⟶ k[x_{o}] by σ^{∗}(x_{o}^{n} · f^{∗}) := x_{o}^{n} · σ(f). Since σ(F_{a_{Y}}) = 0, it follows that σ^{∗} is homogeneous of degree −a_{Y} − 1.

Suppose x_{o}^{n} · f^{∗} · σ^{∗} = 0 for some n ∈ ℕ, f ∈ S. Then we have 0 = (x_{o}^{n} · f^{∗} · σ^{∗})(g^{∗}) = x_{o}^{n} · σ^{∗}(f^{∗}g^{∗}) = x_{o}^{n} · σ^{∗}(x_{o}^{ϵ} · (fg)^{∗}) = x_{o}^{n+ϵ} · σ(fg) with ϵ = ord_{F}f + ord_{F}g − ord_{F}(fg), and hence (f · σ)(g) = σ(fg) = 0 for all g ∈ S. This implies f = 0, because σ is a trace of S/k, and therefore x_{o}^{n} · f^{∗} = 0. Thus we have proved Ann_{S^{∗}}(σ^{∗}) = 0.

“(c)⇒(b)” Since σ^{∗} is homogeneous of degree −a_{Y} − 1, it induces a k-linear map σ : S ⟶ k (f ↦ σ^{∗}(f^{∗})) with σ(F_{a_{Y}}) = 0 (cf. [KK], § 3). First let us show that σ is an S-linear independent element of Hom_{k}(S, k).

Suppose f · σ = 0 for some element f ∈ S. Then σ(fg) = 0 for all g ∈ S, i.e. (f^{∗}σ^{∗})(x_{o}^{n}g^{∗}) = x_{o}^{n+ϵ} · σ^{∗}((fg)^{∗}) = 0 for all n ∈ ℕ, g ∈ S, where ϵ = ord_{F}f + ord_{F}g − ord_{F}(fg). Hence f^{∗} · σ^{∗} = 0, and therefore f = 0. In particular, every nonzero element f ∈ 𝔖_{P} (O_{ℙ}) has σ(f) ≠ 0, since otherwise f · σ = 0.

From this we see that the local socle 𝔖_{P} (O_{ℙ}) is 1-dimensional over k for every point P ∈ Y, because two linearly independent elements f, g ∈ 𝔖_{P} (O_{ℙ}) have a nontrivial linear combination which is in the kernel of σ. The fact that σ maps the local socles 𝔖_{P} (O_{ℙ}) isomorphically onto k implies that σ is a trace of S/k as in the proof of (2.2). q.e.d.

Remark. Let ω_{S^{∗}} be the canonical module of S^{∗}/k. We can describe ω_{S^{∗}} either as ω_{S^{∗}} ≅ Hom_{k}(H_{𝔪}^{1}(S^{∗}), k) where 𝔪 = (x_{o}, …, x_{d}) is the homogeneous maximal ideal of S^{∗}, or as ω_{S^{∗}} ≅ Hom_{k[x_{o}]}(S^{∗}, k[x_{o}])(−1) (cf. [GW], (2.1.2) and (2.2.9)). From the standard exact sequence

0 ⟶ S^{∗} ⟶ Γ_{∗}(O_{Y}) ⟶ H_{𝔪}^{1}(S^{∗}) ⟶ 0

we can compute its Hilbert function H_{ω_{S^{∗}}} as follows:

H_{ω_{S^{∗}}} (n) = dim_{k}(H_{𝔪}^{1}(S^{∗})_{−n})
= deg Y − dim_{k}(S_{−n}^{∗})
= deg Y − H_{Y} (−n).

Note that a_{Y} = max{n ∈ ℤ : H_{𝔪}^{1}(S^{∗})_{n} ≠ 0} = − min{n ∈ ℤ : (ω_{S^{∗}})_{n} ≠ 0}. Proposition (2.3) says that Y is a Gorensteinscheme with the CBP with respect to O_{ℙ}(a_{Y}) if and only if there exists an element σ^{∗} ∈ (ω_{S^{∗}})_{−a_{Y}} with Ann_{S^{∗}}(σ^{∗}) = 0, i.e. with S^{∗} · σ^{∗} ⊆ ω_{S^{∗}}. With this in mind, we can now prove theorem (1.1).

Proof of (1.1): Let Y be arithmetically Gorenstein. Then ω_{S^{∗}} ≅ S^{∗}(a_{Y}) (cf. [GW], (3.1.4)) and every S^{∗}-basis σ^{∗} of ω_{S^{∗}} satisfies the hypothesis of (2.3.c). Moreover, from ω_{S^{∗}} = S^{∗} · σ^{∗} we see that H_{Y} (a_{Y} − n) = dim_{k}(S^{∗} · σ^{∗})_{−n} = H_{ω_{S^{∗}}} (−n) = deg Y − H_{Y} (n), i.e. that H_{Y} is symmetrical.

Conversely, let Y have the CBP with respect to O_{ℙ}(a_{Y}) and a symmetrical Hilbert function. The conditions dim_{k}(coker α_{Y}) = dim_{k} H_{𝔪}^{1}(S^{∗})_{a_{Y}} = dim_{k} H_{ω_{S^{∗}}} (−a_{Y}) = deg Y − H_{Y} (a_{Y}) = H_{Y} (0) = 1 and im α_{Y} ∩𝔖_{P} (O_{ℙ}(a_{Y})) = 0 for P ∈ Y imply that dim_{k} 𝔖_{P} (O_{ℙ}) = dim_{k} 𝔖_{P} (O_{ℙ}(a_{Y})) ≤ 1 for all P ∈ Y, i.e. that Y is a Gorensteinscheme.

By (2.3), we find an S^{∗}-linearly independent element σ^{∗} in (ω_{S^{∗}})_{−a_{Y}}. But for every n ≥ 0 we have dim_{k}(S^{∗} · σ^{∗})_{−a_{Y} +n} = H_{Y} (n) = deg Y − H_{Y} (a_{Y} − n) = dim_{k}(ω_{S^{∗}})_{−a_{Y} +n}, so that indeed S^{∗} · σ^{∗} = ω_{S^{∗}}. Therefore ω_{S^{∗}} is a free S^{∗}-module, which implies that Y is arithmetically Gorenstein. q.e.d.

