GENERATION OF k-JETS ON TORIC VARIETIES

SANDRA DI ROCCO

Abstract. The notion of a k-convex Δ-support function for a toric variety X(Δ) is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the Δ-support function ψ_{L}. Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k.

Introduction

Let X be a non singular algebraic variety. The notion of k-jet ampleness has been introduced by Demailly to describe line bundles L whose global sections can have arbitrarily prescribed k-jets at every single point x ∈ X, see [Dem]. Beltrametti and Sommese generalized it by considering k-jets supported on a finite number of points.

A line bundle L is said to be k-jet ample on X if for any collection of r points, (x_{1}, ⋯, x_{r}), and any r-tuple of positive integers (k_{1}, ⋯, k_{r}), with ∑ k_{i} = k + 1, the natural map

H^{0}(X, L) × X → L/(L ⊗ 𝔪_{x_{1}}^{k_{1}} ⊗ ⋯ ⊗ 𝔪_{x_{r}}^{k_{r}})

is surjective, where 𝔪_{x_{i}} denotes the maximal ideal at x_{i} .

Notice that 0-jet ampleness is equivalent to being spanned by global sections and 1-jet ampleness is equivalent to being very ample.

During the last years many results on k-jet ample line bundles on surfaces have been established, [BeSok, EiLa, BaSz, BaDRSz]. Up to our knowledge the study of k-jets is still quite open for higher dimensional varieties, besides few cases like ℙ^{n} and Fano varieties [BeSok, BeDRSo].

In [Cox1] D. Cox has introduced “homogeneous coordinates” on a toric variety X(Δ). For the points invariant under the torus action the situation looks similar to the projective space case. We use this system of local coordinates to give a description of the fibers of the k-th jet bundle J_{k}(L) on fixed points, see section 2.

According to Oda and Demazure a line bundle L on X(Δ) is generated by global sections (respectively very ample) if the Δ-support function ψ_{L} is convex (respectively strictly convex). This suggests to use a “higher convexity” property for ψ_{L}, in the cases k ≥ 2. In section 3 we introduce the notion of a k-convex Δ-support function, which for k = 0, 1 agrees with being convex and strongly convex. The Δ-support function ψ_{L} being k-convex means that the polyhedron P_{L}, associated to L, is big enough to choose points in it corresponding to sections with arbitrary prescribed k-jets. This translates to the property that the intersection of L with the invariant curves, associated to every edge, is ≥ k. This property can be thought as a generalization of the toric Nakai criterion for ample line bundles.

1991 Mathematics Subject Classification. Primary 14M25, 14J60, 14C20; Secondary 14C25, 14E25.

Key words and phrases. Toric varieties, convex polyhedron, k-jet ampleness.

Partially supported by NFR B/276900.

A key step in the proof is the reduction to the case where the considered points are invariant under the torus action. We are grateful to T. Ekedahl for suggesting to use the Borel’s fixed point theorem, and for pointing out the sufficiency of the reduction argument.

Our result states that in order to check the k-jet ampleness of a line bundle L it is enough to have a bound on the intersection L · C, for all the invariant curves C. This can be applied to the study of “local positivity”. In section 6 we report a series of results on blow-ups and higher adjoint bundles, which in the toric case can be shown by means of a direct checking on intersections. We also state an equivalent criterion for k-jet ampleness in terms of a bound of the Seshadri constant ϵ(L, x). This can be thought as a toric version of the Seshadri criterion for ample line bundles, generalized to k-jet ampleness.

In this paper we prove: Let L be a line bundle on a non singular toric variety X(Δ), then the following statements are equivalent:

• L is k-jet ample;

• L · C ≥ k, for any T -invariant curve C, [Proposition 3.5];

• ψ_{L} is k-convex, [Theorem 4.2];

• the Seshadri constant ϵ(L, x) ≥ k for each x ∈ X, [Proposition 6.5].

It is a pleasure to acknowledge valuable discussions with T. Ekedahl, M. Boij, D. Laksov and D. Cox. This research was initiated during the author’s stay at Mittag-Leffler Institute and KTH and reached a final stage at the Max Planck Institute. To all those institutions we owe thanks for their support.

Notation

We will use standard notation in Algebraic Geometry. The groundfield will always be the field of complex numbers. When not stated X will always denote a smooth n-dimensional toric variety and L a line bundle on it. Cartier divisors, their associated line bundles and the invertible sheaves of their holomorphic sections will be used with no distinction, as well as the multiplicative and additive structure. For basic notions on toric varieties we refer to [Fu, Oda, Ew] and for a nice survey on the resent progress on toric geometry we refer to [Cox2].

1. Toric Varieties

Let N be an n-dimensional lattice and Δ = ∪σ_{i} be a complete and regular fan, meaning:

• supp(Δ) = N_{ℝ} = N ⊗ ℝ and

• for every r-dimensional cone σ ∈ Δ, there exists a ℤ-basis of N_{ℝ}, {ρ_{1}, ⋯, ρ_{n}}, such that the subset {ρ_{1}, ⋯, ρ_{r}} spans σ.

We will denote by X = X(Δ) the associated non singular n-dimensional toric variety and by Δ(t) the set of t-dimensional cones in Δ. Let M = Hom_{ℤ}(N, ℤ) be the dual lattice so that X is obtained by gluing together the affine toric varieties X_{σ} = Spec(ℂ[σ̌ ∩ M]), where σ̌ = {v ∈ M_{ℝ} : ⟨v, σ⟩ ≥ 0}, and σ ∈ Δ. Each m ∈ M can be viewed as a rational function χ^{m} : T = N ⊗ ℂ^{∗} → ℂ^{∗}. There is a 1−1 correspondence between r-dimensional cones τ ∈ Δ(r) and T -invariant codimension r subvarieties of X, which will be denoted by V (τ). Let D_{i} = V (ρ_{i}) be the T -invariant divisors corresponding to the one dimensional cones ρ_{i} ∈ Δ(1). The set {D_{i}}_{ρ_{i}∈Δ(1)} form a set of generators for the Picard group of X and thus every line bundle L can be written as:

L = ∑_{ρ_{i}∈Δ(1)} a_{i}D_{i}

We will denote by P_{L} the associated convex polyhedron:

P_{L} = {m ∈ M_{ℝ} : ⟨m, ρ_{i}⟩ ≥ −a_{i}}

This gives a nice way of expressing the global sections of L:

H^{0}(X, L) = ⨁_{m∈P_{L}∩M} ℂX^{m}

Recently David Cox [Cox1], has introduced the notion of homogeneous coordinates on a toric variety. There is a 1−1 correspondence between the T -invariant divisors D_{i} and linear monomials X_{i} on X. The polynomial ring associated to the toric variety is then defined as:

S = ℂ[X_{i} : ρ_{i} ∈ Δ(1)]

and the grading is given by the group of divisors modulo rational equivalence, P ic(X), i.e. two rationally equivalent divisors D and E are associated to monomials X_{D} and X_{E} of the same degree. Considering the exact sequence:

0 → M → ⊕_{ρ_{i}∈Δ(1)}ℤ · D_{i} → P ic(X) → 0

we associate to each m ∈ M a divisor ∑⟨m, ρ_{i}⟩D_{i} = div(χ^{m}). The global sections of L are generated by the monomials of the form

(Π_{i}X_{i}^{⟨m,ρ_{i}⟩+a_{i}})_{m∈P_{D}∩M}

The notion of Δ-support function will be used constantly throughout this paper:

Definition 1.1. [Oda, 2.1] A real valued function f : ∪_{i}σ_{i} → R is a Δ-linear support function if it is ℤ-valued on N ∩ (∪_{i}σ_{i}) and it is linear on each σ_{i}.

This means that for each σ there exists m_{σ} ∈ M such that f(n) = ⟨m_{σ}, n⟩ for n ∈ σ and ⟨m_{σ}, n⟩ = ⟨m_{τ}, n⟩ when τ is a face of σ. To each divisor L we associate the Δ-support function ψ_{L} defined by:

ψ_{L}(ρ_{i}) := −a_{i}

2. k-jet Bundles

Let D be the diagonal in X × X and p : X × X → X the projection onto the first factor. The k-th jet-bundle associated to L is the vector bundle associated to the sheaf:

p^{∗}L/p^{∗}L ⊗ I_{D}^{k+1}

where I_{D} is the ideal sheaf of D. It is a vector bundle of rank \binom{k+n}{n} whose fiber is

J_{k}(L)_{x} = L_{x}/L_{x} ⊗ m_{x}^{k+1}

where 𝔪_{x}^{k} is the k-th tensor power of the maximal ideal sheaf 𝔪_{x}. For details on jet bundles we refer the reader to [KuSp, Ch.I]. There are natural maps (defined on the sheaf level):

i_{k} : L → J_{k}(L)

sending the germ of a section s at a point x ∈ X to its k-th jet. For s ∈ H^{0}(X, L)

i_{k}(s(x)) ∈ ⨁\binom{k+n}{n}_{1} ℂ is the \binom{k+n}{n}-tuple determined by the coefficients of the terms of degree up to k, in the Taylor expansion of s around x. So if (x_{1}, ⋯, x_{n}) are local coordinates around x_{0} = (0, 0, ⋯, 0) and s = ∑ c_{i_{1},··· ,i_{r}} ∏ x_{i}^{i_{j}} then

i_{k}(s(x_{0})) = (⋯, \frac{∂^{t_{1}}}{∂x_{1}} ⋯ \frac{∂^{t_{r}}}{∂x_{r}} (s), ⋯)|_{x=x_{0}} = (⋯, (constant) · c_{t_{1},··· ,t_{r}}, ⋯)

where t_{1} + ⋯ + t_{r} ≤ k. For example i_{1}(s(x)) consists in the constant and linear terms. The following definition formalizes the property for a linear series |L| on X to generate k-jets on one or more points of X. When more points are considered L is said to generate “simultaneous jets” at those points.

Definition 2.1. Let Z = {x_{1}, ⋯, x_{r}} be a finite collection of distinct points on X. L is said to be k-jet ample on Z (or equivalently the series |L| is said to generate all k-jets on Z) if for any r-tuple of positive integers (k_{1}, ⋯, k_{r}), such that ∑_{1}^{r} k_{i} = k+1 the map:

H^{0}(X, L) → H^{0}(L ⊗ (O_{X}/𝔪_{x_{1}}^{k_{1}} ⊗ ⋯ ⊗ 𝔪_{x_{r}}^{k_{r}})) = ⨁_{i=1}^{r} H^{0}(L ⊗ O_{X}/𝔪_{x_{i}}^{k_{i}})

is surjective. L is k-jet ample on X if it is k-jet ample on each such Z in X.

Clearly from the definition:

• We can rewrite the map above as:

ψ_{Z}^{k_{1},··· ,k_{r}} : H^{0}(X, L) → ⨁_{1}^{r}(J_{k_{i}−1}(L))_{x_{i}}

defined by ψ_{Z}^{k_{1},··· ,k_{r}}(s) = (i_{k_{1}−1}(s(x_{1})), ⋯, i_{k_{r}−1}(s(x_{1}))). We say then that L is k-jet ample on X if the map ψ_{Z}^{k_{1},··· ,k_{r}} is surjective for any Z and any (k_{1}, ⋯, k_{r}) ∈ ℤ_{+}^{r}, such that ∑ k_{i} = k + 1.

• L is 0-jet ample if and only if L is generated by its global sections;

• L is 1-jet ample if and only if using the sections in H^{0}(X, L) we can define an embedding i : X → ℙ^{N} and thus L is very ample.

Using the homogeneous coordinates introduced in the previous section the k-jets at the T -invariant points x(σ) = V (σ) can be better described in terms of the polyhedron associated to L. Let σ = ⟨ρ_{1}, ⋯, ρ_{n}⟩, where σ ∈ Δ(n) and ρ_{i} ∈ Δ(1) are the one dimensional cones generating σ. The point x(σ), lies on the intersection of the T -invariant divisors D_{i}, i = 1, ⋯, n: x(σ) ∈ ∩_{1}^{n}(X_{i} = 0). Then the maximal ideal is generated by the linear monomials in X_{i}:

𝔪_{x(σ)} = ⟨X_{1}, ⋯, X_{n}⟩

and thus

𝔪_{x(σ)}^{k+1} = ⟨Π_{ρ_{i}⊂σ}X_{i}^{t_{i}}|t_{1} + ⋯ + t_{n} = k + 1⟩

i.e. the generators are the monomials of “degree= k + 1” in the variables X_{1}, ⋯, X_{n} ( here by degree we mean the sum of the powers of the variables, i.e. the usual one). Each X^{m}, generator of H^{0}(X, L), can be written in the local coordinates (X_{1}, ⋯, X_{n}) as follows. Fix {ρ_{1}, ⋯, ρ_{n}} as basis of N and let {m_{1}, ⋯, m_{n}} be the dual basis. In this coordinate system m = ∑⟨m, ρ_{i}⟩m_{i} and the germ of X^{m} at x(σ) is:

X^{m}|_{x(σ)} = ∏_{i=1}^{n} X_{i}^{⟨m,ρ_{i}⟩+a_{i}}

Taking its k-th jet means “killing” all the monomials of degree ≥ k+1 in the variables X_{1}, ⋯ X_{n}:

i_{k}(X^{m}(x(σ))) = (⋯, \frac{∂^{t_{1}}}{∂x_{1}} ⋯ \frac{∂^{t_{r}}}{∂x_{r}} (X^{m}), ⋯)|_{x=x(σ)}

where t_{1} + ⋯ + t_{r} ≤ k.

