On generation of jets for vector bundles

Mauro C. Beltrametti, Sandra Di Rocco, and Andrew J. Sommese

Abstract We introduce and study the k-jet ampleness and the k-jet spannedness for a vector bundle, E, on a projective manifold. We obtain different characterizations of projective space in terms of such positivity properties for E. We compare the 1-jet ampleness with different notions of very ampleness in the literature. 1991 Mathematics Subject Classification. Primary 14F05, 14E25; Secondary 14C20. Keywords and phrases. Vector bundles, k-jet ampleness, k-jet spannedness, very ampleness, k-th order embeddings, adjunction theory.

Introduction

Let E be a rank r vector bundle on an n-dimensional projective manifold X. If r = 1, there is a well understood and accepted notion of very ampleness. If E is a vector bundle there are a number of notions of very ampleness that turn out to be different. Restricting to notions with good properties, e.g., direct sums of very ample bundles are very ample, we have two main notions. One definition is that E is very ample if the tautological bundle ξ_{E} on ℙ(E) is very ample, and the second stronger definition is based on the existence of enough sections to separate 1-jets.

In §1, we study these notions and define k-jet ampleness of E, the natural generalization of the second notion to the case when global sections separate k-jets (the first notion does not generalize to the case of k-jets with k > 1).

In §2, we study the behavior of k-jet ampleness under direct sums, tensor products, and blowing up of finite sets. We also point out the analogue of the Chern class inequalities of [3] in the case of k-jet ample vector bundles.

In §3, we give lower bounds on the number of sections and det(E)^{n} for a kjet vector bundle E, and give characterizations of (X, E) when the lower bounds are taken on. We discuss the behavior of k-jet ampleness under adjunction and make some conjectures of what should optimally be true.

In §4 we compare the different definitions used in the preceding sections. We would like to thank the Max-Planck-Institute für Mathematik for its

support. Part of this paper was prepared during short visits of the second author at the University of Genova and the University of Notre Dame, which she thanks for their hospitality. The third author would also like to thank the Alexander von Humboldt Stiftung for its support.

1 Background material

Throughout this paper we deal with complex projective manifolds X. We denote by O_{X} the structure sheaf of X and by K_{X} the canonical bundle. By a vector bundle we mean a locally free sheaf (of O_{X}-modules) of finite rank.

1.1 Notation. In this paper, we use the standard notation from algebraic geometry. Let us only fix the following.

h^{i}(F), the complex dimension of H^{i}(X, F), for any coherent sheaf F on X;

≈ denotes linear equivalence of line bundles;

Γ(E) = H^{0}(E), the space of the global sections of a vector bundle E on X. We say that E is spanned if it is spanned at all points of X by Γ(E);

|V |, the complete linear system associated with a subvector space, V ⊆ Γ(E), for a vector bundle E on X.

If J is an ideal sheaf of X and E is a vector bundle on X, we write E/J for E ⊗ (O_{X}/J).

Line bundles and divisors are used with little (or no) distinction. Hence we freely use the additive notation.

1.2 k-th order embeddings. Let X be a smooth algebraic variety. Let E be a rank r vector bundle on X. For each point x on X let 𝔪_{x} be the maximal ideal sheaf of x in X, i.e., the stalk of 𝔪_{x} at a point y ≠ x is O_{X,y} and at x is the maximal ideal 𝔪_{x}O_{X,x} ⊂ O_{X,x}. Let V ⊆ Γ(E) be a subvector space of Γ(E).

We say that E is k-jet spanned at x with respect to V if V gives global sections with arbitrarily prescribed k-jets at x, i.e., if the evaluation map

X × V → Γ(E ⊗ (O_{X}/𝔪_{x}^{k+1}))

is surjective. We say that E is k-jet spanned with respect to V if E is k-jet spanned at x with respect to V for each point x ∈ X.

We say that E is k-jet spanned at x (respectively E is k-jet spanned) if V = Γ(E) in the above definitions.

Let x_{1}, …, x_{t} be t distinct points of X. Let 𝔪_{i} be the maximal ideal sheaves of the points x_{i} ∈ X, i = 1, …, t. Consider the 0-cycle Z = x_{1} + ⋯ + x_{t}. We say that E is k-jet ample at Z with respect to V if for every t-ple (k_{1}, …, k_{t}) of positive integers such that ∑_{i=1}^{t} k_{i} = k + 1, the evaluation map

X × V → Γ(E ⊗ (O_{X}/ ⊗_{i=1}^{t} 𝔪_{i}^{k_{i}}))(≅ ⊕_{i=1}^{t}Γ(E ⊗ (O_{X}/𝔪_{i}^{k_{i}}))

is onto. Here 𝔪_{i}^{k_{i}} denotes the k_{i}-th tensor power of 𝔪_{i}. We say that E is k-jet ample with respect to V if, for any t ≥ 1 and any 0-cycle Z = x_{1} + ⋯ + x_{t}, where x_{1}, …, x_{t} are t distinct points on X, the vector bundle E is k-jet ample at Z with respect to V.