Remark. In the case Y ⊆ ℙ^{2} the two notions “Y is arithmetically Gorenstein” and “Y is a complete intersection” coincide by a theorem of Serre (cf. [S]), and we recover the result cited in the introduction. Already for Y ⊆ ℙ^{3} the conditions of (1.1) do not guarantee that Y is a complete intersection, as the following example shows.

Example (A.V. Geramita). Let Y ⊆ ℙ^{3} consist of eight reduced points on a twisted cubic. Then H_{Y} (0) = 1, H_{Y} (1) = 4, H_{Y} (2) = 7, and H_{Y} (n) = 8 for n ≥ 3. Thus a_{Y} = 2, H_{Y} is symmetrical, and, by Bezout’s theorem, Y has the CBP with respect to O_{ℙ}(a_{Y}). But Y is not a complete intersection, since every quadric through Y contains the twisted cubic.

Remark. If Y is a Gorensteinscheme with H_{Y} (a_{Y}) = deg Y − 1, then Y has the CBP with respect to O_{ℙ}(a_{Y}) if and only if a_{Z} < a_{Y} for all subschemes Z ⊆ Y.

Using this, theorem (1.1) can be regarded as a special case of Watanabe’s characterization of graded Gorenstein rings (cf. [W], (1.3)).

Proof: Since a_{Z} ≤ a_{Y} for every subscheme Z ⊆ Y, and since every subscheme of Y is contained in a subscheme Z ⊆ Y with deg Z = deg Y −1, the second condition is equivalent with a_{Z} < a_{Y} for subschemes Z ⊆ Y of colength one. This means that the elements f ∈ S defining such subschemes have to have ord_{F}f = a_{Y} + 1, because H_{Z}(a_{Y}) = deg Y − 1 and H_{Z}(n) = H_{Y} (n) for 0 ≤ n < a_{Y}. These elements f ∈ S generate the local socles 𝔖_{P} (O_{ℙ}). Now (2.2) and the subsequent remark show that ord_{F}f = a_{Y} + 1 for these elements f ∈ S is equivalent with the CBP of Y with respect to O_{ℙ}(a_{Y}). q.e.d.

3. On 0-dimensional Zeroschemes in ℙ^{3}

Subsequently let ℙ := ℙ_{k}^{3}, and let E be a locally free O_{ℙ}-module of rank 3. We call E also a vector bundle of rank 3 on ℙ (cf. [Ha1], p. 129). When studying the space H^{o}E of global sections of E, it is useful to introduce the following notions.

Definition 3.1. (a) For s ∈ H^{o}E the scheme (s)_{o} which is defined as the closed subscheme of ℙ whose sheaf of ideals is the image of the dual morphism sˇ : E ˇ ⟶ O_{ℙ}, is called the zeroscheme of s.

(b) For i = 0, .., 3 the set S_{i} = {s ∈ H^{o}E : dim(s)_{o} ≥ i} is called the set of i times degenerated sections of E. The elements of S_{o} \ S_{1} are also called the regular sections of E.

Remark. Obviously H^{o}E ⊇ S_{o} ⊇ ⋯ ⊇ S_{3} = {0}. The elements of H^{o}E \ S_{o} are the global sections of E with an empty zeroscheme. In [K1], (3.3) we showed that if one regards H^{o}E as an affine space over k, then S_{i} is the set of closed points of a closed subset of H^{o}E, which we denote again by S_{i} (i = 0, .., 3).

We expect that for “good” vector bundles the sets of highly degenerated sections should have a small dimension. This can be formalized as follows.

Definition 3.2 (a) We say that E has good sections, if S_{o} = H^{o}E and if codim (S_{i}, H^{o}E) ≥ i for i = 1, 2.

(b) We say that E has very good sections, if E has good sections and for every closed point P ∈ ℙ there is a section s ∈ H^{o}E such that P ∉ (s)_{o}.

Examples. (a) If E splits into a direct sum of line bundles E ≅ O_{ℙ}(n_{1}) ⊕ O_{ℙ}(n_{2}) ⊕ O_{ℙ}(n_{3}), then E has good sections (or very good sections) if and only if n_{1}, n_{2}, n_{3} > 0.

(b) More generally, if E is generated by global sections and E has a regular section, then E has good sections (cf. [K1], (3.7)).

(c) If E splits into a direct sum E ≅ O_{ℙ} ⊕F with a locally free O_{ℙ}-module F of rank 2, then E does not have good sections.

In [K1], § 4 we gave an alternative description of rank 3 bundles with good (resp. very good) sections.

Proposition 3.3. For a locally free O_{ℙ}-module E of rank 3 the following conditions are equivalent.

(a) E has good (resp. very good) sections.

(b) There is a regular section s ∈ H^{o}E and another global section s′ ∈ H^{o}E such that the zeroscheme of the section s∧s′ ∈ H^{o}(⋀^{2} E) is a locally Cohen-Macaulay and generically l.c.i. curve (resp. is a l.c.i. curve).

(c) There is a regular section s ∈ H^{o}E such that for a generic section s′ ∈ H^{o}E the zeroscheme of s ∧ s′ is a locally Cohen-Macaulay and generically l.c.i. curve (resp. is a l.c.i. curve).

In particular, s′ can be chosen regular, too.

Based on this proposition we are now going to characterize 0-dimensional zeroschemes of rank 3 bundles with very good sections. We need the basic properties of reflexive O_{ℙ}- modules (cf. [Ha2], § 1) and the following lemma.