Example 2.2. Let N = ℤ^{2} and Δ be the 2-dimensional fan composed by the following 6 cones, and their edges: σ_{1} = ⟨(0, 1), (1, 1)⟩, σ_{2} = ⟨(1, 1), (1, 0)⟩, σ_{3} = ⟨(1, 0), (0, −1)⟩ σ_{4} = ⟨(0, −1), (−1, −1)⟩, σ_{5} = ⟨(−1, −1), (−1, 0)⟩, σ_{6} = ⟨(−1, 0), (0, 1)⟩ X(Δ) is the equivariant blow up of ℙ^{2} in the 3 fixed points, i.e. a Del Pezzo surface of degree 6. Let L = D_{1} + D_{2} + D_{3} + D_{4} + D_{5} + D_{6} = −K_{X(Δ)}, where the D′_{i}s are associated to the edges in the order given above. Let σ = ⟨(0, 1), (1, 1)⟩ and let {m_{1}, m_{2}} be the basis dual to {(0, 1), (1, 1)}. In this basis P_{L} is the convex hull of the points

{(0, 1), (1, 1), (1, 0), (−1, 0), (−1, −1), (0, −1)}

and thus the generators of H^{0}(X, L) are

{1, X_{1}, X_{2}, X_{1}X_{2}, X_{1}^{2}X_{2}, X_{1}X_{2}^{2}, X_{1}^{2}X_{2}^{2}}

Moreover 𝔪_{x(σ)}^{2} = ⟨X_{1}X_{2}, X_{1}^{2}, X_{2}^{2}⟩ and then

J_{1}(L)_{x(σ)} = ℂ ⊕ ℂX_{1} ⊕ ℂX_{2}

Example 2.3. Let X = ℙ^{n}, then Δ is the fan composed by (n + 1) n-dimensional cones spanned by the (n + 1) edges

• ρ_{i} = (0, ⋯, 0, \underbrace{1}_{i−th}, 0, ⋯, 0) for i = 1, ⋯, n

• ρ_{n+1} = (−1, ⋯, −1) = −ρ_{1} − ⋯ − ρ_{n} Let D_{1}, ⋯, D_{n+1} be the associated T -invariant divisors and let L = D_{1} + ⋯ + D_{k} = O_{ℙ^{n}}(k). Recall that the Picard group is generated by any T -invariant divisor D_{i} and that D_{i} ≡ D_{j} for i ≠ j. So we can think of L as

L = t_{1}D_{1} + ⋯ + t_{n}D_{n}; t_{1} + ⋯ + t_{n} = k

Let σ = ⟨ρ_{1}, ⋯ ρ_{n}⟩, and fix the basis {ρ_{1}, ⋯, ρ_{n}} with dual {m_{1}, ⋯, m_{n}}. In this basis the polyhedron P_{L} is the convex hall of the (n + 1) points

{(−1, ⋯, −1), (k, −1, ⋯, −1), ⋯, (−1, ⋯, −1, k, −1, ⋯, −1), ⋯, (−1, ⋯, −1, k)}

Then for any n-tuple t_{1}, ⋯, t_{n} of positive integers such that ∑_{1}^{n}(t_{i} + 1) ≤ k the lattice point m = ∑_{1}^{n} t_{i}m_{i} ∈ P_{L} and any lattice point m ∈ P_{L} can be written in this form. The situation stays the same if we consider another σ ∈ Δ. It follows that

J_{k}(L)_{x(σ)} = H^{0}(X, L) = ⨁_{t_{1}+···+t_{r}≤k} ℂ(∏_{1}^{r} X_{i}^{t_{i}})

In particular if L = O(1) then J_{1}(L) is trivial. This is in fact a characterization of the projective space, cf.[So].

3. k-convex functions

In order to study positivity properties of line bundles Demazure and Oda introduced the definition of convex and strictly convex Δ-support function.

Theorem 3.1. [Oda, Th. 2.13] A line bundle L on X is globally generated ( i.e. 0-jet ample) if and only if ψ_{L} is convex and it is very ample (i.e. 1-jet ample) if and only if ψ_{L} is strictly convex.

A natural way of generalizing such a criterion to higher jets is to introduce a definition of “higher convexity”.

Definition 3.2. Let ψ be a Δ-linear support function with ψ(v) = ⟨m_{σ}, v⟩ for each v ⊂ σ ∈ Δ. We will say that ψ is k-convex if for any σ ∈ Δ and v ⊄ σ

⟨m_{σ}, v⟩ ≥ ψ(v) + k

Confronting the notion of convex and strictly convex function, see [Oda], it is clear that

• ψ is 0-convex if and only if it is convex;

• ψ is 1-convex if and only if it is strongly convex.

Remark 3.3. It is clear from the definition that:

• If ψ is k convex then it is t-convex for any t ≤ k;

• If ψ_{1} is t_{1}-convex and ψ_{2} is t_{2}-convex, then (ψ_{1} + ψ_{2}) is (t_{1} + t_{2})-convex.

The meaning of convexity and strong convexity of a Δ-support function ψ_{L}, associated to a line bundle L, is quite clear at least at the fixed points x(σ) ∈ X. If ψ_{L} is convex then:

X^{m_{σ}}(x(σ)) = ∏_{i=1}^{n} X_{i}^{a_{i}−a_{i}} ≠ 0

If σ = ⟨ρ_{1}, ⋯, ρ_{n}⟩ and σ′ = ⟨ρ_{0}, ρ_{2}, ⋯, ρ_{n}⟩ then

X^{m_{σ′}}(x(σ)) = ∏_{i=2}^{n} X_{i}^{a_{i}−a_{i}}X_{1}^{⟨m_{σ′},ρ_{1}⟩+a_{1}} = 0

in the case ⟨m_{σ′}, ρ_{1}⟩ + a_{1} > 0, i.e. ψ_{L} strictly convex. In other words if ψ_{L} is convex then for each invariant point there is a non vanishing section, and ψ_{L} strictly convex implies that different invariant points can be separated. The notion of k-convexity generalizes the above property to more points with possible multiplicities.