We say that E is k-jet ample at Z (respectively E is k-jet ample) if V = Γ(E) in the above definitions.

Hence in particular E k-jet ample implies that E is k-jet spanned.

Note that E is 0-jet ample if and only if E is 0-jet spanned, if and only if E is spanned by its global sections. Moreover, for r = 1, i.e., E = L is a line bundle, L is 1-jet ample if and only if L is very ample (see §4 for the case k = 1).

Note also that for a line bundle L, 1-jet spannedness with respect to V is equivalent to V spanning L and the map given by |V | being an immersion.

We refer to [1] and [4] for more on k-jet ampleness in the case of line bundles. We also refer to [9] for results in the case of surfaces.

1.3 The k-th jet bundle. Let X be a smooth algebraic variety of dimension n and let E be a rank r vector bundle on X. By the k-th jet bundle of E, denoted J_{k}(X, E) or J_{k}(E) when no confusion will occur, we mean the vector bundle of rank r\binom{k+n}{n} associated to the sheaf p^{∗}E/(p^{∗} ⊗ J_{Δ}^{k+1}), where p : X × X → X is the projection on the first factor, the tensor product is with respect to O_{X×X} and J_{Δ} is the sheaf of ideals of the diagonal, Δ, of X × X. Note that J_{k}(E) = ∪_{x∈X}J_{k}(E)_{x}, where the fiber over each point x ∈ X is J_{k}(E)_{x} ≅ E_{x}/𝔪_{x}^{k+1}. Note also that J_{0}(E) = E. Moreover there is a natural map j_{k} := j_{k}^{E} : E → J_{k}(E), defined on the sheaf level and which is not a bundle map. It sends a germ of a section to its k-jet.

Interpreting J_{k}(E) as the bundle of k-jets of E, i.e., Taylor expansions of holomorphic sections of E truncated after the k-th term, one has an exact sequence

0 → T_{X}^{∗}^{(k)} ⊗ E → J_{k}(E) → J_{k−1}(E) → 0,  (1)

where T_{X}^{∗}^{(k)} := S^{k}(T_{X}^{∗}) denotes the k-th symmetric power of the cotangent bundle T_{X}^{∗}. In particular there is a surjective natural map J_{k}(E) → J_{k′}(E) → 0 for k′ ≤ k. Then it follows from (1) that

• If J_{k}(E) is spanned by its sections for some k, then J_{k′}(E) is spanned by its sections when k′ ≤ k.

Note that j_{k} gives rise to a natural evaluation map, which we denote by the same symbol, j_{k} : X × Γ(E) → J_{k}(E). This map takes (x, s) ∈ X × Γ(E) to the k-th jet j_{k}(s(x)) ∈ J_{k}(E). Since for each point x ∈ X there is a canonical isomorphism E ⊗ (O_{X}/𝔪_{x}^{k+1}) ≅ J_{k}(E)_{x}, it follows that

• E is k-jet spanned if and only if the evaluation map j_{k} is surjective.

We have the following general result.

Lemma 1.4 Let E be a vector bundle on a smooth projective variety X. Let p : ℙ(E) → X be the bundle projection and let ξ_{E} be the tautological line bundle of ℙ(E). Then J_{1}(E) ≅ p_{∗}J_{1}(ξ_{E}).

Proof. One has a natural surjective map p^{∗}E → ξ_{E} → 0 which induces a surjective map α : J_{1}(p^{∗}E) → J_{1}(ξ_{E}) → 0. We also have a canonical exact sequence

0 → p^{∗}J_{1}(E) → J_{1}(p^{∗}E) → T_{ℙ(E)/X}^{∗} ⊗ p^{∗}E → 0,

where T_{ℙ(E)/X}^{∗} denotes the relative cotangent bundle. Applying p_{∗} to this sequence and noting that p_{∗}(T_{ℙ(E)/X}^{∗}) is the zero sheaf we get J_{1}(E) ≅ p_{∗}J_{1}(p^{∗}E). Thus, it suffices to show that

p_{∗}J_{1}(ξ_{E}) ≅ p_{∗}J_{1}(p^{∗}E).  (2)

To this purpose consider the commutative diagram with exact rows and columns

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & T_{ℙ}^{∗}(E) ⊗ T_{ℙ(E)/X}^{∗} ⊗ ξ_{E} \ar[d] \ar[r] & T_{ℙ(E)}^{∗} ⊗ p^{∗}E \ar[d] \ar[r] & T_{ℙ(E)}^{∗} ⊗ ξ_{E} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K = ker(α) \ar[d] \ar[r] & J_{1}(p^{∗}E) \ar[d] \ar[r] & J_{1}(ξ_{E}) \ar[r] \ar[d] & 0. \\ 0 \ar[r] & T_{ℙ(E)/X}^{∗} ⊗ ξ_{E} \ar[d] \ar[r] & p^{∗}E \ar[d] \ar[r] & ξ_{E} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

(3)