Lemma 3.4. Let F be a reflexive O_{ℙ}-module of rank 2, and let s ∈ H^{o}F have a 1-dimensional zeroscheme C := (s)_{o} with ideal sheaf J_{C} ⊆ O_{ℙ}. Then the Koszul complex

(∗)  0 ⟶ det F ˇ ⟶^{(∧_{s})}ˇ F ˇ ⟶^{s}ˇ J_{C} ⟶ 0

is exact. By applying the functor Hom_{O_{ℙ}}(—, det F ˇ ) to (∗) and using the isomorphisms Hom_{O_{ℙ}}(J_{C}, det F ˇ ) ≅ Hom_{O_{ℙ}}(J_{C}, O_{ℙ}) ⊗ det F ˇ ≅ det F ˇ and Hom_{O_{ℙ}}(F ˇ , det F ˇ ) ≅ Hom_{O_{ℙ}}(F ˇ , O_{ℙ}) ⊗ det F ˇ ≅ F ⊗ det F ˇ we get an exact sequence

0 ⟶ det F ˇ ⟶^{φ} F ⊗ det F ˇ ⟶ O_{ℙ}.

The isomorphism ψ : F ⊗ det F ˇ ⟶^{∼} F ˇ of [Ha2], (1.10) identifies φ with (∧_{s})ˇ .

Proof: The kernel of sˇ : F ˇ ⟶ O_{ℙ} is a reflexive O_{ℙ}-module of rank 1, hence invertible. Using the definition of det F ˇ it follows that (∗) is exact.

Since F is locally free on an open subset U ⊆ ℙ with codim (ℙ\U, ℙ) ≥ 2, and since the restriction H^{o}(V, F ˇ ) ⟶ H^{o}(V ∩ U, F ˇ ) is an isomorphism for every open subset V ⊆ ℙ, it suffices to show ψ_{P} ∘ φ_{P} = (∧_{s})_{P}ˇ for every closed point P ∈ U.

Let {e_{1}, e_{2}} be an O_{ℙ,P} -basis of F_{P}, let {e_{1}ˇ , e_{2}ˇ } be the dual basis of F_{P}ˇ , and write s_{P} = t_{1}e_{1} + t_{2}e_{2} with t_{1}, t_{2} ∈ O_{ℙ,P}. Then the mapping (∧_{s})_{P}ˇ : det F_{P}ˇ ⟶ F_{P}ˇ is given by (∧_{s})_{P}ˇ (e_{1}ˇ ∧ e_{2}ˇ ) = −t_{2}e_{1}ˇ + t_{1}e_{2}ˇ . The dual mapping of s_{P}ˇ : F_{P}ˇ ⟶ J_{C,P} is s_{P} : O_{ℙ,P} ⟶ F_{P} with s_{P} (1) = t_{1}e_{1} + t_{2}e_{2}. Since φ_{P} is constructed by tensoring s_{P} with det F_{P}ˇ , we have φ_{P} (e_{1}ˇ ∧ e_{2}ˇ ) = t_{1}e_{1} ⊗ (e_{1}ˇ ∧ e_{2}ˇ ) + t_{2}e_{2} ⊗ (e_{1}ˇ ∧ e_{2}ˇ ).

The isomorphism ψ_{P} is induced by the pairing F_{P} ⊗ F_{P} ⟶ det F_{P} with e_{i} ⊗ e_{j} ⟼ e_{i} ∧ e_{j} and satisfies therefore ψ_{P} (e_{1} ⊗ (e_{1}ˇ ∧ e_{2}ˇ )) = e_{2}ˇ as well as ψ_{P} (e_{2} ⊗ (e_{1}ˇ ∧ e_{2}ˇ )) = −e_{1}ˇ . Altogether we find ψ_{P} ∘ φ_{P} (e_{1}ˇ ∧ e_{2}ˇ ) = t_{1}e_{2}ˇ − t_{2}e_{1}ˇ = (∧_{s})_{P}ˇ (e_{1}ˇ ∧ e_{2}ˇ ). q.e.d.

Proposition 3.5. Let Y ⊆ ℙ be a 0-dimensional subscheme, and let e ∈ ℤ. The following conditions are equivalent:

(a) Y is the zeroscheme of a regular section of a locally free O_{ℙ}-module E of rank 3 with very good sections and with c_{1}(E) = e + 4.

(b) There is a l.c.i. curve C ⊆ ℙ containing Y such that Y is the zeroscheme of a global section u of ω_{C}(−e), and there is a section v ∈ H^{o}(C, ω_{C}(−e)) which generates ω_{C}(−e) at the points of Y.

Proof: “(a)⇒(b)” Let Y = (s)_{o} be the zeroscheme of s ∈ H^{o}E, and let F := ker sˇ . Then we have an exact sequence

(1)  0 ⟶ F ⟶^{i} E ˇ ⟶^{s}ˇ J_{Y} ⟶ 0,

where J_{Y} is the ideal sheaf of Y. By dualizing this twice and using Ext_{O_{ℙ}}^{1}(J_{Y} , O_{ℙ}) = 0 we see that F is a reflexive O_{ℙ}-module of rank 2, which is locally free outside Y and satisfies Ext_{O_{ℙ}}^{1}(F ˇ , O_{ℙ}) ≅ O_{Y}. By the previous proposition there is another regular section s′ ∈ H^{o}E such that C := (s ∧ s′)_{o} is a l.c.i. curve which contains Y and which is also the zeroscheme of t := H^{o}(iˇ )(s′) ∈ H^{o}F ˇ (cf. [K1], (4.1)).

