VECTOR BUNDLES WITH GOOD SECTIONS

Martin Kreuzer

Fakultät für Mathematik Universität Regensburg Postfach 397 D-W 8400 Regensburg, Germany

After introducing the scheme of zeros of a vector bundle which parametrizes the zeroschemes of its global sections, we give several alternative descriptions. Using results about symmetric algebras, we can characterize local properties of this scheme. If it has the expected dimension, we say that the vector bundle has good sections. Examples show that many interesting bundles share this property which generalizes global generation. Finally, this notion is investigated further in the case of vector bundles of rank 3 on ℙ^{3}.

Introduction

The purpose of this paper is to study algebraic vector bundles E on ℙ^{d} by means of their space of global sections. In section 1 we parametrize the zeroschemes of those sections by the scheme of zeros 𝕊(E) of E, which we describe in several ways: as the fiber product of the zero section and the evaluation homomorphism of E, as the zeroscheme of its universal section, and as the associated vector bundle of the cokernel M of its evaluation homomorphism. We also construct a projectivized version \overline{𝕊}(E) with analogous properties.

In section 2 we obtain a local description of 𝕊(E) as a symmetric algebra. This allows us to characterize local properties of 𝕊(E) in terms of the depths of the Fitting ideals of M by applying results from [1], [12] and [16]. In particular, we characterize global generation of E, correct dimension, integrality and factoriality of 𝕊(E).

In section 3 we show that 𝕊(E) having the expected dimension is also equivalent with “few” global sections of E having zeroschemes of “low” codimension. If this is the case, we say that E has good sections. This notion generalizes global generation of E, and we proceed to establish a “hierarchy of vector bundles”.

Finally we apply our method in the simplest nontrivial case, namely for vector bundles of rank 3 on ℙ^{3}. Here having good sections is characterized by the existence of two sections s, s′ ∈ H^{o}E, s regular, such that (s ∧ s′)_{o} is a curve. Furthermore, that curve is locally Gorenstein if and only if H^{o}E is basepointfree, and we can characterize the 0-dimensional zeroschemes of those bundles.

1. The scheme of zeros of a vector bundle

Let k be an algebraically closed field, ℙ^{d} the d-dimensional projective space over k, and E a locally free O_{ℙ}-module of rank r. By n we denote the dimension of the k-vector space H^{o}E of global sections of E, and by H := (H^{o}E)^{∗} = Hom_{k}(H^{o}E, k) its dual vector space.

For a coherent O_{ℙ}-module F let 𝕍(F) := Spec Sym(F) be the associated vector bundle of F, and let ℙ(F) := Proj Sym(F) be the associated projective space bundle of F. Recall that 𝕍(E ˇ ) is the vector bundle over ℙ^{d} whose sheaf of sections is E. Here E ˇ = Hom_{O_{ℙ}}(E, O_{ℙ}) denotes the dual sheaf of E. Also, 𝕍(H) is an n-dimensional affine space over k whose closed points correspond to the global sections of E via

𝕍(H)_{cl.pts.} ≅ Mor_{k}(Spec k, 𝕍(H)) ≅ Hom_{k-alg}(Sym_{k}(H), k)
≅ Hom_{k}(H, k) ≅ H^{o}E.

By ζ : ℙ^{d} ⟶ 𝕍(E ˇ ) we denote the zero section of E. It can be constructed by applying the functor 𝕍 to the zero morphism 0 : E ˇ ⟶ O_{ℙ}. The O_{ℙ}-homomorphism φ : H^{o}E ⊗_{k} O_{ℙ} ⟶ E given by s ⊗ f ⟼ f · s is called the evaluation homomorphism of E. Let ϱ : ℙ^{d} ⟶ Spec k be the structural morphism. Then the identifications H^{o}E ⊗_{k} O_{ℙ} ≅ ϱ^{∗}(H^{o}E) and 𝕍(H) × ℙ^{d} ≅ 𝕍(ϱ^{∗}H) (cf. [4], 9.4.11) yield a morphism

δ = 𝕍(φˇ ) : 𝕍(H) × ℙ^{d} ⟶ 𝕍(E ˇ )

of vector bundles over ℙ^{d}, which we call the evaluation morphism of E.

Definition 1.1. The fiber product 𝕊(E) of ζ and δ, given by the cartesian diagram

\begin{matrix}𝕊(E) & ⟶^{p} & ℙ^{d} \\ ↓^{ı} & □ & ↓^{ζ} \\ 𝕍(H) × ℙ^{d} & ⟶^{δ} & 𝕍(E ˇ ) \end{matrix}

is called the (affine) scheme of zeros of E.

Note that, since ζ is a closed immersion, ı is one, too, and we can regard 𝕊(E) as a closed subscheme of 𝕍(H) × ℙ^{d}. By π we denote the composition

π : 𝕊(E) ⟶^{ı} 𝕍(H) × ℙ^{d} ⟶^{pr_{1}} 𝕍(H)

of ı with the projection to the first factor.

Remark 1.2. The morphism p : 𝕊(E) ⟶ ℙ^{d} decomposes as follows:

p : 𝕊(E) ⟶^{ı} 𝕍(H) × ℙ^{d} ⟶^{pr_{2}} ℙ^{d}

Proof: Consider the diagram

\begin{matrix}𝕊(E) & ⟶^{p} & ℙ^{d} \\ ↓^{ı} & pr_{2}↗ 1 & ↓^{ζ} \\ 𝕍(H) × ℙ^{d} & ⟶^{δ} & 𝕍(E ˇ ) \end{matrix}

Since δ is a homomorphism of vector bundles over ℙ^{d}, subdiagram 1 is commutative. Hence ζ ∘ pr_{2} ∘ ı = ζ ∘ p, and pr_{2} ∘ ı = p, because ζ is a closed immersion. q.e.d.

Proposition 1.3. For every s ∈ H^{o}E we have

π^{−1}(s) = {s} × (s)_{o} ≅ (s)_{o}

where (s)_{o} is the zeroscheme of s, i.e. the closed subscheme of ℙ^{d} whose sheaf of ideals is the image of the dual homomorphism sˇ : E ˇ ⟶ O_{ℙ}.

In other words, π : 𝕊(E) ⟶ 𝕍(H) parametrizes all zeroschemes of global sections of E.

