Cyclic coverings and higher order embeddings of algebraic varieties

Th. Bauer, S. Di Rocco, T. Szemberg^{∗}

June 24, 1998

Abstract Let Y be a cyclic covering of an algebraic variety X. Given a line bundle on X

we give criteria for its pull-back to Y to define a higher order embedding.

Introduction

In recent years there has been considerable interest in understanding under which circumstances linear series on algebraic varieties restrict surjectively to zero-dimensional subschemes and to collections of fat points. Equivalently, one asks for the order of a given projective embedding of an algebraic variety. The concepts of higher order embeddings are captured by the notions of k-very ampleness, introduced in [5], and k-jet ampleness, studied in [9] and [13]. More geometrically, an algebraic variety embedded in IP^{N} via a k−very ample line bundle has no (k + 1)−secant (k − 1)−plane IP^{k−1} ⊂ IP^{N}. The embedding given by a k−jet ample line bundle has stronger geometrical constraints related to the higher osculating planes.

By now, the situation on surfaces is quite well understood, mainly thanks to the availability of powerful methods such as a Reider type criterion for k-very ampleness [5] and the use of ℚ-divisors in connection with the Kawamata-Viehweg vanishing theorem [17]. In higher dimensions, however, the problem seems to be much more difficult. While there are highly interesting general results on the separation of jets due to Demailly [13], there is still a lack of practical criteria that allow to determine the order of a given embedding. The purpose of this paper is to study higher order embeddings of cyclic coverings π : Y ⟶ X, via line bundles given by pulling back ”sufficiently positive” line bundles on X. Given a line bundle L on X we relate the order of the embedding defined by π^{∗}L to that of L and certain rank 1 summands of the vector bundle L ⊗ π_{∗}O_{Y}. The main results are expressed in the Theorems 2.1 and 3.1.

In Section 1 we recall briefly the concepts of higher order embeddings. Section 2 is devoted to an exposition of the k−jet ample case. In Section 3 we prove our result concerning k−very ampleness. Although the rigorous proof is technically involved, we hope that the ideas behind are fairly transparent. Finally, the examples provided in the last section show that our results are sharp.

∗Supported by a research grant of Polish Academy of Sciences

We work throughout over the field ℂ of complex numbers.

Acknowledgments. The first version of the paper was written during the authors’ stay at the Max-Planck-Institut für Mathematik in Bonn. We would like to thank the Institute for financial support and for providing excellent working conditions. Our stay has been most fruitful and enjoyable.

1 Higher order embeddings

We start by recalling the notions of k-jet ampleness and k-very ampleness that capture the concept of a higher order embedding in two different ways: the first notion requires the simultaneous separation of jets at finitely many points, whereas the second asks for the surjectivity of the restriction to 0-dimensional subschemes of certain length.

A line bundle L on a smooth projective variety X is called k-jet ample, if the evaluation map

H^{0}(X, L) ⟶ H^{0} (X, L ⊗ O_{X}/ (𝔪_{y_{1}}^{k_{1}} ⊗ … ⊗ 𝔪_{y_{r}}^{k_{r}}))

is surjective for any choice of distinct points y_{1}, …, y_{r} in X and positive integers k_{1}, …, k_{r} with ∑ k_{i} = k + 1. Further, L is called k-very ample, if for every zero-dimensional subscheme (Z, O_{Z}) ⊂ (X, O_{X}) of length k + 1 the natural map

H^{0}(X, L) ⟶ H^{0} (X, L ⊗ O_{Z})

is surjective.

As the definitions suggest, k−jet ampleness implies k−very ampleness [9, 2.2] and both notions are equivalent to global generation in the case k = 0 and very ampleness in the case k = 1.

In the case when X = IP^{2} the notion of k-the order embedding is another way of dealing with non “superabundant embeddings”. We refer to [16] and [7] for more details.