Using the method of [Ha2], (4.1) we will show now that Y is the zeroscheme of a global section of ω_{C}(−e). From the lemma we obtain an exact sequence

(2)  0 ⟶ O_{ℙ}(−e − 4) ⟶^{j} F ⟶^{t}ˇ J_{C} ⟶ 0,

because det F ≅ det E ˇ ≅ O_{ℙ}(−e − 4) by (1). Dualizing (2) yields a long exact sequence

(3) 0 ⟶ O_{ℙ} ⟶^{t} F ˇ ⟶^{j}ˇ O_{ℙ}(e + 4) ⟶ Ext_{O_{ℙ}}^{1}(J_{C}, O_{ℙ}) ⟶ Ext_{O_{ℙ}}^{1}(F, O_{ℙ}) ⟶ 0

where Ext_{O_{ℙ}}^{1}(J_{C}, O_{ℙ}) ≅ Ext_{O_{ℙ}}^{2}(O_{C}, O_{ℙ}) ≅ ω_{C}(4). By splitting (1) off the Koszul complex

of sˇ we get an exact sequence 0 ⟶ det E ˇ ⟶^{(∧_{s})}ˇ ⋀^{2}(E ˇ ) ⟶ F ⟶ 0, which shows Ext_{O_{ℙ}}^{1}(F, O_{ℙ}) ≅ coker ∧_{s}. Now, using the Koszul complex

0 ⟶ O_{ℙ} ⟶^{s} E ⟶ ⋀^{2} E ⟶^{∧_{s}} det E ⟶ det E ⊗ O_{Y} ⟶ 0

of s, we find coker (∧_{s}) ≅ det E ⊗ O_{Y} ≅ O_{Y} (e + 4), so that (3) tensored by O_{ℙ}(−e − 4) yields the exact sequence

(4) 0 ⟶ O_{ℙ}(−e − 4) \underrightarrow{_{ȷ}} F ˇ (−e − 4) ⟶ O_{ℙ} ⟶ ω_{C}(−e) ⟶ O_{Y} ⟶ 0.

The isomorphism F ˇ (−e − 4) ≅ F ˇ ⊗ det F identifies j̃ with j by the lemma, so that (4) induces an exact sequence 0 ⟶ O_{C} ⟶^{u} ω_{C}(−e) ⟶ O_{Y} ⟶ 0 which represents Y as the zeroscheme of u ∈ H^{o}ω_{C}(−e).

Starting with the 0-dimensional scheme Y′ := (s′)_{o} ⊆ C we can construct another global section v ∈ H^{o}ω_{C}(−e) with (v)_{o} = Y′ in the same way. By [K1], (4.6) we have Y ∩ Y′ = ∅, i.e. the section v generates ω_{C}(−e) at the points of Y.

“(b)⇒(a)” First of all we want to represent C as the zeroscheme of a global section of a suitable reflexive O_{ℙ}-module F of rank 2 as in [Ha2], (4.1). Since H^{o}ω_{C}(−e) ≅ H^{1}O_{C}(e)^{∗} ≅ Ext_{O_{ℙ}}^{2}(O_{C}(e), ω_{ℙ}) ≅ Ext_{O_{ℙ}}^{1}(J_{C}, O_{ℙ}(−e − 4)), we can represent u by an extension

(5)  0 ⟶ O_{ℙ}(−e − 4) ⟶^{j} F ⟶^{t}ˇ J_{C} ⟶ 0

with a torsionfree coherent O_{ℙ}-module F which is reflexive and locally free outside Y, and with t ∈ H^{o}F ˇ . Now v generates ω_{C}(−e) at the points of Y, and Ext_{O_{ℙ}}^{1}(F, O_{ℙ}) ≅ O_{Y} follows by dualizing (5), i.e. Y is the singular scheme of F. Therefore we can apply [C], (1.1) and get an exact sequence

(6)  0 ⟶ F ⟶^{i} E ˇ ⟶^{s}ˇ J_{Y} ⟶ 0

with a locally free O_{ℙ}-module E of rank 3.

We still need to show that E has very good sections. Dualizing (6) yields an exact

sequence 0 ⟶ O_{ℙ} ⟶^{s} E ⟶^{i}ˇ F ˇ ⟶ 0 which allows us to find a preimage s′ ∈ H^{o}E of t under H^{o}(iˇ ). As in [K1], (4.1) this implies that s ∧ s′ is the image of t in H^{o}(⋀^{2} E) and that C = (s ∧ s′)_{o}. Thus, by the previous proposition, E has very good sections. q.e.d.

Now theorem (1.2) follows easily. Recall that a Cartier divisor Y on a l.c.i. curve C in ℙ is called subcanonical, if Y differs from the canonical divisor of C only by a multiple of the hyperplane section, i.e. if O_{C}(Y) ≅ ω_{C}(−e) for some e ∈ ℤ.

Proof of (1.2): If Y is the zeroscheme of a regular section of a rank 3 bundle with very good sections, then the previous proposition tells us that Y is the associated subscheme of an effective, subcanonical Cartier divisor on some l.c.i. curve C, because Y = (u)_{o} means O_{C}(Y) ≅ ω_{C}(−e). Since u and v generate ω_{C}(−e) everywhere, O_{C}(Y) is globally generated.

On the other hand, if Y = (u)_{o} for some u ∈ H^{o}ω_{C}(−e) and some e ∈ ℤ, and if ω_{C}(−e) is globally generated, then the subspaces {w ∈ H^{o}ω_{C}(−e) : P ∉ (w)_{o}} are properly contained in H^{o}ω_{C}(−e) for every P ∈ Y, so that we can find a section v ∈ H^{o}ω_{C}(−e) which generates this sheaf at the points of Y. Now apply the previous proposition.

q.e.d.

Remark. In the situation of theorem (1.2) we have O_{C}(Y) ≅ ω_{C}(−e) for some e ∈ ℤ, c_{1}(E) = e + 4, c_{2}(E) = deg C and c_{3}(E) = deg Y. The residue theorem for vector bundles (cf. [K2], (4.15)) implies that Y has the CBP with respect to O_{ℙ}(e).

We would also like to point out that it is easy to compute cohomology groups of E from the exact sequences

0 ⟶ O_{ℙ} ⟶^{s} E ⟶ F ˇ ⟶ 0  and  0 ⟶ O_{ℙ} ⟶^{t} F ˇ ⟶ J_{C}(e + 4) ⟶ 0

and their twists. For instance, h^{o}E(n) = 2h^{o}O_{ℙ}(n) + h^{o}J_{C}(e + 4 + n) and h^{1}E(n) = h^{1}J_{C}(e + 4 + n) for all n ∈ ℤ.