For the proof of this proposition we need the following lemma.

Lemma 1.4. Let X be a scheme and M ⟶^{f} N ⟶^{g} C ⟶ 0 an exact sequence of coherent O_{X}-modules. Then we have a cartesian diagram

\begin{matrix}𝕍(C) & ⟶^{ϱ} & X \\ ↓_{𝕍(g)} & □ & ↓^{ζ} \\ 𝕍(N) & ⟶^{𝕍(f)} & 𝕍(M) \end{matrix}

where ϱ : 𝕍(C) ⟶ X is the structural morphism and ζ : X ⟶ 𝕍(M) is the zero section.

Proof: Since 𝕍(f) ∘ 𝕍(g) = 𝕍(g ∘ f) = 0 = ζ ∘ ϱ, the diagram commutes. We have to show that 𝕍(C) has the universal property of the fiber product. Let ψ : Y ⟶ X be a morphism of schemes such that there is a commutative diagram (∗)

Y  ⟶^{ψ} X
↓  ↓^{ζ}
𝕍(N) ⟶^{𝕍(f)} 𝕍(M)

Since Y = Spec O_{Y} and X = Spec O_{X}, we get a commutative diagram of graded O_{X}-algebras

Sym M ⟶^{Sym f} Sym N
↓^{0}  ↓_{h}
O_{X}  ⟶^{ψ^{♯}}  O_{Y}

Restriction to the degree 1 component of Sym M yields a commutative diagram of O_{X}-modules

M ⟶^{f} N
↓^{0}  ↓^{h}
O_{X} ⟶^{ψ^{♯}} O_{Y}

Since h ∘ f = ψ^{♯} ∘ 0 = 0 and C is the cokernel of f, h factors into h : N ⟶^{g} C ⟶^{h′} O_{Y}. By the universal property of the symmetric algebra this extends to a factorization of the O_{X}-algebra homomorphism

h̃ : Sym N ⟶^{Sym g} Sym C ⟶ O_{Y} ,

and therefore we get a factorization Y ⟶ 𝕍(C) ⟶ 𝕍(N). Since everything is defined over X, the composition Y ⟶ 𝕍(C) ⟶^{ϱ} X coincides with ψ.

Hence we see that diagram (∗) factorizes through

𝕍(C) ⟶^{ϱ} X
↓^{𝕍(g)}  ↓^{ζ}
𝕍(N) ⟶^{𝕍(f)} 𝕍(M)

in the obvious way. q.e.d.

Proof of the proposition: Let s ∈ H^{o}E and Y := (s)_{o}. An application of the lemma to X := ℙ^{d} and the exact sequence E ˇ ⟶^{s}ˇ O_{ℙ} ⟶ O_{Y} ⟶ 0 yields a cartesian diagram

\begin{matrix}Y & ⟶ & ℙ^{d} \\ ; ↓ & □ & ↓^{ζ} \\ ℙ^{d} & ⟶^{s} & 𝕍(E ˇ ) \end{matrix}

This says that the image of (s)_{o} in 𝕍(E ˇ ) is the intersection of the section s and the zero section. Now consider the following diagram

\begin{aligned}Y & ⟶^{∼} & \{s\} × Y & ⟶ & 𝕊(E) & ⟶^{p} & ℙ^{d} \\ ; ↓ & □ & ; ↓ & 1 & ↓^{ı} & □ & ↓^{ζ} \\ ℙ^{d} & ⟶^{∼} & \{s\} × ℙ^{d} & ⟶ & 𝕍(H) × ℙ^{d} & ⟶^{δ} & 𝕍(E ˇ ) \end{aligned}

Since Y is the fiber product of the total diagram, {s} × Y makes subdiagram 1 cartesian. But π^{−1}(s) does this by definition, too, so

π^{−1}(s) = {s} × Y = {s} × (s)_{o} q.e.d.

In order to examine the scheme 𝕊(E) more closely, we need alternative descriptions. The morphism

Δ := (pr_{1}, δ) : 𝕍(H) × ℙ^{d} ⟶ 𝕍(H) × 𝕍(E ˇ )

of vector bundles over ℙ^{d} is called the universal section of E. Indeed, it is a section of the vector bundle 𝕍(H) × 𝕍(E ˇ ) ≅ 𝕍(pr_{2}^{∗}E ˇ ) over 𝕍(H) × ℙ.

Proposition 1.5. (Second description of 𝕊(E))

In 𝕍(H) × ℙ^{d} we have 𝕊(E) = (Δ)_{o}.

In particular, if Δ is a regular section of 𝕍(H) × 𝕍(E ˇ ), then 𝕊(E) is a local complete intersection in 𝕍(H) × ℙ^{d}.

Proof: Consider the diagram

\begin{matrix}𝕊(E) & ⟶^{ı} & 𝕍(H) × ℙ^{d} & ⟶^{pr_{2}} & ℙ^{d} \\ ↓^{ı} & 1 & ↓^{ξ} & □ & ↓^{ζ} \\ 𝕍(H) × ℙ^{d} & ⟶^{Δ} & 𝕍(H) × 𝕍(E ˇ ) & ⟶^{_{pr}_{2}} & 𝕍(E ˇ ) \end{matrix}

where ξ is the zero section of the vector bundle 𝕍(H)×𝕍(E ˇ ). Since pr_{2}∘ı = p and p̃r_{2} ∘Δ = δ, the scheme 𝕊(E) is the fiber product of the outer diagram by definition. Therefore also diagram 1 is cartesian, i.e. 𝕊(E) is the zeroscheme of Δ. q.e.d.

Proposition 1.6. (Third description of 𝕊(E))

Let M be the cokernel of the dual of the evaluation homomorphism, i.e.

E ˇ ⟶^{φ}ˇ H ⊗_{k} O_{ℙ} ⟶ M ⟶ 0

is exact. Then 𝕊(E) = 𝕍(M).

Proof: Using the lemma we get a cartesian diagram

\begin{matrix}𝕍(M) & ⟶ & ℙ^{d} \\ ; ↓ & □ & ; ↓ζ \\ 𝕍(H) × ℙ^{d} & ⟶^{𝕍(φˇ )} & 𝕍(E ˇ ) \end{matrix}

Since δ = 𝕍(φˇ ), this implies 𝕍(M) ≅ 𝕊(E) by the definition of 𝕊(E). q.e.d.