More geometrically a Δ-support function ψ is k-convex if for each σ ∈ Δ the graph of the defining linear function ⟨m_{σ}, ⟩ is “very” high compared to the graph of ψ. Recall that if ψ_{L} is convex then the polyhedron P_{L} is the convex hull of the points m_{σ} in M_{ℝ}. If ψ_{L} is strictly convex then there is a correspondence between the faces in Δ and the set of non empty faces of P_{L} (cf. [Oda, 2.12]). Any face F ⊂ P_{L} corresponds to

F^{∗} = {n ∈ N_{ℝ}| < m, n >= ψ_{L}(n), for any m ∈ F } ∈ Δ

and any cone σ ∈ Δ corresponds to

σ^{∗} = {m ∈ M_{ℝ}| < m, n >= ψ_{L}(n), for any n ∈ σ} ⊂ P_{L}

Remark 3.4. By abuse of terminology for each τ = σ_{i} ∩ σ_{j} ∈ Δ(n − 1) we will call the “length” of the associated edge τ^{∗} the integer

l(τ^{∗}) = |m_{σ_{i}} − m_{σ_{j}}|

Assume ψ_{L} k-convex and let σ_{i} = ⟨ρ_{1}, ⋯, ρ_{n}⟩ and σ_{j} = ⟨ρ_{2}, ⋯, ρ_{n+1}⟩. Then, using the basis {m_{1}, ⋯, m_{n}} dual to {ρ_{1}, ⋯, ρ_{n}}, we have that m_{σ_{i}} = (−a_{1}, ⋯, −a_{n}) and m_{σ_{j}} = (−a_{1} + l_{i,j}, ⋯, −a_{n}), where l_{i,j} ≥ k.

Then ψ_{L} being k-convex means that the “length” of the edges of the polyhedron, corresponding to τ = σ_{i} ∩ σ_{j}, is bigger or equal to k. The following Proposition formulates an equivalent criterion for ψ_{L} to be k-convex in terms of the intersections of the divisor L with the T -invariant rational curves associated to each τ ∈ Δ(n − 1). In fact the polyhedron P_{L} having edges of “length” at least k translates to the restriction of L to each curve, corresponding to such edges, being at least k. This is in a way a generalization of the “toric Nakai criterion”, cfr. [Oda, Th. 2.18].

Proposition 3.5. Let L be a line bundle on a smooth n-dimensional toric variety X. Then ψ_{L} is k-convex if and only if the restriction L|_{V (τ)} has degree ≥ k, for every τ ∈ Δ(n − 1), i.e. L_{V (τ)} = O_{ℙ^{1}}(a) with a ≥ k for every τ ∈ Δ(n − 1).

Let τ = σ_{0} ∩ σ_{1} and assume σ_{i} = ⟨τ, n_{i}⟩ for i = 0, 1. Then, since we are assuming X to be non singular, there exists a ℤ-basis ⟨n_{i}, n_{2}, ⋯, n_{n}⟩ and (n − 1) integers (s_{2}, ⋯, s_{n}) such that:

n_{0} + n_{1} − ∑_{2}^{n−2} s_{i}n_{i} = 0

Write L = − ∑_{i} ψ_{L}(n_{i})D_{i} where D_{0}, D_{1} are the T -invariant divisors associated to the edges n_{0}, n_{1} and D_{i} is the one associated to n_{i}, i = 2, ⋯, n. then

• D_{1} · V (τ) = D_{0} · V (τ) = 1

• D_{i} · V (τ) = −s_{i} for i = 2, ⋯, n

• D_{j} · V (τ) = 0 otherwise

L · V (τ) = ∑(−ψ_{L}(n_{i}))D_{i} · V (τ) =
= −ψ_{L}(n_{0}) − ψ_{L}(n_{1}) + ∑_{2}^{n} ψ_{L}(n_{i})s_{i}
= ⟨m_{σ_{1}}, n_{0}⟩ − ψ_{L}(n_{0})

It follows that L · V (τ) ≥ k for all τ ∈ Δ(n − 1) if and only if for any σ ∈ Δ and ⟨ρ_{j}, σ ∩ σ′⟩ = σ′ the inequality

⟨m_{σ}, ρ_{j}⟩ − ψ_{L}(ρ_{j}) ≥ k

holds. In other words L · V (τ) ≥ k if and only if the support function ψ_{L} is kconvex.

Remark 3.6. If L is a k-jet ample line bundle then the restriction L|_{V (τ)} to every τ ∈ Δ(n − 1) is k-jet ample, i.e. L_{V (τ)} = O_{ℙ^{1}}(a) with a ≥ k for every τ ∈ Δ(n − 1). Proposition 3.5 then implies that ψ_{L} is k-convex.

4. The Main Result

The main purpose of this paper is to prove the equivalence between k-jet ampleness of L and k-convexity of ψ_{L}. In this section we show the implication left to prove after 3.6 i.e. that ψ_{L} k-convex implies L k-jet ample. The following Lemma implies that if ψ_{L} is k-convex, the polyhedron P_{L} is “big enough” to choose points in it, corresponding to sections with an arbitrary prescribed k-jet.

Lemma 4.1. Let P_{L} be the polyhedron associated to L and assume ψ_{L} is k-convex. Let σ_{1}, ⋯, σ_{r} be the n-dimensional cones in Δ. Then for each integer partition (t_{1}^{1}, ⋯, t_{n}^{1}, t_{1}^{2}, ⋯, t_{n}^{2}, ⋯, t_{n}^{r}) where t_{j}^{i} ≥ 0, ∑_{j=1}^{n} t_{j}^{i} = k_{i} − 1 and ∑_{1}^{r} k_{i} = k + 1, and for any σ_{i} we can find m ∈ P_{L} such that

• ⟨m, ρ_{l}⟩ = −a_{l} + t_{l}^{i} for all ρ_{l} ⊂ σ_{i}

• ⟨m, ρ_{j}⟩ ≥ −a_{j} + t_{j}^{i} for all ρ_{j} ⊄ σ_{i} with equality only if t_{j}^{i} = 0

Proof. . Fix σ_{i} = ⟨ρ_{1}, ⋯, ρ_{n}⟩ for simplicity of notation and let σ_{l} = ⟨ρ_{1}, ⋯, ρ̌_{l}, ⋯, ρ_{n}, \overline{ρ_{l}}⟩ the n-cone so that σ_{i} ∩ σ_{l} = τ_{l} = ⟨ρ_{1}, ⋯, ρ̌_{l}, ⋯, ρ_{n}⟩. By 3.4 l(τ_{l}^{∗}) ≥ k. Choose the t_{l}^{i}-th lattice point next to m_{σ_{i}} traveling on τ_{l}^{∗} towards m_{σ_{l}}, i.e