The rest of this section is devoted to applications of (1.2). Already the most trivial case shows that the representation of Y as a zeroscheme is in general not unique.

Example. Let Y be a closed point of ℙ.

(a) By choosing a line C through Y and applying the theorem we get a representation Y = (s)_{o} with s ∈ H^{o}E, where E is a rank 3 bundle with very good sections and with Chern classes c_{1}(E) = c_{2}(E) = c_{3}(E) = 1. From h^{o}E = 2 + h^{o}J_{C}(e + 4) = 4 we see that E is irreducible, because a direct summand O_{ℙ}(a) of E with a > 0 already has at least 4 linearly independent global sections.

(b) By choosing a twisted cubic C through Y and applying the theorem we get a representation Y = (s)_{o} with s ∈ H^{o}O_{ℙ}(1)^{3}.

Example. Let Y consist of two closed points of ℙ.

(a) If C is the line through Y, then the hypotheses of (1.2) are satisfied and we get a rank 3 bundle E with very good sections and with c_{1}(E) = c_{2}(E) = c_{3}(E) = 2, h^{o}E = 9. Let us show that E is irreducible.

Suppose E ≅ O_{ℙ}(a) ⊕ E′ with a > 0 and a locally free O_{ℙ}-module E′ of rank 2. Then 2 = c_{1}(E) = c_{1}(E′) + a, 2 = c_{2}(E) = ac_{1}(E′) + c_{2}(E′) and 2 = c_{3}(E) = ac_{2}(E′). This is impossible.

(b) If C consists of two disjoint lines, one through each point of Y, then O_{C}(Y) ≅ ω_{C}(3) and we get a 3-bundle E with c_{1}(E) = 1, c_{2}(E) = c_{3}(E) = 2, h^{o}E = 2. This example is the reason why we did not require codim (S_{3}, H^{o}E) ≥ 3 in definition (3.2). Obviously E is irreducible and not globally generated.

Example. Sometimes also singular curves C come in handy to represent Y as a zeroscheme. For instance, if Y consists of four reduced, complanar points, no three of which are collinear, then after a suitable change of coordinates Y can be represented as the intersection of C := V(X_{o}^{2} − X_{1}X_{3}, X_{o}X_{1} − X_{2}X_{3}) with a hyperplane which misses the singular point of C. Since ω_{C} ≅ O_{C}, the hypotheses of (1.2) are satisfied and we get Y = (s)_{o}, where s is a global section of a rank 3 bundle E with very good sections and with c_{1}(E) = 3, c_{2}(E) = c_{3}(E) = 4.

Example. Suppose that for every n ≥ 1 we want to construct an irreducible 3-bundle E with very good sections and with c_{1}(E) = c_{2}(E) = c_{3}(E) = n. A look at the above examples indicates to us what to do: Take Y to be n reduced points on a line, and let C be a plane curve of degree n through Y. The conditions of (1.2) are satisfied, and from the Chern classes we read off that we have indeed found irreducible bundles. In this way, (1.2) can be used to construct “good” vector bundles with given invariants.

4. On 0-dimensional Complete Intersections in ℙ^{3}

In this section we let again ℙ denote ℙ_{k}^{3}, and E a locally free O_{ℙ}-module of rank 3. Recall that Horrocks’ splitting criterion (cf. [Ho]) says that E is the direct sum of three line bundles if and only if h^{1}E(n) = h^{2}E(n) = 0 for all n ∈ ℤ. We need the following version of this criterion.

Proposition 4.1. Let e := c_{1}(E) − 4. The following conditions are equivalent: (a) E splits into a direct sum of three line bundles.

(b) We have h^{1}E(n) = 0 for all n ∈ ℤ, and there is an m ∈ ℤ and a global section s ∈ H^{o}E(m) such that the zeroscheme Y := (s)_{o} of s is 0-dimensional, arithmetically Gorenstein and satisfies a_{Y} = 3m + e.

Proof: “(a)⇒(b)” For m >> 0 we have E(m) ≅ O_{ℙ}(a_{1}) ⊕ O_{ℙ}(a_{2}) ⊕ O_{ℙ}(a_{3}) with a_{1}, a_{2}, a_{3} > 0. Let s ∈ H^{o}E(m) be a regular section and Y := (s)_{o}. Then Y is a 0-dimensional complete intersection with a_{Y} = a_{1} + a_{2} + a_{3} − 4 = 3m + e.

“(b)⇒(a)” By splitting the Koszul complex of sˇ and twisting it with O_{ℙ}(n) for n ∈ ℤ we get exact sequences

(∗) 0 ⟶ (det E ˇ )(−3m + n) ⟶ (⋀^{2} E ˇ )(−2m + n) ⟶ K(n) ⟶ 0 and (∗∗) 0 ⟶ K(n) ⟶ E ˇ (−m + n) ⟶ J_{Y} (n) ⟶ 0 where K is a coherent O_{ℙ}-module and J_{Y} ⊆ O_{ℙ} the ideal sheaf of Y. Using the isomorphism ⋀^{2} E ˇ ≅ E ⊗ det E ˇ ≅ E(−e − 4) we find h^{1}(⋀^{2} E ˇ )(n) = h^{1}E(n − e − 4) = 0 for all n ∈ ℤ. Then (∗) implies h^{1}K(n) = 0 for all n ∈ ℤ. By Serre duality, we also have h^{2}E ˇ (n) = h^{1}E(−n − 4) = 0 for all n ∈ ℤ. Substituting everything in the long exact cohomology sequences of (∗) and (∗∗) yields