Corollary 1.7. If E is generated by global sections, then 𝕊(E) is a vector bundle of rank n − r over ℙ^{d}.

In particular, in this case 𝕊(E) is an integral, nonsingular scheme.

In this situation the scheme 𝕊(E) has been used already in the literature, cf. e.g. [11], 5.1 or [8], 1.4.b.

Proof: Since E is generated by global sections, the evaluation homomorphism φ is surjective, and its kernel is a locally free sheaf of rank n − r. Thus M, the dual of this kernel, is locally free of rank n − r, too. q.e.d.

Although, as we shall see, the scheme 𝕊(E) has many good properties, it has the drawback of not being a projective scheme over k. We are now going to remedy this situation by introducing a projective scheme \overline{𝕊}(E) ⊆ ℙ(H)×ℙ^{d} with analogous properties.

Definition 1.8. Let \overline{𝕊}(E) ⊆ ℙ(H) × ℙ^{d} be the closed subscheme defined by the sheaf of homogeneous ideals J of Sym (H ⊗_{k} O_{ℙ}) which is generated by the image of the morphism φˇ : E ˇ ⟶ H ⊗_{k} O_{ℙ}. We call \overline{𝕊}(E) the projective scheme of zeros of E. Remarks 1.9.

a) Since J is also the sheaf of homogeneous ideals of Sym (H ⊗_{k} O_{ℙ}) which is generated by the image of Sym (E ˇ )_{+} = ⨁_{i≥1} Sym^{i} E ˇ under

Sym φ : Sym E ˇ ⟶ Sym (H ⊗_{k} O_{ℙ}),

it follows that \overline{𝕊}(E) is precisely the closed subscheme of ℙ(H) × ℙ^{d} outside which Sym φˇ defines a morphism

Φ : ℙ(H) × ℙ^{d} \ \overline{𝕊}(E) ⟶ ℙ(E ˇ )

(cf. [5], sect. 3.5).

b) Using the definition of J it is easy to check that the closed points of \overline{𝕊}(E) are the pairs ([s], P) for which P ∈ (s)_{o}.

c) Let M be the cokernel of φˇ . Then Sym M ≅ Sym (H ⊗_{k} O_{ℙ})/J, and hence \overline{𝕊}(E) ≅ ℙ(M). Therefore 𝕊(E) ≅ 𝕍(M) is the affine cone over \overline{𝕊}(E), considered as schemes over ℙ^{d} in the sense of [5], sect. 8.3.

d) For s ∈ H^{o}E \ {0} and λ ∈ k \ {0} we have (s)_{o} = (λs)_{o}, so that (s)_{o} only depends on the equivalence class [s] ∈ ℙ(H)_{cl.pts.}. Therefore also the fibers of the morphism

\overline{π} : \overline{𝕊}(E) ⟶ ℙ(H) × ℙ^{d} ⟶^{pr_{1}} ℙ(H)

over closed points of ℙ(H) are the zero sections of the global sections of E. Note that \overline{π} is a morphism of projective schemes over k, a fact which we will exploit later.

2. Local properties of the scheme of zeros

In Proposition 1.6 we have identified 𝕊(E) as the spectrum of the symmetric algebra of the O_{ℙ}-module M given by

E ˇ ⟶^{φ}ˇ H ⊗_{k} O_{ℙ} ⟶ M ⟶ 0.

Many properties of Sym M can be characterized in terms of the depths of the sheaves of Fitting ideals of M.

Notations 2.1. Let {s_{1}, …, s_{n}} be a k-basis of H^{o}E. Let U ⊆ ℙ^{d} be an affine open subset such that E|_{U} is a free O_{U}-module. Choose an O_{U}-basis {e_{1}, …, e_{r}} of E|_{U} and write

(φ(U))(s_{i} ⊗ 1) = t_{i1}e_{1} + … + t_{ir}e_{r},

where t_{ij} ∈ O(U) for i = 1, …, n and j = 1, …, r. Then 𝔄 := (t_{ij})^{transp.} is the matrix of φˇ |_{U} : E ˇ |_{U} ⟶ H ⊗_{k} O_{U} with respect to the dual bases {e_{1}ˇ , …, e_{r}ˇ } and {s_{1}^{∗} ⊗ 1, …, s_{n}^{∗} ⊗ 1}.

Definition 2.2. For U ⊆ ℙ^{d} as above let F^{i}M(U) be the i^{th} Fitting ideal of M(U), i.e. the ideal of O(U) generated by the (n−i)×(n−i)-minors of 𝔄. Since the formation of Fitting ideals localizes, these ideals glue together to give a sheaf of ideals F^{i}M ⊆ O_{ℙ}, which we call the i^{th} Fitting ideal of M.

For i < max{0, n−r} we let F^{i}M := 0, and for i ≥ n we let F^{i}M := O_{ℙ}.

Remark 2.3. The Fitting ideals of M form an ascending chain

0 = F^{−1}M ⊆ F^{o}M ⊆ ⋯ ⊆ F^{n}M = O_{ℙ}

of ideals of O_{ℙ}. We have (Ann M)^{n} ⊆ F^{o}M ⊆ Ann M, and φˇ is injective if and only if F^{n−r}M ≠ 0.

Proposition 2.4. The following conditions are equivalent:

a) E is generically globally generated, i.e. if K denotes the (constant) sheaf of quotient fields of O_{ℙ}, then

φ ⊗ id : H^{o}E ⊗_{k} K ⟶ E ⊗ K

is surjective.

b) The morphism φˇ : E ˇ ⟶ H ⊗_{k} O_{ℙ} is injective.

c) The image of ⋀^{r}(H^{o}E) ⟶ H^{o}(⋀^{r} E) is nonzero.

In particular, in this case we have n ≥ r.

Proof: “(a)⇔(b)” Let C be the cokernel of φ. Then φ ⊗ id is surjective if and only if Supp C is a proper closed subset of ℙ^{d}. Now use

Supp C = V(Ann C) = V(F^{o}C) = V(F^{n−r}M)

and the above remark.