Since the higher derived functors p_{(i)}(T_{ℙ(E)/X}^{∗} ⊗ ξ_{E}) are zero for all i ≥ 0, we conclude that there is an injection 0 → p_{(1)}K → p_{(1)}J_{1}(p^{∗}E) if and only if there is an injection

0 → p_{(1)}(T_{ℙ(E)}^{∗} ⊗ p^{∗}E) → p_{(1)}J_{1}(p^{∗}E).  (4)

To show (4), consider the diagram with an exact row and column

\xymatrix{ &  &  & 0 \ar[d] \\  &  &  & T_{ℙ(E)/X}^{∗} ⊗ E \ar[d] \\ 0 \ar[r] & p^{∗}T_{X} ⊗ E \ar[r] & J_{1}(E) \ar[r] & J_{1/X}(E) \ar[r] \ar[d] & 0, \\  &  &  & E \ar[d] \\  &  &  & 0}

where J_{1/X}(E) denotes the relative 1-jet bundle (defined by the exact sequence above). Since p_{(1)}(p^{∗}T_{X}^{∗} ⊗ E) = 0, to get a surjection

π_{∗}J_{1}(E) → p_{∗}E → 0,  (5)

it suffices to have a surjection p_{∗}J_{1/X}(E) → p_{∗}E → 0. This last surjection follows from the decomposition, which comes from the p^{−1}O_{X} (the topological inverse image of O_{X}) splitting given by j_{1}, p_{(i)}J_{1/X}(E) ≅ p_{(i)}(E) ⊕ p_{(i)}(T_{ℙ(E)/X}^{∗} ⊗ E), i ≥ 0. By looking at diagram (3), we see that surjection (5) implies (4), which in turn gives the desired isomorphism (2). Q.E.D.

For more general properties of jet bundles we refer to [8] and [13].

2 Some general properties

In this section we will examine various functorial properties of the definitions given in §1. In this section X always denotes a smooth projective variety.

Proposition 2.1 Let E_{j} be a b_{j}-jet ample vector bundle on X, j = 1, …, m. Then E_{1} ⊕ ⋯ ⊕ E_{m} is a min_{j=1,...,m}{b_{j}}-jet ample vector bundle on X.

Proof. Assume e.g., that b_{1} = min_{j=1,...,m}{b_{j}}. Let {x_{1}, …, x_{t}} be a collection of t distinct points of X and let (k_{1}, …, k_{t}) ∈ ℤ_{+}^{t} such that ∑_{i=1}^{t} k_{i} = b_{1} + 1. Then, by assumption, we have surjective maps σ_{j} : H^{0}(E_{j}) → H^{0}(E_{j}/⊗_{i=1}^{t}𝔪_{x_{i}}^{k_{i}}), j = 1, …, m. Thus, by composition, we get a surjective map

H^{0}(⊕_{j=1}^{m}E_{j}) = ⊕_{j=1}^{m}H^{0}(E_{j}) ⟶^{⊕_{j=1}^{m}σ_{j}} ⊕_{j=1}^{m}H^{0}(E_{j}/⊗_{i=1}^{t}𝔪_{x_{i}}^{k_{i}}) ≅ H^{0}((⊕_{j=1}^{m}E_{j})/⊗_{i=1}^{t}𝔪_{x_{i}}^{k_{i}}).

Q.E.D.

Proposition 2.2 Let E be a k-jet ample vector bundle on X. Then E/F is a k-jet ample vector bundle for each subvector bundle F of E.

Proof. Let {x_{1}, …, x_{t}} be a collection of t distinct points of X and let (k_{1}, …, k_{t}) ∈ ℤ_{+}^{t} such that ∑_{i=1}^{t} k_{i} = k + 1. Note that, by tensoring with O_{X}/ ⊗_{i=1}^{t} 𝔪_{x_{i}}^{k_{i}}, the surjection E → E/F → 0 induces a surjective map E/ ⊗_{i=1}^{t} 𝔪_{x_{i}}^{k_{i}} → (E/F)/ ⊗_{i=1}^{t} 𝔪_{x_{i}}^{k_{i}}. Thus the assertion follows from the commutativity of the diagram

H^{0}(E) ⟶  H^{0}(E/ ⊗_{i=1}^{t} 𝔪_{x_{i}}^{k_{i}})
↓  ↓
H^{0}(E/F) ⟶ H^{0}((E/F)/ ⊗_{i=1}^{t} 𝔪_{x_{i}}^{k_{i}})

where the upper horizontal arrow is surjective by assumption and the vertical arrows are surjective by construction. Q.E.D.

Proposition 2.3 Let E be an a-jet ample vector bundle on X and let F be a b-jet ample vector bundle on X. Then E ⊗ F is an (a + b)-jet ample vector bundle on X.

Proof. It is a straightforward modification of the proof of [4, Lemma (2.2)]. Q.E.D.

As an immediate consequence of (2.2) and (2.3) we have the following.