\overline{m_{l}} = (−a_{1}, ⋯, −a_{l} + t_{l}^{i}, ⋯, −a_{n}) = m_{σ_{i}} + (\frac{t_{l}^{i}}{l_{l}})(m_{σ_{l}} − m_{σ_{i}})

in the basis dual to {ρ_{1}, ⋯, ρ_{n}}, where ⟨m_{σ_{l}}, ρ_{l}⟩ = −a_{l} + l_{l} ≥ −a_{l} + k by hypothesis. Traveling on the n edges next to m_{σ_{i}} we get

m = m_{σ_{i}} + ∑_{l=1}^{n}(\frac{t_{l}^{i}}{l_{l}})(m_{σ_{l}} − m_{σ_{i}})

Rewriting it in the form

m = (1 − ∑_{l=1}^{n}(\frac{t_{l}^{i}}{l_{l}}))m_{σ_{i}} + ∑_{l=1}^{n}(\frac{t_{l}^{i}}{l_{l}})m_{σ_{l}}

it is clear that m is a convex combination of {m_{σ_{i}}, m_{σ_{1}}, ⋯, m_{σ_{n}}}, since 0 ≤ ∑_{1}^{n}(\frac{t_{l}^{i}}{l_{l}}) ≤ \frac{1}{k} ∑_{1}^{n} t_{l}^{i} ≤ 1 and therefore m ∈ P_{L} = Conv(m_{σ})_{σ∈Δ(n)}. Moreover

• ⟨m, ρ_{l}⟩ = ⟨\overline{m_{l}}, ρ_{l}⟩ = −a_{l} + t_{l}^{i} for l = 1, ⋯, n

• if ρ_{j} ⊄ σ_{i} then

⟨m, ρ_{j}⟩ = (1 − ∑(\frac{t_{l}^{i}}{l_{l}}))⟨m_{σ_{i}}, ρ_{j}⟩ + ∑_{l}[(\frac{t_{l}^{i}}{l_{l}})⟨m_{σ_{l}}, ρ_{j}⟩]

> (1 − ∑(\frac{t_{l}^{i}}{l_{l}}))(−a_{j} + k) + ∑[(\frac{t_{l}^{i}}{l_{l}})(−a_{j})]

= −a_{j} − k(∑ \frac{t_{l}^{i}}{l_{l}}) + k = −a_{j} + k(1 − ∑ \frac{t_{l}^{i}}{l_{l}})

Then 1 − ∑ \frac{t_{l}^{i}}{l_{l}} < 1 if ∑ \frac{t_{l}^{i}}{l_{l}} ≥ 1, i.e., 1 − k_{l} = ∑ t_{l}^{i} ≥ k, since l_{l} ≥ k. This implies

t_{j}^{i} = 0 and ⟨m, ρ_{j}⟩ = −a_{j}.

Theorem 4.2. A line bundle L generates k-jets on X if and only if the Δ-support function ψ_{L} is k-convex.

We need the following reduction step: Claim If L is k-jet ample on any r-tuple of fixed points {x(σ_{1}), ⋯, x(σ_{r})}, σ_{i} ≠ σ_{j}, then it is k-jet ample on X.

Proof. of the claim. To every (x_{1}, ⋯, x_{k+1}) ∈ X^{k+1} we can associate Z = (x_{1}, ⋯, x_{r}), a collection of r ≤ k + 1 distinct points of X, and (k_{1}, ⋯, k_{r}), an r-tuple of positive integers such that ∑ k_{i} = k + 1, simply counting the multiplicities of each x_{i}:

(\underbrace{x_{1}, ⋯, x_{1}}_{k_{1}}, \underbrace{x_{2}, ⋯, x_{2}}_{k_{2}}, ⋯, \underbrace{x_{r}, ⋯, x_{r}}_{k_{r}}) = [Z = (x_{1}, ⋯, x_{r}), (k_{1}, ⋯, k_{r})]

Then to each \underline{x} ∈ X^{k+1} we can associate the map:

ψ\underline{x} = ψ_{Z}^{k_{1},··· ,k_{r}} : H^{0}(X, L) → ⨁_{i=1}^{r}(J_{k_{i−1}}(L))_{x_{i}}

Let C = \{\underline{x} ∈ X^{k+1} such that coker(ψ\underline{x}) ≠ 0\}. Since ψ\underline{x} is an equivariant map, C inherits the torus action from X, i.e. it is an invariant closed subvariety of X^{k+1}, and hence proper. If L is not k-jet ample on X, then ψ\underline{x} is not surjective for some \underline{x} = [Z, (k_{1}, ⋯, k_{r})], which means C ≠ ∅. But this implies C^{T} ≠ ∅, where C^{T} is the set of the fixed points in C. To see this one can apply Borel’s fixed point theorem ( see [Hum, 21.1]) or more directly observe that C is a lower dimensional toric variety and thus it must contain fixed points. It follows that there exists \underline{x} ∈ X^{k+1}, fixed by the torus action, for which ψ\underline{x} is not surjective. Such \underline{x} must have all the components fixed so it is of the form (x(σ_{1}), ⋯, x(σ_{r})), which is a contradiction.

Proof. of the theorem. If L is a k-jet ample line bundle then ψ_{L} is k-convex by 3.6. Assume now that ψ_{L} is k-convex. By the reduction step it suffices to prove that the map ψ_{Z}^{(k_{1},··· ,k_{r})} is surjective for each Z = {x(σ_{1}), ⋯, x(σ_{r})}, with k_{1} + ⋯ + k_{r} = k + 1. This follows immediately from Lemma 4.1. For each k_{i} and for each partition t_{1}^{i} + ⋯ + t_{n}^{i} = k_{i} − 1 we can choose m ∈ P_{L} such that

• X^{m} = ∏_{ρ_{j}⊂σ_{i}} X_{j}^{t_{j}^{i}} around x(σ_{i}) and

• X^{m} = ∏_{ρ_{j}⊂σ_{l}} X_{j}^{t_{j}^{l}+c_{j}^{l}} around x(σ_{l}) ≠ x(σ_{i}), with c_{j}^{l} > 0 for some j and for any

partition ∑ t_{j}^{l} ≤ k_{l} This means that

(i_{k_{i}}(X^{m}(x(σ_{1})), ⋯, i_{k_{r}}(X^{m}(x(σ_{r}))) = (0, ⋯, 1, 0, ⋯, 0)

the non zero term corresponding to \frac{∂^{t_{1}}}{∂x_{1}} ⋯ \frac{∂^{tr}}{∂x_{r}} (X^{m})|_{x=x(σ_{i})}. This is enough to prove the surjectivity.