“(a)⇒(c)” If E|_{U} is generated by the sections s|_{U}, s ∈ H^{o}E, then there are indices i_{1}, …, i_{r} ∈ {1, …, n} such that (s_{i_{1}} ∧ … ∧ s_{i_{r}})_{P} ∉ 𝔪_{P} detE_{P} for some closed point P ∈ U.

“(c)⇒(a)” If s_{i_{1}} ∧ … ∧ s_{i_{r}} ≠ 0 for {i_{1}, …, i_{r}} ⊆ {1, …, n}, then (s_{i_{1}} ∧ … ∧ s_{i_{r}})_{P} ∉ 𝔪_{P} detE_{P} for all closed points P in some open subset U ⊆ ℙ^{d}, i.e. {s_{i_{1}}|_{U}, …, s_{i_{r}}|_{U}} generates E|_{U}. q.e.d.

Remark 2.5. (Local description of 𝕊(E))

Let U ⊆ ℙ^{d} be as above. From the presentation of M(U) as the cokernel of φˇ (U) we deduce a presentation

Sym_{O(U)} M(U) ≅ ≅ O(U)[X_{1}, …, X_{n}]/(t_{11}X_{1} + … + t_{n1}X_{n}, …, t_{1r}X_{1} + … + t_{nr}X_{n}).

Using Proposition 1.6 we see that the open subset 𝕊_{U}(E) := 𝕊(E)∩(𝕍(H)×U) of 𝕊(E) lying over U is the spectrum of Sym_{O(U)} M(U).

Thus we have found a local description of the scheme 𝕊(E) to which we can apply commutative algebra results about the symmetric algebra of a module. First we give a converse to Corollary 1.7.

Proposition 2.6. The following conditions are equivalent:

a) E is generated by global sections.

b) 𝕊(E) is a vector bundle of rank n − r over ℙ^{d}.

c) 𝕊(E) is a nonsingular scheme and E is generically globally generated.

d) F^{n−r}M = O_{ℙ}

e) The sections of the form σ_{1} ∧ … ∧ σ_{r} with σ_{i} ∈ H^{o}E generate det E.

Proof: “(a)⇒(b)” See Corollary 1.7. “(b)⇒(c)” This is clear.

“(c)⇒(b)” Let P ∈ ℙ^{d} be a closed point, let U ⊆ ℙ^{d} be an open neighbourhood of P as in 2.1, and let

S_{P} := O_{ℙ,P} [X_{1}, …, X_{n}]/((t_{11})_{P} X_{1} + … + (t_{n1})_{P} X_{n}, … …, (t_{1r})_{P} X_{1} + … + (t_{nr})_{P} X_{n}).

Then the localization of S_{P} in its unique homogeneous maximal ideal (𝔪_{P} , X_{1}, …, X_{n}) is a regular local ring by assumption, so also S_{P} is a regular ring by [15], 2.1. Since S_{P} = Sym_{O_{P,ℙ}} M_{P}, it follows from [15], 2.2 that M_{P} is a free O_{P,ℙ}-module. By Proposition 2.4, M is a locally free O_{ℙ}-module of rank n − r.

“(b)⇒(d)” Since 𝕊(E) = 𝕍(M), we see that M is locally free of rank n − r. Hence F^{n−r}M = O_{ℙ}.

“(d)⇒(a)” For U ⊆ ℙ^{d} as in 2.1 the O(U)-linear map φ(U) is surjective by assumption. Therefore the O_{U}-homomorphism O_{U}^{n} ⟶ E|_{U} (e_{i} ⟼ s_{i}) is surjective, i.e. {s_{1}, …, s_{n}} generates E.

“(d)⇔(e)” Let U ⊆ ℙ^{d} be as above. The O_{ℙ,P} -linear map φ_{P} is surjective for all closed points P ∈ U if and only if for every P ∈ U there is a subset {i_{1}, …, i_{r}} ⊆ {1, …, n} such that (s_{i_{1}} ∧ … ∧ s_{i_{r}})_{P} ∉ 𝔪_{P} detE_{P}, i.e. if and only if det E|_{U} is generated by the sections of the form (σ_{1} ∧ … ∧ σ_{r})|_{U} with σ_{i} ∈ H^{o}E. q.e.d.

Definition 2.7. For a sheaf of ideals J ⊆ O_{ℙ} we call

dpt J := inf_{x∈ℙ_{cl.pts.}} dpt (J_{x}) ∈ ℕ ∪ {∞}

the depth of J. Here dpt I stands for the length of a maximal regular sequence in a proper ideal I of a local ring R, and dpt R := ∞.

From the results of [1], [12] and [16] we derive the following propositions about local properties of 𝕊(E) relative ℙ^{d}. We say that 𝕊(E) has property P locally relative ℙ^{d}, if the affine coordinate ring of 𝕊_{U}(E) has property P for every subset U ⊆ ℙ^{d} as in 2.1.

Proposition 2.8. The following conditions are equivalent:

a) dim 𝕊(E) = d + n − r

b) Δ is a regular section of 𝕍(H) × 𝕍(E ˇ ).

c) For k = 1, …, r we have dpt (F^{n−k}M) ≥ r − k + 1.

In this case E is generically globally generated, 𝕊(E) is locally a complete intersection relative ℙ^{d} and a local complete intersection in 𝕍(H) × ℙ^{d}.

Proof: “(a)⇔(b)” This follows from Proposition 1.5.

“(a)⇔(c)” Since dim 𝕊(E) = sup_{U⊆ℙ as in 2.1} dim 𝕊_{U}(E), it suffices to show this locally relative ℙ^{d}. We can apply [16], 2.2 and 2.3. Note that (a) implies rk E = n − m in [16] and that (c) means t_{o} = 1 there. The supplementary claims follow from Remark 2.3, Proposition 2.4 and [16], 2.1.

q.e.d.

Proposition 2.9. The following conditions are equivalent:

a) 𝕊(E) is an integral scheme of dimension d + n − r.

b) 𝕊(E) is an integral scheme and E is generically globally generated.

c) For k = 1, …, r we have dpt (F^{n−k}M) ≥ r − k + 2.

Proof: “(a)⇒(b)” See Proposition 2.8.

“(b)⇒(c)” For U ⊆ ℙ^{d} as in 2.1 we are given that Sym_{O(U)} M(U) is an integral domain and that φˇ (U) is injective (cf. Remark 2.3). Now apply [12], 1.1.