For the purposes of the proof of Theorem 3.1 below, it will be useful to have a slightly more general definition of k-very ampleness:

A subspace V ⊂ H^{0}(X, L) will be called k-very ample on an open subset U ⊂ X, if for every zero-dimensional subscheme (Z, O_{Z}) ⊂ (U, O_{U}) of length k + 1 the natural map

ev_{Z} : V ⟶ H^{0} (X, L ⊗ O_{Z})

is surjective. It is then obvious that if V ⊂ H^{0}(X, L) is k-very ample on U, and if (Z, O_{Z}) ⊂ (U, O_{U}) is a 0-dimensional subscheme of length ℓ, then the subspace

ev_{Z}^{−1}(g) ⊂ V

is (k − ℓ)-very ample on U \ supp(Z) for every g ∈ H^{0}(X, L ⊗ O_{Z}).

2 Coverings and k-jet ampleness

Theorem 2.1 Let X be a smooth projective variety and B ⊂ X a smooth divisor. Let M be a line bundle on X such that O_{X}(dM) ≅ O_{X}(B) and let π : Y ⟶ X be the cyclic covering of degree d defined by M. Let L be a line bundle on X and k a non-negative integer. If L − qM is (k − q)-jet ample for q = 0, …, min(k, d − 1), then π^{∗}L is k-jet ample.

Proof. Let \overline{M} be the total space of M and p : \overline{M} ⟶ X the bundle projection. Let s_{B} ∈ H^{0}(X, O_{X}(B)) be a section whose divisor of zeros is B, and let τ ∈ H^{0}(\overline{M}, p^{∗}M) be the tautological section. As usual, Y may be viewed as the divisor of zeros of the section p^{∗}s_{B} − τ^{d}. Letting t be the restriction of τ to Y and π : Y ⟶ X the restriction of p, the projection formula gives the decomposition

H^{0}(Y, π^{∗}L) = ⨁_{q=0}^{d−1} t^{q}π^{∗}H^{0}(X, L − qM)  (1)

corresponding to the eigen-values of the action of the primitive covering automorphism φ.

Let y_{1}, …, y_{r} ∈ Y be points and k_{1}, …, k_{r} positive integers with ∑ k_{i} = k + 1. Given a simultaneous jet

J ∈ ⨁_{i=1}^{r} H^{0}(Y, π^{∗}L ⊗ O_{Y} /𝔪_{y_{i}}^{k_{i}}) = H^{0}(Y, π^{∗}L ⊗ O_{Y} /𝔪_{y_{1}}^{k_{1}} ⊗ … ⊗ 𝔪_{y_{r}}^{k_{r}}) ,

we decompose J into a sum of simultaneous jets J = J_{1} + … + J_{r}, where J_{i} = (j_{1i}, …, j_{ii}, …, j_{ri}), with j_{li} the zero jet of order k_{l} at y_{l} for l ≠ i. It is enough to find sections s_{i} ∈ H^{0}(Y, π^{∗}L) such that

s_{i} mod 𝔪_{y_{1}}^{k_{1}} ⊗ … ⊗ 𝔪_{y_{r}}^{k_{r}} = J_{i} for i = 1, …, r ,

since then s = s_{1} + … + s_{r} has the desired jet J. We may therefore assume J = J_{1}. Now, the idea is to construct a section s explicitly out of sections in the line bundles L − qM, q ≥ 0. We distinguish three cases.

Case 1. Suppose that y_{1} ∉ R and that none of the points y_{2}, …, y_{r} lies in the orbit π^{−1}(π(y_{1})). This is the easiest case. The desired sections is obtained as a pull-back of a section in L. More precisely we can view the jet j_{11} as a jet at π(y − 1), via the isomorphism of local rings O_{Y,y_{1}} ≅ O_{X,π(y_{1})} induced by π. Since L is k−jet ample, there is a section s ∈ H^{0}(X, L) such that

s mod 𝔪_{π(y_{1})}^{k_{1}} = j_{11}

and

s mod 𝔪_{π(y_{i})}^{k_{i}} = 0 for i = 2, …, r .