Corollary 2.4 Let E be a k-jet ample vector bundle of rank r on X. Then ∧^{m}E and S^{m}(E) are rk-jet ample vector bundles. In particular det(E) is a rk-jet ample line bundle on X.

Remark 2.5 Note that if in (2.1) we assume that the vector bundles E_{j} are b_{j}-jet spanned for each j and similarly in (2.2), (2.3), (2.4) we assume that the vector bundle E is k-jet spanned, then all the corresponding conclusions hold true for jet spannedness as well. This is an immediate consequence of the above proofs, by noting also that the proof of the key-lemma [4, (2.2)] makes only use of jet spannedness.

Proposition 2.6 Let 0 → E → F → G → 0 be a short exact sequence of vector bundles on X. Assume that E is k-jet ample and G is k-jet ample with respect to V := Im(Γ(F) → Γ(G)). Then F is k-jet ample.

Proof. Let x_{1}, …, x_{t} be t distinct points and let k_{1}, …, k_{t} be positive integers such that ∑_{i=1}^{t} k_{i} = k + 1. Let 𝔪_{i} be the ideal sheaves of the points x_{i}, i = 1, …, t. We have the commutative diagram with exact rows

0 ⟶  X × Γ(E)  ⟶  X × Γ(F)  ⟶  X × V  ⟶ 0
↓ e_{E}  ↓ e_{F}  ↓ e_{G}
0 ⟶ Γ(E/ ⊗_{i=1}^{t} 𝔪_{i}^{k_{i}}) ⟶ Γ(F/ ⊗_{i=1}^{t} 𝔪_{i}^{k_{i}}) ⟶ Γ(G/ ⊗_{i=1}^{t} 𝔪_{i}^{k_{i}}) ⟶ 0,

where the vertical arrows are the evaluation maps. By hypothesis e_{E}, e_{G} are surjective. By a diagram chase we see that e_{F} is surjective, so we are done.Q.E.D.

The following is the rank > 1 version of [6, Proposition (3.5)] (compare also with [3, Corollary (1.1)] and [4, Lemma (3.1)]).

Proposition 2.7 Let E be a k-jet ample vector bundle on X. Let x_{1}, …, x_{t} be t distinct points on X and let a_{1}, …, a_{t} be t positive integers. Let π : X̃ → X be the blowing up of X at x_{1}, …, x_{t}. Let E_{i} := π^{−1}(x_{i}) be the exceptional divisors,

i = 1, …, t. Then π^{∗}E ⊗O_{X}(− ∑_{i=1}^{t} a_{i}E_{i}) is μ := min{k−∑_{i=1}^{t} a_{i}, a_{1}, …, a_{t}}-

jet ample.

Proof. It is a straightforward modification of the proof of [6, (3.5)]. Q.E.D.

In the setting of Proposition (2.7), if we blow up a single point x ∈ X, π : X̃ → X, we have that π^{∗}E ⊗ O_{X}(−kE) is spanned by global sections only assuming that E is k-jet spanned. Therefore, comparing with Corollary (1.5) of [3], we see that a formal extension of the arguments of [3] gives us the following lower bounds for the Chern classes and the Segre classes of E.

Corollary 2.8 Let X be an n-dimensional smooth projective variety and let E be a k-jet spanned vector bundle of rank r on X. Then

1. c_{i_{1}}(E) ⋯ c_{i_{t}}(E) ≥ k^{n}\binom{r}{i_{1}} ⋯ \binom{r}{i_{t}} for 0 ≤ i_{j} ≤ r, i_{1} + ⋯ + i_{t} = n;

2. s_{i_{1}}(E) ⋯ s_{i_{t}}(E) ≥ k^{n}\binom{r+i_{1}−1}{i_{1}} ⋯ \binom{r+i_{t}−1}{i_{t}} for 0 ≤ i_{j} ≤ r, i_{1} + ⋯ + i_{t} = n.

In particular c_{n}(E) ≥ k^{n}\binom{r}{n} and s_{n}(E) ≥ k^{n}\binom{r+n−1}{n}.

Remark 2.9 It is worthwhile to note that the same argument as in the proof of Lemmas (3.2), (3.3) in [6] gives us the following result.

Let X_{1}, …, X_{t} be t smooth projective varieties and let E_{1}, …, E_{t} be t vector bundles on X_{1}, …, X_{t} respectively. Let p_{i} : X_{1} × ⋯ × X_{t} → X_{i} be the projections on each factor, i = 1, …, t. For i = 1, …, t assume that E_{i} is k_{i}-jet ample and let k := min{k_{1}, …, k_{t}}. Then p_{1}^{∗}E_{1} ⊗ ⋯ ⊗ p_{t}^{∗}E_{t} is a k-jet vector bundle on the product X_{1} × ⋯ × X_{t}.