Remark 4.3. From 3.3 it follows immediately that:

• if L is k-jet ample then it is t-jet ample for any t ≤ k;

• If L is k-jet ample and E is t-jet ample then the line bundle E ⊗ L is (k + t)-jet

ample. In fact ψ_{E⊗L} = ψ_{L} + ψ_{E}. It is worth observing that in the toric case the notion of k-jet ampleness is equivalent to the notion of k-very ampleness (which is weaker in general). For basic properties of k-very ample line bundles we refer to [BeSoB].

Definition 4.4. L is said to be k-very ample if for every zeroscheme (Z, O_{Z}) of length h^{0}(O_{Z}) = k + 1 the map H^{0}(X, L) → H^{0}(Z, L ⊗ O_{Z}) is onto.

Proposition 4.5. A line bundle L on a smooth toric variety X is k-very ample if and only if it is k-jet ample.

If L is k-jet ample then it is k-very ample ( see [BeSok, Prop. 2.2]). If L is k-very ample then the degree of L restricted to any irreducible curve must be ≥ k, i.e. L · V (τ) ≥ k for all τ ∈ Δ(n − 1) and thus it is k-jet ample by 3.5.

5. Examples

In this section we work out few examples, for which the k-jet ampleness has been studied without the use of toric geometry. In each case our criterion is of course equivalent to the already known one. This should convince the reader that our criteria provide the “natural” and “global” way to look at k-jets on those varieties.

Example 5.1. The projective space ℙ^{n}. Notation as in 2.3. Let L = t_{1}D_{1} + ⋯+t_{n+1}D_{n+1} ≡ (t_{1}+⋯+t_{n+1})D_{1}. By 3.5 L is k-jet ample if and only if V (τ_{i})·L ≥ k where τ_{i} is the (n − 1)-dimensional cone ⟨ρ_{1}, ⋯ ρ̌_{i}, ⋯, ρ̌_{n+1}⟩ and D_{i} is the divisor associated to the edge ρ_{i}. Since ρ_{n+1} + ρ_{i} + ∑_{j≠i} ρ_{j} = 0

V (τ_{i}) · L = t_{1} + · + t_{n+1} ≥ k

In other words O_{ℙ^{N}}(a) is k-jet ample if and only if it is k-very ample if and only if a ≥ k, as proven in [BeSok].

Example 5.2. The Hirzebruch surface 𝔽_{n}. Let {e_{1}, e_{2}} be the standard basis for ℝ^{2}. The Hirzebruch surface 𝔽_{n} is the toric surface associated to the fan Δ spanned by the following 2-cones:

σ_{1} = ⟨e_{1}, e_{2}⟩, σ_{2} = ⟨e_{2}, −e_{1}⟩, σ_{3} = ⟨−e_{2}, −e_{1} + ne_{2}⟩, σ_{4} = ⟨−e_{1} + ne_{2}, e_{2}⟩

Let D_{1}, ⋯, D_{4} be the divisors associated respectively to e_{2}, e_{1}, −e_{2}, −e_{1} + ne_{2}. Then we have the following intersection matrix:

(\begin{matrix}D_{1}^{2} & D_{1} · D_{2} & D_{1} · D_{3} & D_{1} · D_{4} \\ D_{2} · D_{1} & D_{2}^{2} & D_{2} · D_{3} & D_{2} · D_{4} \\ D_{3} · D_{1} & D_{2} · D_{3} & D_{3}^{2} & D_{3} · D_{4} \\ D_{4} · D_{1} & D_{2} · D_{4} & D_{4} · D_{3} & D_{4}^{2} \end{matrix}) = (\begin{matrix}−n & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & n & 1 \\ 1 & 0 & 1 & 0 \end{matrix})

Recall that D_{3} ≡ D_{1} +nD_{2} and D_{2} ≡ D_{4}. Let L = a_{1}D_{1} +⋯+a_{4}D_{4} = (a_{1} +a_{3})D_{1} + (a_{4} + a_{2} + na_{3})D_{2}. Then by 3.5 L is k-jet ample if and only if it is k-very ample if and only if

• L · D_{1} = a_{4} − na_{1} + a_{2} ≥ k

• L · D_{2} = a_{1} + a_{3} ≥ k

• L · D_{3} = a_{2} + na_{3} + a_{4} ≥ k

• L · D_{4} = a_{1} + a_{2} ≥ k Let E_{0} be the section of minimal selfintersection −n ( i.e. D_{1}) and F be the general fiber of the projection onto ℙ^{1} (i.e. D_{2}). Then, from the inequalities above, we have that L = aE_{0} + bF is k-jet ample if and only if

• a = a_{1} + a_{3} ≥ k

• −an + b = −na_{1} − na_{3} + a_{4} + na_{3} + a_{2} = a_{4} − na_{1} + a_{2} ≥ k This conditions have been given by Beltrametti-Sommese in [BeSok], using a decomposition argument.

Example 5.3. Del Pezzo surfaces. The toric Del Pezzo surfaces are ℙ^{2}, 𝔽_{1} and the equivariant blow up of ℙ^{2} in 2 or 3 points. The most interesting one is the last one. Let S be the equivariant blow up of ℙ^{2} in the 3 invariant points as described in 2.2. The T -invariant divisors D_{1}, ⋯, D_{6} are the 6 (−1)-curves on the surface, i.e., the three exceptional divisors and the pull back of the three lines passing through two of the 3 points blown up. Proposition 3.5 says that L is k-jet ample if and only if it is k-very ample if and only if the intersection with all the (−1)-curves on the surface is ≥ k. This criterion has been given for k-very ampleness in [DR] using a generalization of Reider’s theorem. Note that the equivalence between k-jet ampleness and k-very ampleness is not always true for Del Pezzo surfaces. In fact it is not hard to see that if S is the blow up of ℙ^{2} in 7 points in general position ( so it is not toric), then the line bundle L = −2K_{S} is 2-very ample but it is not 2-jet ample.

6. Local positivity applications

In this section we report some nice applications of k-jet ampleness to the study of “local positivity” of line bundles. Most of it is a survey on well known results. We think it is interesting to show how these results can be established rather easily in the case of toric varieties.

Blow ups. A k-jet ample line bundle carries its positivity along blow ups at a finite number of points. The following property has been proved by Beltrametti and Sommese in [BeSo96] and it has a very “visible” proof in the toric case. We refer to [Oda] for notation and the definition of equivariant blow ups.

Proposition 6.1. Let p : X(Δ′) → X(Δ) be the equivariant blow up of X(Δ) at r points, x_{1}, ⋯, x_{r}, and let L be a k-jet ample line bundle on X(Δ). Then p^{∗}(L)−∑ ϵ_{i}E_{i} is min(k −∑ ϵ_{i}, ϵ_{1}, ⋯, ϵ_{r})-jet ample on X(Δ), where the E_{i}’s are the exceptional divisors.