Then of course π^{∗}s mod 𝔪_{y_{1}}^{k_{1}} ⊗ … ⊗ 𝔪_{y_{r}}^{k_{r}} = J_{1}.

Case 2. Suppose that y_{1}, …, y_{l} (with l ≥ 2) lie in the orbit π^{−1}(π(y_{1})) and that none of the points y_{l+1}, …, y_{r} does. This case is more difficult since we have to separate points in a fiber of the covering. The construction of s builds upon the cyclic group at hand. Let k′ = max(k_{1}, …, k_{l}). Observe that k′ ≤ k + 1/(l − 1) − ∑_{i=l+1}^{r} k_{i}. Let ĵ_{11} be any preimage of the jet j_{11} under the surjective map

O/𝔪_{y_{1}}^{k′} ⟶ O/𝔪_{y_{1}}^{k_{1}} .

Using now again the identification O_{Y,y_{1}} ≅ O_{X,π(y_{1})} as well as the facts that t(y_{1}) ≠ 0 and that the line bundle L − qM is (k − q)-jet ample for 0 ≤ q ≤ l − 1, we can find sections s_{1}, …, s_{l} such that

π^{∗}s_{1} mod 𝔪_{y_{1}}^{k′} = α_{1} · ĵ_{11}
t · π^{∗}s_{2} mod 𝔪_{y_{1}}^{k′} = α_{2} · ĵ_{11}
⋮
t^{l−1} · π^{∗}s_{l} mod 𝔪_{y_{1}}^{k′} = α_{l} · ĵ_{11}

and

t^{q} · π^{∗}s_{q+1} mod 𝔪_{y_{i}}^{k_{i}} = 0 for 0 ≤ q ≤ l − 1 and l + 1 ≤ i ≤ r .

for any given complex numbers α_{1}, …, α_{l}. Now, let (α_{1}, …, α_{l}) be a solution of the system of linear equations

α_{1} + … + α_{l} = 1 ; α_{1} + ε^{β_{2}}α_{2} + … + ε^{(l−1)β_{2}}α_{l} = 0 ; ⋮ ; α_{1} + ε^{β_{l}}α_{2} + … + ε^{(l−1)β_{l}}α_{l} = 0  (2)

where β_{2}, …, β_{l} are integers such that y_{i} = φ^{β_{i}}(y_{1}). (Note that the determinant of the system (2) is just the Vandermonde determinant of the numbers 1, ε^{β_{2}}, …, ε^{β_{l}}.) Consider the section s = π^{∗}s_{1} + … + t^{l−1}π^{∗}s_{l}. We have s mod 𝔪_{y_{1}}^{k′} = ĵ_{11}, which implies

s mod 𝔪_{y_{1}}^{k_{1}} = j_{11} ,

as required. Moreover we have

s mod 𝔪_{y_{i}}^{k′} = (φ^{β_{i}})^{∗}s mod 𝔪_{y_{1}}^{k′}
= π^{∗}s_{1} + ε^{β_{i}}tπ^{∗}s_{2} + … + ε^{(l−1)β_{i}}t^{l−1}π^{∗}s_{l} mod 𝔪_{y_{1}}^{k′}
= (α_{1} + ε^{β_{i}}α_{2} + … + (ε^{β_{i}})^{l−1}α_{l})ĵ_{11} = 0 ,

so that s mod 𝔪_{y_{i}}^{k_{i}} = 0 for i = 2, …, l, and of course

s mod 𝔪_{y_{i}}^{k_{i}} = 0 for i = l + 1, …, r .

Case 3. Suppose y_{1} ∈ R. Since B (and hence R) is smooth, there are local coordinates u_{1}, …, u_{n} at y_{1} and v_{1}, …, v_{n} at π(y_{1}) such that π is locally given as

π(u_{1}, …, u_{n}) = (u_{1}^{d}, u_{2}, …, u_{n}) = (v_{1}, …, v_{n}) .