3 Characterizations of projective space

Let E be a k-jet ample or k-jet spanned vector bundle on a smooth projective variety X of dimension n. In this section we give a lower bound for h^{0}(E) and the degree of det(E). We also study the “boundary cases” when either h^{0}(E) or (det(E))^{n} reaches the the lowest possible value. For any merely ample vector bundle E we also consider the special case when the k-th jet bundle J_{k}(E) is trivial for a given k (compare with [13] for the line bundle case). This gives different characterizations of projective space. As a consequence of these results we obtain an adjunction type result concerning the jet spannedness of K_{X} + det(E) for a k-jet spanned vector bundle E.

Theorem 3.1 Let E be an ample vector bundle of rank r on a smooth projective variety X of dimension n. Assume that J_{k}(E) is trivial for some k > 0. Then (X, E) ≅ (ℙ^{n}, ⊕^{r}O_{ℙ^{n}}(k)) (hence in particular E is k-jet ample).

Proof. By dualizing and by tensoring with E the exact sequence (1) we get a surjective map J_{k}(E)^{∗} ⊗ E → T_{X}^{(k)} ⊗ E^{∗} ⊗ E → 0.

Notice that we have an injection O_{X} → E^{∗}⊗E. By dualizing and by tensoring with T_{X}^{(k)} we obtain a surjective map T_{X}^{(k)} ⊗ E^{∗} ⊗ E → T_{X}^{(k)} → 0.

Thus by composition we get a surjective map J_{k}(E)^{∗} ⊗E → T_{X}^{(k)} → 0. Since J_{k}(E) is trivial and E is k-jet ample, we conclude that T_{X}^{(k)} is ample. It thus follows that −K_{X} = det(T_{X}) is ample, so that X is a Fano manifold. Therefore Mori’s theory applies to say that X contains an extremal ray R = ℝ_{+}[γ], where γ is a possibly singular reduced irreducible rational curve satisfying the numerical condition −K_{X} · γ ≤ n + 1. Let f : ℙ^{1} → C ⊂ X be the normalization of any irreducible reduced rational curve C ⊂ X, in particular any deformation of γ. Note that f^{∗}(T_{X}^{(k)}) ≅ (f^{∗}T_{X})^{(k)} is ample since T_{X}^{(k)} is ample. Thus f^{∗}T_{X} is ample on ℙ^{1}. Therefore, by using Mori’s proof of the Hartshorne conjecture (see e.g., [11, §4] for a discussion) we conclude that X ≅ ℙ^{n}. By dualizing the exact sequence (1) we get a surjective map J_{k}(E)^{∗} → T_{X}^{(k)} ⊗ E^{∗} → 0. Since J_{k}(E) is trivial it follows that T_{X}^{(k)} ⊗ E^{∗} is spanned. Take a line ℓ ⊂ ℙ^{n} ≅ X. From the normal sequence of ℓ in ℙ^{n},

0 → T_{ℓ} ≅ O_{ℙ^{1}}(2) → T_{X|ℓ} → N_{ℓ/X} ≅ (n − 1)O_{ℙ^{1}}(1) → 0,

we get T_{X}^{(k)}_{|ℓ} ≅ (T_{X|ℓ})^{(k)} ≅ ((n − 1)O_{ℙ^{1}}(1) ⊕ O_{ℙ^{1}}(2))^{(k)}. Therefore we see that

O_{ℙ^{1}}(k) occurs as a direct summand of T_{X}^{(k)}_{|ℓ}.

The restriction of E to ℓ is E_{ℓ} ≅ ⊕_{i=1}^{r}O_{ℙ^{1}}(a_{i}) for some positive integers a_{i}, i = 1, …, r. Since, by the above, the restriction (T_{X}^{(k)} ⊗ E^{∗})_{ℓ} is spanned and contains O_{ℙ^{1}}(k − a_{i}) as direct summands we conclude that a_{i} ≤ k, i = 1, …, r.

For a rank r vector bundle E on X it follows that

det(J_{k}(E)) ≈ \binom{k + n}{n + 1} (krK_{X} + (n + 1) det(E)) .

Since the assumption on J_{k}(E) implies that det(J_{k}(E)) is trivial and X ≅ ℙ^{n}, we get det(E) ≈ O_{ℙ^{n}}(kr). On the other hand, det(E) ≈ O_{ℙ^{n}}(a_{1} + ⋯ + a_{r}). Then we infer that a_{1} + ⋯ + a_{r} = kr. Since a_{i} ≤ k, i = 1, …, r, we then conclude that a_{i} = k for i = 1, …, r.

Therefore (E ⊗ O_{ℙ^{n}}(−k))_{ℓ} ≅ ⊕^{r}O_{ℓ}. From [12, (3.2.1), p. 51] it follows that E ⊗ O_{ℙ^{n}}(−k) ≅ ⊕^{r}O_{ℙ^{n}}, i.e., E ≅ ⊕^{r}O_{ℙ^{n}}(k). Q.E.D.

The result above has the following consequences.