“(c)⇒(a)” By Proposition 2.8 we have dim 𝕊(E) = d + n − r and E is generically globally generated. By [12], 1.1, we see that the schemes 𝕊_{U}(E) with U ⊆ ℙ^{d} as above are integral. But these schemes cover 𝕊(E) and have pairwise nonempty intersections, whence 𝕊(E) is integral, too. q.e.d.

Proposition 2.10. The following conditions are equivalent:

a) 𝕊(E) is a locally factorial scheme and E is generically globally generated.

b) 𝕊(E) is locally factorial at its vertex (considered as the cone over \overline{𝕊}(E) as in Remark 1.9) and E is generically globally generated.

c) 𝕊(E) is locally factorial relative ℙ^{d} and E is generically globally generated.

d) For k = 1, …, r we have dpt (F^{n−k}M) ≥ r − k + 3.

Proof: “(a)⇒(b)” This is clear.

“(b)⇒(d)” Let (0, P) be a closed point of the vertex, and let U ⊆ ℙ^{d} be an open neighbourhood of P as in 2.1. Then O_{𝕊(E),(0,P)} is the localization of

S_{P} := O_{ℙ,P} [X_{1}, …, X_{n}]/((t_{11})_{P} X_{1} + … + (t_{n1})_{P} X_{n}, … …, (t_{1r})_{P} X_{1} + … + (t_{nr})_{P} X_{n})

in its homogeneous maximal ideal (𝔪_{ℙ,P} , X_{1}, …, X_{n}). As in [7], sect. 10 we find that S_{P} is a factorial domain, too, and then [1], prop. 6 shows dpt (F^{n−k}M) ≥ r − k + 3 for k = 1, …, r. Note the different enumeration of Fitting ideals in [1].

“(d)⇒(c)” This follows from Remark 2.3, Proposition 2.4 and [1], prop. 6. “(c)⇒(a)” This is clear. q.e.d.

3. Properties of vector bundles with good sections

Definition 3.1. For i = 0, …, r the set

S_{i} := {s ∈ H^{o}E : dim (s)_{o} ≥ d − r + i}

is called the set of i times degenerated sections of E.

Remark 3.2. We have a decreasing chain

H^{o}E ⊇ S_{o} ⋯ ⊇ S_{r} = {0}.

Here H^{o}E \S_{o} is the set of global sections of E with empty zeroscheme (we use dim ∅ = −∞) and S_{o}\S_{1} is the set of regular sections of E. Since (s)_{o} = (λs)_{o} for s ∈ H^{o}E, λ ∈ k \ {0}, the sets S_{i} are conical subsets of the set of closed points of 𝕍(H).

Proposition 3.3. For i = 0, …, r the set S_{i} is the set of closed points of a closed subset of 𝕍(H), which we denote again by S_{i}. More precisely, for i = 0, …, r − 1 the set S_{i} is the affine cone over a closed subset \overline{S}_{i} of ℙ(H).

Proof: We have noted already that S_{i} is the cone over a subset \overline{S}_{i} of the set of closed points of ℙ(H) for i = 1, …, r − 1. Consider the morphism \overline{π} : \overline{𝕊}(E) ⟶ ℙ(H) constructed in Remark 1.9. By Chevalley’s semicontinuity theorem (cf. [6], 13.1.5) we find that

\overline{S}_{i} = \{[s] ∈ ℙ(H) : dim \overline{π}^{−1}([s]) ≥ d − r + i\}

is a closed subset of ℙ(H). q.e.d.

Definition 3.4. We say that E has good sections, if S_{o} = H^{o}E and for i = 0, …, r − 1 we have codim (S_{i}, 𝕍(H)) ≥ i.

Remark 3.5. Let E have good sections.

a) E does not have a global section with an empty zeroscheme.

b) We have n ≥ r − 1. The condition codim (S_{r}, 𝕍(H)) ≥ r is equivalent with n ≥ r.

c) The generic global section of E is regular.

d) We have d ≥ r, since E has a regular global section.

Proposition 3.6. The following conditions are equivalent:

a) E has good sections and n ≥ r.

b) dim 𝕊(E) = d + n − r and E has a regular global section.

Proof: “(a)⇒(b)” Since 𝕍(H) = ⋃_{i=0}^{r−1}(S_{i} \ S_{i+1}) ∪ {0} and dim π^{−1}(s) = d − r + i for s ∈ S_{i} \ S_{i+1} as well as dim (S_{i} \ S_{i+1}) ≤ dim S_{i} ≤ n − i, it follows that dim 𝕊(E) ≤ max_{i=0,...,r−1}(d − r + i) + (n − i) = d + n − r. The other inequality is obvious in view of Proposition 1.5.

“(b)⇒(a)” Suppose codim (S_{i}, 𝕍(H)) ≤ i − 1 for some i ∈ {1, …, r − 1}, i.e. suppose dim S_{i} ≥ n − i + 1. For s ∈ S_{i} we have dim π^{−1}(s) = dim (s)_{o} ≥ d − r + i, so 𝕊(E) contains a closed subset of dimension at least (n − i + 1) + (d − r + i) = d + n − r + 1, a contradiction.

Suppose S_{o} is a proper closed subset of 𝕍(H). Then dim S_{o} < n and the nonempty open subset π^{−1}(𝕍(H)\S_{1}) of 𝕊(E) has dimension < d+n−r. But by Proposition 1.5 and our assumption, 𝕊(E) is equidimensional of dimension d + n − r, a contradiction again. Hence S_{o} = H^{o}E. q.e.d.

Proposition 3.7. (The hierarchy of vector bundles)

Suppose that E has a regular global section. Then we have the following implications:

E is generated by global sections ⇔ 𝕊(E) is a vector bundle of rank n − r over ℙ^{d} ⇔ 𝕊(E) is nonsingular and E is generically globally generated ⇒ 𝕊(E) is locally factorial and E is generically globally generated ⇒ 𝕊(E) is integral and E is generically globally generated ⇒ dim 𝕊(E) = d + n − r ⇔ E has good sections and n ≥ r.

Proof: Combine the results of section 2 and the previous proposition. q.e.d.