h^{1}J_{Y} (n) = h^{1}E ˇ (−m + n) + h^{2}K(n)
= h^{1}E ˇ (−m + n) + h^{2}(⋀^{2} E ˇ )(−2m + n) + h^{3}(det E ˇ )(−3m + n)
− h^{3}(⋀^{2} E ˇ )(−2m + n) + h^{3}K(n)
= h^{1}E ˇ (−m + n) + h^{2}E(−2m + n − e − 4) + h^{3}O_{ℙ}(−3m + n − e − 4)
− h^{3}E(−2m + n − e − 4) + h^{3}E ˇ (−m + n) − h^{3}J_{Y} (n)
= h^{1}E ˇ (−m + n) + h^{1}E ˇ (2m − n + e) + h^{o}O_{ℙ}(3m − n + e)
− h^{o}E ˇ (2m − n + e) + h^{o}E(m − n − 4) + h^{3}O_{ℙ}(n)
= h^{1}E ˇ (−m + n) + h^{1}E ˇ (2m − n + e) + h^{o}O_{ℙ}(3m − n + e)
− h^{o}J_{Y} (3m − n + e) − h^{o}K(3m − n + e) + h^{o}E(m − n + 4)
− h^{o}O_{ℙ}(−n − 4)
= h^{1}E ˇ (−m + n) + h^{1}E ˇ (2m − n + e) + H_{Y} (3m − n + e)
+ h^{o}(det E ˇ )(−n + e) + h^{o}E(m − n + 4) − h^{o}O_{ℙ}(−n − 4)
= h^{1}E ˇ (−m + n) + h^{1}E ˇ (2m − n + e) + H_{Y} (a_{Y} − n).

Since H_{Y} is symmetrical, we get h^{1}E ˇ (−m+n)+h^{1}E ˇ (2m−n+e) = h^{1}J_{Y} (n)−H_{Y} (a_{Y} −n) = deg Y −H_{Y} (n)−H_{Y} (a_{Y} −n) = 0, and therefore h^{2}E(m−n−4) = h^{1}E ˇ (−m+n) = 0 for all n ∈ ℤ. Now an application of Horrocks’ splitting criterion finishes the proof. q.e.d.

Remark. If the number m in (4.1.b) can be chosen large enough, then the condition a_{Y} = 3m + e is automatically satisfied. Thus proposition (4.1) says that we can drop the condition “h^{2}E(n) = 0 for all n ∈ ℤ” in Horrocks’ splitting criterion, if we can find a regular section of a high enough twist of E whose zeroscheme is arithmetically Gorenstein. An analogous result holds for vector bundles of rank 2 on ℙ^{2}.

By applying (4.1) to (1.2), we can now prove the desired characterization of 0-dimen-sional complete intersections in ℙ^{3}.

Proof of (1.3): If Y ⊆ ℙ is a 0-dimensional complete intersection of type (a_{1}, a_{2}, a_{3}), then Y is the zeroscheme of a regular section of E := O_{ℙ}(a_{1}) ⊕ O_{ℙ}(a_{2}) ⊕ O_{ℙ}(a_{3}). Since E has very good sections, there is another regular section s′ ∈ H^{o}E such that C := (s ∧ s′)_{o} is a l.c.i. curve which contains Y as the associated subscheme of an effective Cartier divisor with O_{C}(Y) ≅ ω_{C}(−e) globally generated, and with e = c_{1}(E) − 4 = a_{1} + a_{2} + a_{3} − 4. By assumption, also a_{Y} = a_{1} + a_{2} + a_{3} − 4. From the exact sequence

(∗)  0 ⟶ det E ˇ ⟶ ⋀^{2} E ˇ ⟶ F ⟶ 0

(cf. proof of (3.5)) we get h^{1}F(n) = 0 for all n ∈ ℤ, and from

(∗∗)  0 ⟶ O_{ℙ}(−e − 4) ⟶ F ⟶ J_{C} ⟶ 0

(cf. (3.5)(2)) we then find h^{1}J_{C}(n) = 0 for all n ∈ ℤ, i.e. C is projectively normal and all conditions are satisfied.

As for the converse, by theorem (1.2) there is a locally free O_{ℙ}-module E of rank 3 with very good sections, and a global section s ∈ H^{o}E such that Y = (s)_{o}. We want to use (4.1) to show that E splits, because this implies that Y is a complete intersection. Since a_{Y} = e = c_{1}(E)−4 by assumption and (3.5), we still need h^{1}E(n) = 0 for all n ∈ ℤ. But the exact sequences (∗) and (∗∗) above yield h^{1}E(n) = h^{1}(⋀^{2} E ˇ )(n + e + 4) = h^{1}J_{C}(n + e + 4) = 0 for all n ∈ ℤ, because C is projectively normal. q.e.d.

Let us first illustrate this theorem in some very simple cases. Suppose that we are given a 0-dimensional subscheme Y of ℙ and we want to represent it as a 0-dimensional complete intersection.

Example. If Y consists of n ≥ 1 collinear reduced points, then Y has the following Hilbert function: H_{Y} (m) = m + 1 for 0 ≤ m ≤ n − 2 and H_{Y} (m) = n for m ≥ n − 1. Thus a_{Y} = n − 2 and H_{Y} is symmetrical. Hence we are looking for a splitting 3-bundle E with c_{1}(E) = a_{Y} + 4 = n + 2 and c_{3}(E) = deg Y = n. This implies that we need a projectively normal curve C through Y with deg C = c_{2}(E) = 2n + 1 and p_{a}(C) = \frac{1}{2}(deg ω_{C} + 2) = \frac{1}{2}(deg O_{C}(Y) ⊗ O_{C}(n − 2) + 2) = \frac{1}{2}(deg Y + (n − 2) deg C + 2) = n(n − 1). We see that a divisor of type (n, n + 1) on a nonsingular quadric is a good candidate for C, and in fact, it is an easy exercise to check that all conditions of the theorem can be satisfied.

Example. If Y consists of four complanar points, no three of which are collinear, then we need a projectively normal curve C with deg C = 8 and p_{a}(C) = 7 through Y. Again it is a straightforward exercise to show that a divisor of type 8l −3e_{1} −3e_{2} −3e_{3} −3e_{4} −2e_{5} −2e_{6} on a suitable nonsingular cubic surface does the job.

The bigger deg Y gets, the harder it seems to find suitable curves satisfying the conditions of the theorem. The following corollary helps us out.