Proposition 3.2 Let E be a vector bundle of rank r on a smooth projective variety X of dimension n. We have:

1. If E is k-jet spanned at a given point x ∈ X, then h^{0}(E) ≥ r\binom{k+n}{n};

2. If equality happens and E is k-jet spanned, then (X, E) ≅ (ℙ^{n}, ⊕^{r}O_{ℙ^{n}}(k)).

Proof. For a given point x ∈ X, the map H^{0}(E) → H^{0}(E ⊗ O_{X}/𝔪_{x}^{k+1}) is onto and therefore

h^{0}(E) ≥ h^{0}(E ⊗ O_{X}/𝔪_{x}^{k+1}) = rh^{0}(O_{X}/𝔪_{x}^{k+1}) = r\binom{k + n}{n}.

Since E is k-jet spanned, the evaluation map j_{k} : X × Γ(E) → J_{k}(E) is surjective. If h^{0}(E) = r\binom{k+n}{n} = rank(J_{k}(E)) it thus follows that j_{k+1} is in fact an isomorphism and therefore J_{k}(E) is trivial. Then Theorem (3.1) applies to give the result. Q.E.D.

Proposition 3.3 Let E be a k-jet spanned vector bundle of rank r on a smooth projective variety X of dimension n. Then

1. det(E)^{n} = (rk)^{n} if and only if (X, E) ≅ (ℙ^{n}, ⊕^{r}O_{ℙ^{n}}(k));

2. det(E)^{n} ≥ (rk)^{n−1}(rk + 1) otherwise.

Proof. Since det(E) is rk-jet ample by Corollary (2.4) and Remark (2.5), we have (det(E))^{n} = (rk)^{n} if and only if (X, det(E)) ≅ (ℙ^{n}, O_{ℙ^{n}}(rk)) by [4, Corollary (3.1)]; note in fact that the proof of Theorem (3.1) and Corollary (3.1) in [4] makes only use of the jet spannedness. Now, the same argument as in the proof of (3.1) shows that E ≅ ⊕^{r}O_{ℙ^{n}}(k) in this case. This proves 1).

Thus we can assume (X, det(E)) ≅ (ℙ^{n}, O_{ℙ^{n}}(rk)), so that 2) follows again from [4, Corollary (3.1)]. Q.E.D.

Building up on the argument of [10, §1], we can now prove the following result for jets supported on a single point (compare with [3, (2.4)] and [14] for related results in the case of a very ample vector bundle).

Theorem 3.4 Let E be a k-jet spanned vector bundle of rank r on a smooth projective variety of dimension n. Assume kr ≥ n. Then K_{X} + det(E) is (kr − n)-jet spanned unless (X, E) ≅ (ℙ^{n}, ⊕^{r}O_{ℙ^{n}}(k)).

Proof. We have to show that for every x ∈ X, the map

H^{0}(K_{X} ⊗ det(E)) → H^{0}((K_{X} ⊗ det(E))/𝔪_{x}^{rk−n+1})

is surjective. Let π : Y → X be the blowing up of X at x with E = ℙ^{n−1} the exceptional divisor. Notice that det(E) is rk-jet spanned by (2.4) and Remark (2.5). Hence we can apply [4, Lemma (3.1)], whose proof only uses jet spannedness, to conclude that π^{∗} det(E) − rkE is spanned. Moreover, by (3.3), one has (π^{∗} det(E) − rkE)^{n} = (det(E))^{n} − (rk)^{n} > 0 unless (X, E) ≅ (ℙ^{n}, ⊕^{r}O_{ℙ^{n}}(k)).

Thus by Leray’s spectral sequence and the Kawamata-Viehweg vanishing theorem (see e.g., [7]), we have

H^{1}(K_{X} ⊗ det(E) ⊗ 𝔪_{x}^{rk−n+1}) = H^{1}(π^{∗}(K_{X} + det(E)) − (rk − n + 1)E)
= H^{1}(K_{Y} + π^{∗} det(E) − rkE) = 0

Q.E.D.

The above result gives some evidence to expect the following conjectures to be true in the line bundle case.

Conjecture 3.5 Let L be a k-jet ample line bundle on a smooth n-dimensional projective variety X, n ≥ 2. Assume k ≥ n. Then K_{X} + L is (k − n)-jet ample unless (X, L) ≅ (ℙ^{n}, O_{ℙ^{n}}(k)).

Note that the following conjecture is true for surfaces in the classical case k = 1.

Conjecture 3.6 Let L be a k-jet ample line bundle on a smooth n-dimensional projective variety X, n ≥ 2. Assume k ≥ n − 1. Then K_{X} + L is (k − n + 1)-jet spanned (perhaps even (k−n+1)-jet ample) unless either (X, L) ≅ (ℙ^{n}, O_{ℙ^{n}}(a)), with a = k, k + 1, or (X, L) ≅ (Q, O_{Q}(k)), Q hyperquadric in ℙ^{n+1}, or X is a ℙ^{n−1}-bundle, p : X → C, over a smooth curve C and L_{F} ≈ O_{ℙ^{n−1}}(k) for any fiber F ≅ ℙ^{n−1} of p.