In these coordinates t is given by u_{1} and j_{11} may be written as

j_{11} = ∑_{i_{1}+...+i_{n}≤k_{1}} a_{i_{1},...,i_{n}}u_{1}^{i_{1}} · . . . · u_{n}^{i_{n}} = ∑_{l=0}^{d−1} u_{1}^{l} ∑_{\begin{matrix}i_{1}≡l mod d \\ i_{1}+...+i_{n}≤k_{1} \end{matrix}} a_{i_{1}...i_{n}}(u_{1}^{d})^{\frac{i_{1}−l}{d}} u_{2}^{i_{2}} · . . . · u_{n}^{i_{n}}.

This splitting reflects the fact that only jets containing powers of u_{1} divisible by d arise as pull-backs of jets on X. Since L − qM is (k − q)-jet ample for q = 0, …, min(d − 1, k), there are sections s_{q} ∈ H^{0}(X, L − qM) such that

s_{q} mod 𝔪_{π(y_{1})}^{k_{1}−q} = ∑_{\begin{matrix}i_{1}≡l mod d \\ \frac{i_{1}−q}{d} +i_{2}+...+i_{n}≤k_{1}−q \end{matrix}} a_{i_{1}...i_{n}}v1^{\frac{i_{1}−l}{d}} v_{2}^{i_{2}} · . . . · v_{n}^{i_{n}} .

and

s_{q} mod 𝔪_{π(y_{i})}^{k_{i}} = 0 for i = 2, …, r .

It is now easy to check that the section

s = ∑_{q≥0} t^{q}π^{∗}s_{q}

has the prescribed jets.

Working as in cases 1 and 2 of the above proof, one gets immediately the same result for unbranched coverings:

Theorem 2.2 (Unbranched case) Let π : Y ⟶ X be an unbranched cyclic covering of degree d, defined by a line bundle M with O_{X}(dM) = O_{X}. Let L be a line bundle on X and k a non-negative integer. If L − qM is (k − q)-jet ample for q = 0, …, min(k, d − 1), then π^{∗}L is k-jet ample.

3 Coverings and k-very ampleness

In Theorem 3.1 below we will assert the k-very ampleness of a pullback of an ample line bundle L under suitable hypotheses on the line bundles L − qM, q ≥ 0. In order to get a useful criterion, one needs to formulate a delicate numerical hypothesis. To this end, given positive integers k and ℓ, we introduce the abbreviations γ(k, ℓ) = 1 if k/ℓ is an integer, and γ(k, ℓ) = 0 otherwise, and we set

τ(k, ℓ) = k − ⌊\frac{k}{ℓ} ⌋ − ℓ + γ(k, ℓ) + 1 .

Further, we use the notation

σ(k, d, q) = max {τ(k + 1, ℓ) | q + 1 ≤ ℓ ≤ min(d, k + 1)} − 1

if q > 0 and σ(k, d, 0) = k.

Our result can then be stated as follows:

Theorem 3.1 Let X be a smooth projective variety and B ⊂ X a smooth divisor. Let M be a line bundle on X such that O_{X}(dM) ≅ O_{X}(B) and let π : Y ⟶ X be the cyclic covering of degree d defined by M. Let L be a line bundle on X and k a non-negative integer. If L − qM is σ(k, d, q)-very ample for q = 0, …, min(k, d − 1) then π^{∗}L is k-very ample.

One can prove the k-very ampleness of π^{∗}L under the assumption that L − qM is (k − q)-very ample for 0 ≤ q ≤ min(k, d + 1) by arguments very similar to the ones used in the proof of Theorem 2.1. The point however here is that k-very ampleness holds already under the much weaker (but also more intricate) hypotheses involving the numbers σ(k, d, q). For instance, if L is 2-very ample, then π^{∗}L is also 2-very ample as soon as both L − M and L − 2M are globally generated. If π is of degree 2 it is enough to check that L − M is globally generated. We provide a list of explicit values of σ(k, d, q) in Remark 3.5.