Examples 3.8. (Vector bundles with good sections)

a) A line bundle E has good sections if and only if E ≅ O_{ℙ}(n), n > 0.

b) A rank-2-bundle E has good sections if and only if the generic section of E is regular.

c) A splitting bundle E ≅ O_{ℙ}(n_{1})⊕⋯⊕O_{ℙ}(n_{r}) has good sections if and only if r ≤ d and n_{1}, …, n_{r} > 0.

d) If E is globally generated, then E has good sections.

e) Let E be the first twist of the Horrocks-Mumford bundle of rank 3 on ℙ^{5} which has a nontrivial global section. Then E has good sections, but is not generated by global sections (cf. [3]).

f) Many examples of rank-3-bundles on ℙ^{3} with good sections can be constructed using the methods of section 4.

4. Applications to vector bundles of rank 3 on ℙ^{3}

In this section let E be a locally free O_{ℙ}-module of rank 3 on ℙ := ℙ^{3}. We want to describe in yet another way what it means for E to have good sections. For this we need some preparations.

Lemma 4.1. Let s ∈ H^{o}E be a regular section, Y := (s)_{o} its zeroscheme and F := ker sˇ . In other words, we have an exact sequence

0 ⟶ F ⟶^{i} E ˇ ⟶^{s}ˇ J_{Y} ⟶ 0

where J_{Y} is the ideal sheaf of Y. Then the following is true:

a) 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}.

b) If s′ ∈ H^{o}E is another global section of E and t := H^{o}(iˇ )(s′) ∈ H^{o}F ˇ , then we have (t)_{o} = (s ∧ s′)_{o}.

Proof: “(a)” Using codim (Y, ℙ) = 3 and the cohomological characterization of depth we get Ext_{O_{ℙ}}^{1}(J_{Y} , O_{ℙ}) = 0. Therefore the dual of the above sequence is the exact sequence

(∗)  0 ⟶ O_{ℙ} ⟶^{s} E ⟶^{i}ˇ F ˇ ⟶ 0,

and dualizing once more yields an exact sequence

0 ⟶ F ˇ ˇ ⟶^{iˇ} ˇ E ˇ ⟶^{s}ˇ O_{ℙ} ⟶ Ext_{O_{ℙ}}^{1}(F ˇ , O_{ℙ}) ⟶ 0.

Thus F ≅ F ˇ ˇ and Ext_{O_{ℙ}}^{1}(F ˇ , O_{ℙ}) ≅ O_{Y} as claimed.

“(b)” The Koszul complex of s gives us an exact sequence

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

Splitting off (∗) from this sequence yields an injection F ˇ ⟶^{ı} ⋀^{2} E such that ∧_{s} = ı ∘ iˇ . Therefore H^{o}(ı)(t) = s ∧ s′ and by dualizing ı we see tˇ ∘ ıˇ = (s ∧ s′)ˇ . Since also the Koszul complex

0 ⟶ det E ˇ ⟶ ^{⋀2}E ˇ ⟶ E ˇ ⟶^{s}ˇ J_{Y} ⟶ 0

is exact, and since F is the kernel of sˇ , we find that ıˇ is surjective. Thus im tˇ = im (s ∧ s′)ˇ , i.e. (t)_{o} = (s ∧ s′)_{o}. q.e.d.

Lemma 4.2. If s, s′, s′′ ∈ H^{o}E are k-linearly independent sections, and if dim(s ∧ s′)_{o} ≤ 1, then s ∧ s′ ∧ s′′ ≠ 0 in H^{o}(det E).

Proof: Suppose s ∧ s′ ∧ s′′ = 0. Let η be the generic point of ℙ. Then (s ∧ s′ ∧ s′′)_{η} = 0 implies that we can find functions φ, φ′φ′′ ∈ O_{ℙ,η} such that φs_{η} + φ′s′_{η} + φ′′s′′_{η} = 0. Since (s ∧ s′)_{η} ≠ 0, this is only possible if φ′′ ≠ 0. Now find an affine open subset U = D_{+}(F) ⊆ ℙ where F is a homogeneous polynomial of degree δ, and find functions f, f′, f′′ ∈ O_{ℙ}(U) with φ = f_{η}, φ′ = f′_{η} and φ′′ = f′′_{η} such that f(s|_{U}) + f′(s′|_{U}) + f′′(s′′|_{U}) = 0. Then there is an N ∈ ℕ such that F^{N}f, F^{N}f′, F^{N}f′′ extend to global sections g, g′, g′′ of O_{ℙ}(Nδ). From (gs + g′s′ + g′′s′′)|_{U} = 0 it follows that gs + g′s′ + g′′s′′ = 0 in H^{o}E(Nδ).

W.l.o.g. we can assume that g, g′, g′′ do not have a common factor, and that deg g′′ = Nδ > 0. On the hypersurface H = V(g′′) we have g|_{H} · s|_{H} + g′|_{H} · s′|_{H} = 0, and hence (g|_{H})(s ∧ s′)|_{H} = 0 as well as (g′|_{H})(s ∧ s′)|_{H} = 0. Since g, g′, g′′ do not have a common factor, there is a nonempty open subset of H where g or g′ is nonzero, i.e. which is contained in (s ∧ s′)_{o}. This contradicts the assumption dim (s ∧ s′)_{o} ≤ 1. q.e.d.

Proposition 4.3. The following conditions are equivalent:

a) E has good sections.

b) We have dim 𝕊(E) = max{n, 3}, n ≥ 2 and E has a regular global section.

c) There is a regular section s ∈ H^{o}E and another global section s′ ∈ H^{o}E such that dim (s ∧ s′)_{o} = 1.

d) There is a regular section s ∈ H^{o}E and a nonempty open subset V ⊆ 𝕍(H) such that for every s′ ∈ V_{cl.pts.} we have dim (s ∧ s′)_{o} = 1.

Proof: “(a)⇒(d)” Choose a regular global section s ∈ H^{o}E. Since we have codim (S_{2}, 𝕍(H)) ≥ 2, there is a nonempty open subset V ⊆ 𝕍(H) such that the line {s′ + λ · s : λ ∈ k} does not meet S_{2} for every s′ ∈ V_{cl.pts.}. In particular, s and s′ are k-linearly independent and dim (s ∧ s′)_{o} ≤ 2.