Remark 3.7 Note that if L is a k-jet ample line bundle on a smooth surface X with k ≥ n + 1, then K_{X} + L is very ample by Reider’s theorem. Indeed, from [4, Theorem (3.1)] we have L^{2} ≥ (n + 1)^{2} + (n + 1) = 12. If K_{X} + L is not very ample we know from Reider’s theorem (see e.g., [5, (8.5.1)]) that there exists an effective curve C on X such that

L · C − 2 < C · C < L · C/2 < 2.

Thus C · C = 1 and L · C ≤ 3. This contradicts the Hodge inequality (C · C)(L · L) ≤ (L · C)^{2}.

4 Comparing definitions

In this section we compare the case k = 1, i.e., the 1-jet ampleness, with some others positivity properties given in the literature. The following two notions of “very ample” vector bundles have been introduced for “ad hoc” settings and, as we will see, are not equivalent. We heard the second notion to be attributed to Lazarsfeld (see e.g., [11, §3]).

Let X be a smooth projective variety. A vector bundle E on X is said to be very ample if the tautological line bundle ξ_{E} of ℙ(E) is very ample.

A vector bundle E on X is said to be strongly very ample if there exists a very ample line bundle L on X such that E ⊗ L^{−1} is generated by its global sections. For example, the tangent bundle of ℙ^{n} is strongly very ample.

Note that for a line bundle, M, very ampleness and strong very ampleness coincide since M ⊗ M^{−1} = O_{X} is spanned (see also •) below).

We want to show that 1-jet ampleness is a “middle point” definition between the previous two, i.e.,

• strongly very ample ⟹ 1-jet ampleness ⟹ very ampleness.

Proposition 4.1 Let E be a strongly very ample vector bundle on a smooth projective variety X. Then E is 1-jet ample.

Proof. Since E is strongly very ample, there exists a very ample line bundle L such that E ⊗ L^{−1} is spanned. Therefore Proposition (2.3) implies that E = (E ⊗ L^{−1}) ⊗ L is 1-jet ample (recall that for line bundles, 1-jet ampleness is equivalent to very ampleness). Q.E.D.

Proposition 4.2 Let E be a 1-jet ample vector bundle on a smooth projective variety X. Then E is very ample.

Proof. Let r := rank(E) and let ξ_{E} be the tautological line bundle of the ℙ^{r−1}- bundle p : ℙ(E) → X. We have to show that ξ_{E} is very ample.

Since E is spanned by global sections, it follows that ξ_{E} is spanned by global sections. Since no less than all the sections of O_{ℙ^{r−1}}(1) span O_{ℙ^{r−1}}(1), we see that the mapping ϕ : ℙ(E) → ℙ_{ℂ}, associated to |ξ_{E}|, embeds all fibers of p. Thus we must show that

1. given two distinct points x, y of ℙ(E), it follows that ϕ(x) ≠ ϕ(y);

2. given a point x ∈ ℙ(E) and a nonzero tangent vector τ_{x} at x, then the differential dϕ(τ_{x}) is not zero.

Let x, y be two distinct points of ℙ(E), and choose trivializations of E in neighborhoods of p(x) and p(y). Then we can write x = (x′, u), y = (y′, v), where x′, y′ ∈ X, u, v ∈ ℙ^{r−1}. Note that an element e ∈ E_{x′} gives a section of ξ_{E|p^{−1}(x′)} ≈ O_{ℙ^{r−1}}(1) under the isomorphisms

H^{0}(X, E) ≅ H^{0}(X, p_{∗}ξ_{E}) ≅ H^{0}(ℙ(E), ξ_{E}).

Thus regarding e as a linear homogeneous form, we have that the value at x = (x′, u) of the section of ξ_{E|p^{−1}(x′)} associated to e is e · u.

First take any two distinct points x, y of ℙ(E). Since ϕ embeds fibers, we can assume that x, y belong to different fibers of p, i.e., x′ ≠ y′. Since E is 1-jet ample on X we can find sections s′_{1}, s′_{2} ∈ H^{0}(X, E) such that s′_{1}(x′) = s′_{2}(y′) = 0 and s′_{1}(y′) = e_{y′}, s′_{2}(x′) = e_{x′} with e_{y′} · v ≠ 0 and e_{x′} · u ≠ 0. This shows that s_{1}, s_{2} separate the points x, y.

Now take any point x ∈ ℙ(E) and any nonzero tangent vector τ_{x} at x with dϕ(τ_{x}) ≠ 0. Since ϕ embeds fibers, we can assume that dp(τ_{x}) ≠ 0. Using the fact that E is 1-jet ample, choose a section s_{1} of E with s_{1}(x′) =: e_{1,x′} and such that e_{1,x′} · u ≠ 0, and also such that all first derivatives of s_{1} are zero at x′ relative to the chosen trivialization. Similarly choose a section s_{2} of E such that s_{2}(x′) = 0, but such that, relative to the chosen trivialization, the derivative of s_{2} in the direction dp(τ_{x}) evaluated at x′ is an element e_{2,x′} ∈ E_{x′} with e_{2,x′} · u ≠ 0.