Corollary 4.2. Let Y ⊆ ℙ be a 0-dimensional subscheme which has the Hilbert function of a complete intersection of type (a_{1}, a_{2}, a_{3}) and the CBP with respect to O_{ℙ}(a_{1}+a_{2}+a_{3}−4), and let I ⊆ k[W, X, Y, Z] be the homogeneous ideal of Y. Suppose 1 ≤ a_{1} ≤ a_{2} ≤ a_{3}. Then we can choose homogeneous polynomials f_{i} ∈ I_{a_{i}} (i = 1, 2, 3) such that f_{1} ≠ 0, f_{2} ∉ (f_{1}) and f_{3} ∉ (f_{1}, f_{2}). The following conditions are equivalent:

(a) Y is a complete intersection of type (a_{1}, a_{2}, a_{3}).

(b) There are homogeneous polynomials g_{i} ∈ k[W, X, Y, Z]_{a_{i}} (i = 1, 2, 3) such that the ideal of 2 × 2-minors of \binom{f_{1} f_{2} f_{3}}{g_{1} g_{2} g_{3}} defines a l.c.i. curve C, and for every triple (g_{1}, g_{2}, g_{3}) like this, Y is the zeroscheme of a global section of ω_{C}(−a_{1} − a_{2} − a_{3} + 4).

(c) There are homogeneous polynomials g_{i} ∈ k[W, X, Y, Z]_{a_{i}} (i = 1, 2, 3) such that the ideal of 2 × 2-minors of \binom{f_{1} f_{2} f_{3}}{g_{1} g_{2} g_{3}} defines a l.c.i. curve C which contains Y as the zeroscheme of a global section of ω_{C}(−a_{1} − a_{2} − a_{3} + 4).

In this case, ω_{C}(−a_{Y}) is generated by its 2-dimensional space of global sections, and Y is the zeroscheme of a unique global section of ω_{C}(−a_{Y}).

Proof: Notice that f_{1}, f_{2}, f_{3} can be chosen as stated, because we assumed that Y has the Hilbert function of a complete intersection of type (a_{1}, a_{2}, a_{3}).

“(a)⇒(b)” It follows from our assumptions that {f_{1}, f_{2}, f_{3}} is a regular sequence defining Y. Choose g_{i} ∈ k[W, X, Y, Z]_{a_{i}} (i = 1, 2, 3) such that Y ∩ V(g_{1}, g_{2}, g_{3}) = ∅. Then, locally at each point P ∈ C, one of the entries of the matrix \binom{(f_{1})_{P} (f_{2})_{P} (f_{3})_{P}}{(g_{1})_{P} (g_{2})_{P} (g_{3})_{P}} is a unit, and therefore C is a l.c.i. at P (cf. [BV], p. 10).

Now let (g_{1}, g_{2}, g_{3}) be any triple as above defining a l.c.i. curve C. Then the homogeneous ideal I_{C} = (f_{1}g_{2} − f_{2}g_{1}, f_{2}g_{3} − f_{3}g_{2}, f_{3}g_{1} − f_{1}g_{3}) of C is necessarily perfect (cf. [BV], (2.7)) and has the following minimal free resolution

(∗) 0 ⟶ R(−a_{1} − a_{2} − a_{3})^{2} ⟶^{𝔄^{tr.}} R(−a_{1} − a_{2}) ⊕ R(−a_{2} − a_{3}) ⊕ R(−a_{3} − a_{1}) ⟶ I_{C} ⟶ 0

where R = k[W, X, Y, Z] and 𝔄 = \binom{f_{3} f_{1} f_{2}}{g_{3} g_{1} g_{2}}. Let S := R/I_{C} be the projective coordinate ring of C. By dualizing (∗) we get that the canonical module ω_{S} ≅ Ext_{R}^{2}(S, R)(−4) of S/k has the following presentation

R(a_{1} + a_{2} − 4) ⊕ R(a_{2} + a_{3} − 4) ⊕ R(a_{3} + a_{1} − 4) ⟶^{𝔄} R(a_{1} + a_{2} + a_{3} − 4)^{2} ⟶ ω_{S} ⟶ 0,

i.e. ω_{S} ≅ (R^{2}/ < (f_{1}, g_{1}), (f_{2}, g_{2}), (f_{3}, g_{3}) > )(a_{1} + a_{2} + a_{3} − 4). Since ω_{C} = ω̃_{S} is the associated sheaf of O_{C}-modules, we find H^{o}ω_{C}(−a_{Y}) ≅ k^{2}. Denote the preimage of (λ, μ) ∈ k^{2} under this isomorphism by s_{(λ,μ)}. We claim that Y is the zeroscheme of s_{(0,1)}.

The homogeneous ideal of the zeroscheme of s_{(λ,μ)} is the image of the dual homomorphism

\begin{matrix}s_{(λ,μ)}ˇ : Hom_{S}(R^{2}/ < (f_{1}, g_{1}), (f_{2}, g_{2}), (f_{3}, g_{3}) > & , S) ⟶ S \\ l ⟼ l(λe_{1} + μe_{2}) \end{matrix}

where e_{1}, e_{2} are the residue classes of the standard basis vectors of R^{2}. Here l has to satisfy f_{i}l(e_{1}) + g_{i}l(e_{2}) = 0 (i = 1, 2, 3) in order to be well-defined. Obviously l_{1}(e_{1}) := −g_{1}, l_{1}(e_{2}) := f_{1}, l_{2}(e_{1}) := −g_{2}, l_{2}(e_{2}) := f_{2}, and l_{3}(e_{1}) := −g_{3}, l_{3}(e_{2}) := f_{3} specify welldefined elements of Hom_{S}(ω_{S}(−a_{Y}), S). Thus f_{1}, f_{2} and f_{3} lie in the ideal of (s_{(0,1)})_{o}, i.e. we have (s_{(0,1)})_{o} ⊆ Y. In particular, s_{(0,1)} is a regular section of ω_{C}(−a_{Y}), and using the formulas given in [PS], (3.1) we get deg(s_{(0,1)})_{o} = deg ω_{C}(−a_{Y}) = 2p_{a}(C) − 2 − (a_{1} + a_{2} + a_{3} − 4) deg C = a_{1}a_{2}a_{3} = deg Y. Hence Y = (s_{(0,1)})_{o}, as was to be shown.