Suppose there is a surface H = V(f) contained in (s ∧ s′)_{o}, where f is an irreducible homogeneous polynomial of degree m. This means that the section s ∧ s′ of ⋀^{2} E is in the image of the multiplication map μ_{f} : H^{o}(⋀^{2} E)(−m) ⟶ H^{o}(⋀^{2} E). As in Lemma 4.1 we get a commutative diagram with exact rows

0 ⟶ F ˇ (−m) ⟶^{ı} (^{⋀2}E)(−m) ⟶^{∧_{s}} (det E)(−m)
↓^{μ_{f}}  ↓^{μ_{f}}  ↓^{μ_{f}}
0 ⟶  F ˇ  ⟶^{ı}  ^{⋀2}E  ⟶^{∧_{s}}  det E
t  ⟼  s ∧ s′

which shows that t is in the image of the map μ_{f} : H^{o}F ˇ (−m) ⟶ H^{o}F ˇ . Now we construct the following commutative diagram with exact rows and columns

0 ⟶ O_{ℙ}(−m) ⟶^{s} E(−m) ⟶^{i}ˇ F ˇ (−m) ⟶ 0
↓^{μ_{f}}  ↓^{μ_{f}}  ↓^{μ_{f}}
0 ⟶  O_{ℙ}  ⟶^{s}  E  ⟶^{i}ˇ  F ˇ  ⟶ 0
↓ ; ↓  ↓ ; ↓  ↓ ; ↓
0 ⟶ K ⟶  O_{H}  ⟶^{s|_{H}} E|_{H} ⟶ F ˇ |_{H} ⟶ 0

where K := ker (s|_{H}). For P ∉ (s)_{o} we use Lemma 4.1 to find that K_{P} ≅ Tor_{O_{ℙ,P}}^{1} (F_{P̌} , O_{H,P}) ≅ Tor_{O_{ℙ,P}}^{1} (O_{ℙ,P}^{2} , O_{H,P}) = 0 and H ⊆ (s ∧ s′)_{o} = (t)_{o}. Therefore the image of t in H^{o}(F ˇ |_{H}) vanishes. Since t|_{H} = (iˇ |_{H})(s′|_{H}), this implies that s′|_{H} ∈ H^{o}(s|_{H} · O_{H}). But Supp K ⊆ (s)_{o} is 0-dimensional, so s|_{H} : H^{o}O_{H} ⟶ H^{o}(s|_{H} ·O_{H}) is surjective, and hence s′|_{H} = λ·s|_{H} for some λ ∈ k. But then s′ −λs ∈ im μ_{f} in contradiction to {s′ +λs : λ ∈ k}∩S_{2} = ∅.

Altogether this shows that dim (s ∧ s′)_{o} ≤ 1 for s′ ∈ V_{cl.pts.}. In this case (s ∧ s′)_{o} is a determinantal curve, which is locally Cohen-Macaulay and generically locally Gorenstein.

“(d)⇒(c)” This is clear.

“(c)⇒(b)” For n ≥ 3 we can show dim 𝕊(E) = n using Proposition 2.8. Regularity of s implies dpt F^{n−1}M ≥ 3 and dim (s ∧ s′)_{o} = 1 implies dpt F^{n−2}M ≥ 2. If we choose s′′ ∈ H^{o}E such that {s, s′, s′′} is k-linearly independent, then s ∧ s′ ∧ s′′ ≠ 0 by (4.2), and therefore dpt F^{n−3}M ≥ 1. Now Proposition 2.8 proves the claim.

In the case n = 2 we still have dpt F^{n−1}M ≥ 3 and dpt F^{n−2}M ≥ 2, so that [16], 2.2 shows dim 𝕊(E) = 3 again. Actually the only component of dimension 3 of 𝕊(E) is {0} × ℙ.

“(b)⇒(a)” For n ≥ 3 this follows from Proposition 3.6. For n = 2 we have codim (S_{1}, 𝕍(H)) ≥ 1, because E has a regular section. Let us show S_{2} = {0}. If s ∈ S_{2}, then (s)_{o} contains a hypersurface V(f), where f is a homogeneous polynomial of degree m ≥ 1. Thus s = f · s̃ with s̃ ∈ H^{o}E(−m). But h^{o}E(−m) ≤ ⋯ ≤ h^{o}E(−1) and h^{o}E(−1) ≥ 1 implies h^{o}E ≥ 4, a contradiction.

Finally we need to show S_{o} = H^{o}E. Let s be a regular section of E. Suppose there is another section s′ ∈ H^{o}E with an empty zeroscheme. Using Lemma 4.1 we construct an exact sequence

0 ⟶ O_{ℙ} ⟶^{s} E ⟶ F ˇ ⟶ 0

with a reflexive O_{ℙ}-module F of rank 2 such that Ext_{O_{ℙ}}^{1}(F ˇ , O_{ℙ}) ≅ O_{(s)_{o}}, and a global section t ∈ H^{o}F ˇ such that (t)_{o} = (s ∧ s′)_{o}. In the same way, starting with s′ we can also construct an exact sequence

0 ⟶ O_{ℙ} ⟶^{s′} E ⟶ (F′)ˇ ⟶ 0

with a locally free O_{ℙ}-module F′ of rank 2, and a global section t′ ∈ H^{o}(F′)ˇ such that (t′)_{o} = (s ∧ s′)_{o}. Since H^{0}(F′)ˇ (−1) = 0, the scheme (s ∧ s′)_{o} is a local complete intersection curve C with ω_{C} ≅ O_{C}(4 − c_{1}(E)). Now [9], 4.1 shows F ≅ F′, in contradiction to c_{3}(F) = deg (s)_{o} ≠ 0 = c_{3}(F′). q.e.d.

Definition 4.4. We say that E has very good sections, if E has good sections and if H^{o}E is basepointfree, i.e. for every closed point P ∈ ℙ there is a section s ∈ H^{o}E such that s_{P} ∉ 𝔪_{P} E_{P}.

Remark 4.5. The second condition is equivalent with dpt F^{n−1}M = ∞, since we are requiring for every P ∈ ℙ that not all entries of the matrix 𝔄_{P} lie in the maximal ideal 𝔪_{P} of O_{ℙ,P} (cf. sect. 2).