Proof. Use induction on r. Assume the number of edges in Δ is e. If r = 1, let x = x(σ), σ = ⟨ρ_{1}, ⋯, ρ_{n}⟩ and E_{1} = E be the divisor associated to the edge \overline{ρ} = ρ_{1} + ⋯ + ρ_{n}. In the new fan Δ′ there are n new n-cones σ_{i} = ⟨\overline{ρ}, ρ_{1}, ⋯, ρ̌_{i}, ⋯, ρ_{n}⟩. Moreover let \overline{D_{i}} be the T -invariant divisors in P ic(X(Δ′)) corresponding to the edges ρ_{i}, then

• \overline{D_{i}} = p^{∗}(D_{i}) − E for i = 1, ..., n;

• \overline{D_{i}} = D_{i} for i = n + 1, ..., e. If L = ∑ a_{i}D_{i} then

H = p^{∗}(L) − ϵE = ∑_{1}^{e} a_{i}p^{∗}(D_{i}) − ϵE = ∑_{1}^{e} a_{i}\overline{D_{i}} − (ϵ + ∑_{i}^{n} a_{i})E

Let τ ∈ Δ′(n−1). If τ ∈ Δ(n−1) then clearly H·V (τ) = L·V (τ) ≥ k. If τ ∈ Δ′ −Δ then it is one of the following:

(a) σ_{i} ∩ σ_{j} = ⟨\overline{ρ}, ρ_{1}, ⋯, ρ̌_{i}, ρ̌_{j}, ⋯, ρ_{n}⟩

(b) σ_{i} ∩ ⟨ρ_{n+1}, ρ_{1}, ⋯, ρ̌_{i}, ⋯, ρ_{n}⟩ Following the lines of 3.5: In case (a), since ρ_{i} + ρ_{j} − \overline{ρ} + ∑_{l≠i,j} ρ_{l} = 0

H · V (τ) = −a_{i} − a_{j} + ϵ + ∑_{1}^{n} a_{l} − ∑_{l≠i,j} a_{l} = ϵ

In case (b), assume ρ_{n+1} + ρ_{i} − ∑_{j≠i} s_{j}ρ_{j} = 0, then ρ_{n+1} + \overline{ρ} − ∑_{j≠i}(s_{j} − 1)ρ_{j} = 0 and

H · V (τ) = −a_{n+1} − ϵ − ∑_{1}^{n} a_{j} + ∑_{j≠i} s_{j} + ∑_{j≠i} a_{i} = L · V (τ′) − ϵ

where τ′ = ⟨ρ_{1}, ⋯, ρ̌_{i}, ⋯, ρ_{n}, ρ_{n+1}⟩ ∩ σ ∈ Δ(n − 1) If r > 1, iterating this process, after r blow-ups H = p_{r}^{∗}(L′) − ϵ_{r}E, where p_{r} : X(Δ′) → X(Δ_{r−1}) is the r-th blow up map and Δ_{r−1} is the fan associated to the toric surface gotten after (r − 1) equivariant blow-ups of X(Δ). By induction L′ is min(k − ∑_{1}^{r−1} ϵ_{1}, ϵ_{r}, ⋯, ϵ_{r−1})-jet ample on X(Δ_{r−1}). Clearly from our previousely arguments

H · V (τ) ≥ min(L′ · V (τ′) − ϵ_{r}, ϵ_{r}) ≥ min(k − ∑_{1}^{n} ϵ_{i}, ϵ_{1}, ⋯, ϵ_{r})

for any τ ∈ Δ′(n − 1).

Toric Seshadri criterion. An ample line bundle on a smooth projective variety, X, is characterized by the positive value of its Seshadri constant at each point. Let L be a nef line bundle on X. For every irreducible curve C ⊂ X, m_{x}(C) denotes the multiplicity of C at the point x ∈ C and

m(C) = sup_{x∈C}{m_{x}(C)}.

Theorem 6.2. (Seshadri [Ha, 7.1]) A line bundle L on X is ample if and only if there exists a positive real number ϵ > 0 such that L·C ≥ ϵ·m(C) for every irreducible curve C ⊂ X.

As for the Nakai criterion we can generalize the Seshadri criterion to k-jet ampleness on toric varieties. Let us first reformulate the Seshadri’s theorem in the “modern language” of Seshadri constants. For a nef line bundle L the Seshadri constant of L at a point x ∈ X is the real number

ϵ(L, x) = inf_{x∈C} \frac{L · C}{m_{x}(C)} = sup\{ϵ ∈ ℝ|p^{∗}(L) − ϵL is nef \}

where the inf is taken over all the irreducible curves containing x and p is the blow-up map of X at x. Then one can immediately see that the Seshadri criterion says that L is ample if and only if ϵ(L, x) > 0 for every x ∈ X.

Demailly showed that the Seshadri constant is a measure of the highest degree jets that can be generated by the global sections of L:

Proposition 6.3. [Dem] Let s(L, x) be the largest integer such that |L| generates s-jets at x. Then

ϵ(L, x) = lim sup_{n→∞} \frac{s(nL, x)}{n}

Geometrical arguments yeld immediately that ϵ(L, x) ≥ 1 for every ample and spanned line bundle. Since every ample line bundle on a toric variety is very ample then we can think of a “toric Seshadri criterion” as saying that L is ample if and only if ϵ(L, x) ≥ 1 for every x ∈ X(Δ). More generally:

Proposition 6.4. A line bundle L on a non singular toric variety X(Δ) is k-jet ample if and only if

L · V (τ) ≥ k · m(V (τ))

for every invariant curve V (τ).

Proof. Assume L is k-jet ample, where k is the biggest integer such that the property is true. Then by 6.3 ϵ(L, x) = k, since nL is (nk)-jet ample by 4.3. It follows that \frac{L·V (τ)}{m(V (τ))} ≥ k for every τ ∈ Δ. Assume now that L · V (τ) ≥ k · m(V (τ)) for any τ ∈ Δ. Let τ = σ_{i} ∩ σ_{j}, we have to prove that L · V (τ) ≥ k. Consider the invariant point x = V (σ_{i}) ∈ V (τ), clearly m_{x}(V (τ)) = 1. Then L · V (τ) ≥ k · m(V (τ)) ≥ k · m_{x}(V (τ)) ≥ k.

Corollary 6.5. A line bundle L on a non singular toric variety is k-jet ample if and only if ϵ(L, x) ≥ k for every x ∈ X.