“(c)⇒(a)” We want to apply theorem (1.3). By assumption and (1.1), Y is arithmetically Gorenstein and has a_{Y} = a_{1} + a_{2} + a_{3} − 4. As above we see that I_{C} is perfect. Hence C is projectively normal. We see also that H^{o}ω_{C}(−a_{Y}) ≅ k^{2} generates ω_{C}(−a_{Y}), because it even generates ω_{S}(−a_{Y}). From (1.3) and the Hilbert function of Y we then conclude that Y is a complete intersection of type (a_{1}, a_{2}, a_{3}). q.e.d.

Suppose we are given a 0-dimensional subscheme Y ⊆ ℙ which has the Hilbert function of a complete intersection of type (a_{1}, a_{2}, a_{3}) and the CBP with respect to O_{ℙ}(a_{Y}). For instance, this situation arises when we study hyperplane sections of arithmetically Cohen-Macaulay curves in ℙ^{4}. Our final examples demonstrate how we can use corollary (4.2) to prove or disprove that Y is a complete intersection of type (a_{1}, a_{2}, a_{3}).

Example. Let Y consist of the eight reduced points P_{1} = (1 : 1 : 1 : 1), P_{2} = (1 : −1 : 1 : −1), P_{3} = (1 : 2 : 4 : 8), P_{4} = (1 : −2 : 4 : −8), P_{5} = (1 : 3 : 9 : 27), P_{6} = (1 : −3 : 9 : −27), P_{7} = (1 : 4 : 16 : 64), and P_{8} = (1 : −4 : 16 : −64). We suppose that we know already that Y has the Hilbert function of a complete intersection of type (2, 2, 2) and the CBP with respect to O_{ℙ}(2). By solving one system of linear equations, we find that the space of quadrics through Y is k(X^{2} − W Y) ⊕ k(Y^{2} − XZ) ⊕ k(XY − W Z). Hence we can take f_{1} = X^{2} − W Y, f_{2} = Y^{2} − XZ, f_{3} = XY − Z, g_{1} = W^{2}, g_{2} = Y^{2} and g_{3} = Z^{2}. Then V(g_{1}, g_{2}, g_{3}) is concentrated at the point (0 : 1 : 0 : 0), which does not satisfy f_{1}, so C := V(f_{1}g_{2} − f_{2}g_{1}, f_{2}g_{3} − f_{3}g_{2}, f_{3}g_{1} − f_{1}g_{3}) is an arithmetically Cohen-Macaulay l.c.i. curve.

Now either Y = (s_{(0,1)})_{o} or (s_{(0,1)})_{o} contains an irreducible component of C. Since every hyperplane meets every irreducible component of C, we can check which is the case by looking at the hyperplane section C∩V(W) = V(W, X^{2}Y^{2}, X^{2}Z^{2}, Y^{2}Z^{2}−XZ^{3}−XY^{3}). Its support consists of the points P = (0 : 1 : 0 : 0), Q = (0 : 0 : 1 : 0) and R = (0 : 0 : 0 : 1). Since R ∈ V(f_{1}, f_{2}, f_{3}), it follows that R ∈ (s_{(0,1)})_{o} \ Y and that s_{(0,1)} is not a regular section of ω_{C}(−a_{Y}). Hence Y is not the zeroscheme of a global section of ω_{C}(−a_{Y}), and therefore not a complete intersection of type (2, 2, 2).

Example. Let Y consist of the twelve points P_{1} = (1 : 1 : 1 : 1), P_{2} = (1 : j^{2} : 1 : j), P_{3} = (1 : j : 1 : j^{2}), P_{4} = (1 : −1 : 1 : 1), P_{5} = (1 : −j^{2} : 1 : j), P_{6} = (1 : −j : 1 : j^{2}), P_{7} = (1 : i : −1 : 1), P_{8} = (1 : ij^{2} : −1 : j), P_{9} = (1 : ij : −1 : j^{2}), P_{10} = (1 : −i : −1 : 1), P_{11} = (1 : −ij^{2} : −1 : j), and P_{12} = (1 : −ij : −1 : j^{2}), where char k ≠ 2 and i, j ∈ k\{0, 1} with i^{2} = −1, j^{3} = 1. Suppose again that we know that Y has the Hilbert function of a complete intersection of type (2, 2, 3) and the CBP with respect to O_{ℙ}(3). By solving two systems of linear equations we find that we can take f_{1} = X^{2} − Y Z, f_{2} = Y^{2} − W^{2} and f_{3} = Z^{3} − W^{3}. Therefore we can use g_{1} = W^{2}, g_{2} = Y^{2} and g_{3} = Z^{3} to obtain an arithmetically Cohen-Macaulay l.c.i. curve C = V(f_{1}g_{2} − f_{2}g_{1}, f_{2}g_{3} − f_{3}g_{2}, f_{3}g_{1} − f_{1}g_{3}).

Here C ∩V(Z +W) = V(Z +W, X^{2}Y^{2} −Y^{3}Z −Y^{2}Z^{2} +Z^{4}, Y^{2}Z^{3} +Z^{5}, X^{2}Z^{3} −Y Z^{4} − 2Z^{5}) is concentrated at six points which do not lie on V(f_{1}, f_{2}, f_{3}). We conclude that s_{(0,1)} is a regular section of ω_{C}(−a_{Y}) with zeroscheme Y, and therefore Y is a complete intersection of type (2, 2, 3).

Acknowledgements. The present paper is based on part of the author’s doctoral thesis [K2], submitted at Universität Regensburg under the supervision of Prof. E. Kunz, to whose continuing interest and support I owe my deepest gratitude. I would also like to thank Queen’s University, Kingston, and in particular Prof. A.V. Geramita, for their kind hospitality during the final stages of this work, which was partially supported by an NSERC grant.

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