Proposition 4.6. The following conditions are equivalent:

a) E has very good sections.

b) There is a regular section s ∈ H^{o}E and another section s′ ∈ H^{o}E such that (s ∧ s′)_{o} is a locally Gorenstein curve.

c) There is a regular section s ∈ H^{o}E and a nonempty open subset V ⊆ 𝕍(H) such that for all s′ ∈ V_{cl.pts.} the scheme (s ∧ s′)_{o} is a locally Gorenstein curve.

Proof: “(a)⇒(c)” By the previous proposition there is a regular section s ∈ H^{o}E and a nonempty open subset Ṽ ⊆ 𝕍(H) such that (s ∧ s′)_{o} is a locally Cohen-Macaulay curve for every s′ ∈ Ṽ_{cl.pts.}. For a closed point P ∈ (s)_{o} the vector space {s̃ ∈ H^{o}E : s̃_{P} ∉ 𝔪_{P} E_{P} } is a proper subspace of H^{o}E by assumption. Therefore there is a nonempty open subset V ⊆ Ṽ such that no section s′ ∈ V_{cl.pts.} is in any of these finitely many subspaces. Then (s)_{o} ∩(s′)_{o} = ∅, and the ideal defining (s∧s′)_{o} is a local complete intersection ideal, because one of the local coordinate functions of s or s′ at P is always a unit of O_{ℙ,P} (cf. [2], page 10).

“(c)⇒(b)” This is clear.

“(b)⇒(a)” By the previous proposition E has good sections. Suppose there is a closed point P ∈ (s)_{o} ∩ (s′)_{o}. Let {e_{1}, e_{2}, e_{3}} be a basis of E_{P} and write s_{P} = t_{1}e_{1} + t_{2}e_{2} + t_{3}e_{3} as well as s′_{P} = t′_{1}e_{1} + t′_{2}e_{2} + t′_{3}e_{3} with t_{i}, t′_{i} ∈ O_{ℙ,P}. By assumption the ideal I := (t_{1}t′_{2} −t_{2}t′_{1}, t_{2}t′_{3} −t_{3}t′_{2}, t_{3}t′_{1} −t_{1}t′_{3}) is a complete intersection ideal of height 2 in O_{ℙ,P}, so that we can assume that I = (t_{1}t′_{2} − t_{2}t′_{1}, t_{2}t′_{3} − t_{3}t′_{2}) because of Nakayama’s Lemma. Since I ⊆ (t_{2}, t′_{2}), also {t_{2}, t′_{2}} is a regular sequence in O_{ℙ,P} and [10], 3.3 tells us that the sequence

0 ⟶ O_{ℙ,P}^{2} \underrightarrow{({t_{3} \atop t′_{3}} {t_{1} \atop t′_{1}} {t_{2} \atop t′_{2}} )^{transp.}} O_{ℙ,P}^{3} ⟶^{(a_{1},a_{2},Δ)} O_{ℙ,P} ⟶ O_{ℙ,P} /I ⟶ 0

with a_{1} = t_{1}t′_{2} − t_{2}t′_{1}, a_{2} = t_{2}t′_{3} − t_{3}t′_{2} and Δ = t_{3}t′_{1} − t_{1}t′_{3} is exact. Now all coefficients in this sequence lie in 𝔪_{P}, hence μ(I) = 3, a contradiction.

Thus (s)_{o} ∩ (s′)_{o} = ∅, and for every closed point P ∈ ℙ we have either s_{P} ∉ 𝔪_{P} E_{P} or s′_{P} ∉ 𝔪_{P} E_{P}. q.e.d.

In [13] this characterization of rank-3-bundles on ℙ^{3} with very good sections was used to characterize their 0-dimensional zeroschemes as follows.

Proposition 4.7. For a 0-dimensional subscheme Y ⊆ ℙ the following conditions are equivalent:

a) Y is the zeroscheme of a regular global section of a rank-3-bundle E with very good sections.

b) There is a locally Gorenstein curve C containing Y as a Cartier divisor such that the complete linear system |Y | is subcanonical and basepointfree.

Furthermore, this characterization can be made into a characterization of 0-dimensional complete intersections in ℙ^{3} by applying a suitable splitting criterion to E (cf. [13]).

ACKNOWLEDGMENTS This paper is based on appendix C of the author’s thesis [14] under the

supervision of Prof. E. Kunz, whose patience and guidance have been an invaluable source of encouragement and inspiration for me. The final stages of this work profited also from the kind hospitality of Queen’s University, Kingston, in particular of Prof. A.V. Geramita, and were partially supported by an NSERC grant.

REFERENCES

[1] L.L. Avramov, Complete intersections and symmetric algebras, J. Algebra 73 (1981), 248–263.

[2] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math. 1327, Springer, Berlin-Heidelberg-New York 1988.

[3] W. Decker and F.O. Schreyer, Sections of the Horrocks bundle on ℙ^{5}, preprint 1990.

[4] A. Grothendieck, Eléments de géométrie algébrique I, Springer, Heidelberg 1971.

[5] A. Grothendieck, Eléments de géométrie algébrique II, Publ. Math. IHES 8 (1961).

[6] A. Grothendieck, Eléments de géométrie algébrique IV, troisième partie, Publ. Math. IHES 28 (1966).

[7] R.M. Fossum, The divisor class group of a Krull domain, Springer, Berlin-Heidelberg-New York 1973.

[8] R. Hartshorne, Stable vector bundles of rank 2 on ℙ^{3}, Math. Ann. 238 (1978), 229–280.

[9] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121– 176.

[10] J. Herzog, Certain complexes associated to a sequence and a matrix, Manus. Math. 12 (1974), 217–248.

[11] G. Horrocks and D. Mumford, A rank 2 vector bundle on ℙ^{4} with 15000 symmetries, Topology 12 (1973), 63–81.

[12] C. Huneke, On the symmetric algebra of a module, J. Algebra 69 (1981), 113–119.

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

[14] M. Kreuzer, Vektorbündel und der Satz von Cayley-Bacharach, Dissertation, Regensburger Mathematische Schriften 21, Universität Regensburg 1989.

[15] J. Matijevic, Three local conditions on a graded ring, Trans. Am. Math. Soc. 205 (1975), 275–284.

[16] A. Simis and W.V. Vasconcelos, On the dimension and integrality of symmetric algebras, Math. Zeitschrift 177 (1981), 341–358.