Using the two sections s_{1}, s_{2} regarded as sections of ξ_{E}, it is straightforward to check that dϕ(τ_{x}) ≠ 0. We thus conclude that E is very ample. Q.E.D.

We provide examples to show that the three “very ampleness” notions considered above are in fact not equivalent.

Let us start showing that the 1-jet ampleness is stronger that very ampleness.

Example 4.3 There are examples of scrolls V = ℙ(E) in ℙ^{5}, where E is a rank 2 vector bundle on a surface X, such that the tautological line bundle ξ_{E} is very ample, and embeds V in ℙ^{5}, but E is not 1-jet ample.

From the lists of low degree threefolds in ℙ^{5} (see e.g., [2, Chapter 6]), consider the following examples.

1. V = ℙ(E) is of degree 6 and E is a rank 2 vector bundle over X = ℙ^{2} given by the exact sequence

0 → O_{ℙ^{2}} → E → J_{Z}(4) → 0,

where Z = x_{1}+⋯+x_{10} is a reduced 0-cycle of ℙ^{2} supported on 10 distinct points;

2. V = ℙ(E) is of degree 7 and E is a rank 2 vector bundle over the blowing up X = ℙ^{2}(x_{1}, …, x_{6}) of ℙ^{2} at six points x_{1}, …, x_{6};

3. V = ℙ(E) is a ℙ^{1}-bundle of degree 9 over a minimal K3 surface X.

Note that in each case E is not 1-jet ample. Assume otherwise. Then h^{0}(X, E) ≥ 6 by (3.2), 1). Since h^{0}(ℙ(E), ξ_{E}) = h^{0}(X, E) = 6 in each of the examples 1), 2), 3), we would have X ≅ ℙ^{2} and E ≅ O_{ℙ^{2}}(1) ⊕ O_{ℙ^{2}}(1) by (3.2), 2), which is not the case.

More generally, ℙ^{d}-bundles of “middle dimension” in ℙ^{n} are other examples.

Example 4.4 Consider a ℙ^{d}-bundle V = ℙ(E), E a rank d + 1 vector bundle on a smooth projective variety X, embedded in ℙ^{n} with either dim V = \frac{n}{2} if n is even or dim V = \frac{n+1}{2} if n is odd. Thus E is very ample but is not 1-jet ample. Indeed, otherwise, we would have by (3.2), 1), the numerical contradictions

n + 1 = h^{0}(V, ξ_{E}) = h^{0}(X, E) ≥ (d + 1) (\frac{n}{2} + 1) ≥ n + 2, for n even, or

n + 1 = h^{0}(V, ξ_{E}) = h^{0}(X, E) ≥ (d + 1) (\frac{n + 1}{2} + 1) ≥ n + 3, for n odd.

Let us show that strong very ampleness implies 1-jet ampleness.

Example 4.5 We construct here an example of 1-jet ample vector bundle which is not strongly very ample. Let C be a nonhyperelliptic curve of genus g = 5 and let D be a non-effective divisor of degree 3 on C. Consider the vector bundle E := K_{C} ⊕ K_{C}(D).

Note that K_{C} and K_{C}(D) are very ample line bundles on C and hence they are 1-jet ample (see (1.2)). From Proposition (2.1) it thus follows that E is a 1-jet ample rank two vector bundle on C. Take a very ample line bundle L on C. Since there are no smooth curves of genus 5 in ℙ^{2}, we must have that Γ(L) embeds C in ℙ^{N} with N ≥ 3. Notice that, by Castelnuovo’s bound on the genus, L must be of degree ≥ 7.

If E ⊗ L^{−1} = (K_{C} ⊗ L^{−1}) ⊕ (K_{C}(D) ⊗ L^{−1}) is spanned then, since deg(K_{C} ⊗ L^{−1}) ≤ 1, we conclude that L ≈ K_{C}. Thus E ⊗ L^{−1} ≅ O_{C} ⊕ O_{C}(D) is not spanned since D is not effective. Therefore E is not strongly very ample.

Remark 4.6 The previous example shows that strong very ampleness is not a functorial property. In fact E = K_{C} ⊕ K_{C}(D) as in (4.5) is not strongly very ample, but it is the direct sum of two very ample, i.e., strongly very ample, line bundles.

Note also that if E, F are strongly very ample vector bundles on a variety X, then, as it clearly follows from the definition, E ⊗ F is strongly very ample.

References

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[14] Y.-G. Ye and Q. Zhang, On ample vector bundles whose adjunction bundles are not numerically effective, Duke Math. J. 60 (1990), 671–687.

Mauro C. Beltrametti  Sandra Di Rocco
Dipartimento di Matematica  Department of Mathematics
Via Dodecaneso 35  KTH, Royal Insitute of Technology
I-16146 Genova,Italy  S-100 44 Stockolm, Sweden
beltrame@dima.unige.it  sandra@math.kth.se

Andrew J. Sommese Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556, U.S.A sommese@nd.edu http://www.nd.edu/∼sommese/index.html