Proof. If L is k-jet ample then ϵ(L, x) = k by 6.3. If ϵ(L, x) ≥ k for every x ∈ X then in particular L · V (τ) ≥ k · m(V (τ)) for every invariant curve V (τ) and thus L is k-jet ample by 6.4.

Higher adjoint series. By use of bounds on the Seshadri constant of L at a sufficiently general point Ein and Lazarsfeld showed that:

Proposition 6.6. [Laz, 5.14] Let L be an ample line bundle on a smooth surface S. Then the adjoint series |K_{S} + (k + 3)L| generates k-jets at a sufficiently general point x ∈ S.

The fact that for a line bundle on a toric variety being ample is equivalent to being very ample implies a simple generalization.

Proposition 6.7. Let S be a non singular toric surface and let L be an ample line bundle on it such that L^{2} > 1. Then

(a) |K_{S} + (k + 2)L| generates k-jets at every point x ∈ S;

(b) |K_{S} + (2k + 2)| generates k-jets on S.

Proof. (a) By the Nakai toric criterion L is in fact very ample and by 4.3 (k + 2)L is (k + 2)-jet ample. Let x ∈ S, using the long exact sequence:

→ H^{0}(K_{S} +(k+2)L) → H^{0}((K_{S} +(k+2)L)/𝔪_{x}^{k+1})) → H^{1}((K_{S} +(k+2)L)⊗𝔪_{x}^{k+1}) →

the vanishing of H^{1}((K_{S} + (k + 2)L) ⊗ 𝔪_{x}^{k+1}) would imply the result. Let p : \overline{S} → S be the blow up of S at x with E = p^{−1}(x), then by Leray spectral sequence and Serre duality

H^{1}((K_{S} + (k + 2)L) ⊗ 𝔪_{x}^{k+1}) = H^{1}(K\overline{S} + [p^{∗}((k + 2)L) − (k + 2)E])

Kawamata vanishing theorem applies since p^{∗}((k +2)L)−(k +2)E is nef and big and thus K_{S} + (k + 2)L is k-jet ample at any point x ∈ S.

(b) Consider now simoultaneus jets supported on {x_{1}, ⋯ x_{r}} ∈ S and (k_{1}, ⋯, k_{r}) ∈ ℤ_{+}^{r} such that ∑ k_{1} = k + 1. Let p : \overline{S} → S the blow up of S at x_{1}, ⋯, x_{r} with E_{i} = p^{−1}(x_{i}). Using the same exact sequence as above it suffices to prove that:

H^{1}((K_{S}+(2k+2)L)⊗(𝔪_{x_{1}}^{k_{1}}⊗⋯⊗𝔪_{x_{r}}^{k_{r}})) = H^{1}(K\overline{S}+[p^{∗}((2k+2)L)−∑(k_{i}+1)E_{i}]) = 0

Since (2k+2)L is (2k+2)-jet ample and ∑(k_{i}+1) ≤ 2k+2, p^{∗}((2k+2)L)−∑(k_{i}+1)E_{i} is spanned by 6.1. Moreover

(p^{∗}((2k + 2)L) − ∑(k_{i} + 1)E_{i})^{2} > (2k + 2 − ∑(k_{1} + 1))(2k + 2 + ∑(k_{1} + 1)) > 0

Then Kawamata vanishing theorem applies to give the needed vanishing.

The k-reduction. In the case of surfaces we can make some further remarks about the “k-reduction” process (see [BeSo]). Let S be a non singular toric surface and L a k-very ample line bundle on it. Since k-jet very ampleness and k-very ampleness are equivalent we will use freely the property of being k-very ample according to the criterion given in 4.2. Assume the adjoint bundle is not k-very ample i.e., the surface contains (−1)-curves whose intersection with L is exactly k. If the k-adjoint bundle kK_{S} + L is nef we can contract down those curves and get the k-reduction (S′, L′).

Notice that if L = −kK_{S} then −K_{S} is ample and hence S is a toric Del Pezzo surface. We can then compute directly the nefness of the k-adjoint bundle obtaining the same result as in [BeSo]. Using the same notation as in 5.1,5.2 we have that:

Proposition 6.8. Let L ≠ −kK_{S} be a k-very ample line bundle on S. Then kK_{S}+L is nef unless:

• S = ℙ^{2} and L = aD_{1} with a < 3k ;

• S = 𝔽_{r} and L = aD_{1} + bD_{2} with a < 2k.

Proof. Let L = ∑ a_{i}D_{i} and D_{i}^{2} = −s_{i}, then

(kK_{S} + L) · D_{i} = L · D_{i} + k(s_{i} − 2) ≥ 0 if s_{i} ≥ 1

Recall that S is isomorphic to ℙ^{2}, 𝔽_{n} or their equivariant blow up in a finite number of points. If S is minimal then intersecting L with the basic generators of P ic(S) and imposing at least one intersection to be less then k gives the cases in the statement. Assume now S not minimal. If S = Bl_{r}(ℙ^{2}) (i.e. the equivariant blow up of ℙ^{2} in r points) let D_{1}, D_{j}, D_{l} be the divisors associated to the edges (0, 1), (1, 0), (−1, −1) respectively. If r ≥ 2, for each D_{i} generator of P ic(S), the corresponding weight D_{i}^{2} = −s_{i} ≤ −1 unless possibly only one among (s_{1}, s_{j}, s_{l}), say s_{1} = −1. But in this case D_{1} ≡ D_{2} + ∑_{1}^{r} D_{i}, the D_{i}’s being the exceptional divisors, L · D_{1} ≥ (r + 1)k and

(kK_{S} + L)D_{1} ≥ (r + 1)k − 3k ≥ 0

If r = 1 then S ≅ 𝔽_{1}. If S = Bl_{r}(𝔽_{n}), let D_{l}, D_{j}, D_{3}, D_{h} be the divisors corresponding to the edges (0, 1), (1, 0), (0, −1), (−1, n) respectively. We can assume the weights s_{i} ≥ 1, unless possibly s_{3} = −n + s < 1, since Bl_{1}𝔽_{0} ≅ Bl_{2}(ℙ^{2}) and D_{j} ≡ −D_{h}. But in this case D_{3} ≡ D_{l} + nD_{j} + ∑_{1}^{r−s} D_{i}, L · D_{3} ≥ (n + 1 + r − s)k and

(kK_{S} + L)D_{3} ≥ (n + r + 1 − s)k − (n − s + 2)k = (r − 1)k ≥ 0

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KTH, Department of Mathematics, S-100 44 Stockholm

  E-mail address: sandra@math.kth.se

Current address: Max-Planck-Institute für Mathematik, Gottfried-Claren-Str. 26 D-53225 Bonn