In order to simplify the exposition of the proof of the theorem, we start by stating two lemmas. We begin with an elementary observation on local rings:

Lemma 3.2 Let (O, 𝔪) be a local ring, and let I_{1}, …, I_{ℓ}, ℓ ≥ 2, be ideals contained in 𝔪 such that length O/I_{1} ≥ length O/I_{i} for all i. Let k = ∑_{i=1}^{ℓ} length O/I_{i}. If k < ∞, then

length O/I_{2} ∩ … ∩ I_{ℓ} ≤ τ(k, ℓ) .

Proof. We have the inequality

length O/I_{2} ∩ … ∩ I_{ℓ} ≤ ∑_{i=1}^{ℓ} length O/I_{i} − length O/I_{2} + (I_{3} ∩ … ∩ I_{ℓ})

− … − length O/I_{2} + … + I_{ℓ} ≤ k − length O/I_{1} − (ℓ − 2) .

The assertion follows now from the fact that length O/I_{1} ≥ ⌊k/ℓ⌋ + 1 − γ(k, ℓ).

We will also need the following Lemma 3.3 Let ℓ_{1}, …, ℓ_{m}, K_{1}, …, K_{r} and q be positive integers, and let K = ∑_{i=1}^{r} K_{i}. If ℓ_{i} ≥ 2 for all i, then

∑_{i=1}^{m} τ(K_{i}, ℓ_{i}) + ∑_{i=m+1}^{r} ⌊\frac{K_{i}}{q + 1}⌋ ≤ max \{τ(K, ℓ) | min(ℓ_{1}, …, ℓ_{q}, q + 1) ≤ ℓ\} .

Proof. First we note that it is enough to prove

∑_{i=1}^{m} τ(K_{i}, ℓ_{i}) + ∑_{i=m+1}^{r} ⌊\frac{K_{i}}{q + 1}⌋ ≤ τ(K, ℓ)  (3)

with ℓ = max(ℓ_{1}, …, ℓ_{m}). Due to the fact that

⌊\frac{K_{i}}{q + 1}⌋ ≤ K_{i} − ⌊\frac{K_{i}}{2} ⌋ − 1 + γ(K_{i}, 2) = τ(K_{i}, 2) .

we may assume r = m (by setting l_{i} = 2 for m < i ≤ r). Then (3) reads ∑_{i=1}^{r} τ(K_{i}, ℓ_{i}) ≤ τ(K, ℓ) and it suffices to prove it for r = 2. We may also assume ℓ_{1} ≤ ℓ_{2} = ℓ. From

⌊\frac{K_{1}}{ℓ_{1}} ⌋ ≥ ⌊\frac{K_{1}}{ℓ} ⌋ + γ(K_{1}, ℓ_{1}) · min(1, ℓ − ℓ_{1})

and

⌊\frac{K_{1}}{ℓ} ⌋ + ⌊\frac{K_{2}}{ℓ} ⌋ ≥ ⌊\frac{K}{ℓ} ⌋ − 1 + γ(K_{1}, ℓ) + γ(K_{2}, ℓ) − γ(K_{1}, ℓ) · γ(K_{2}, ℓ)

we get

\begin{aligned}τ(K_{1}, ℓ_{1}) + τ(K_{2}, ℓ_{2}) & ≤ & K − ⌊\frac{K}{ℓ} ⌋ − ℓ + γ(K_{1}, ℓ) · γ(K_{2}, ℓ) + 1+ \\ & & +[2 − ℓ_{1}] + [γ(K_{1}, ℓ_{1}) − γ(K, ℓ) − γ(K_{1}, ℓ_{1}) · min(1, ℓ − ℓ_{1})]. \end{aligned}

Now, the assertion follows by observing that γ(K_{1}, ℓ) · γ(K_{2}, ℓ) ≤ γ(K, ℓ) and the terms in the square brackets are non-positive.

Proof of Theorem 3.1. Let (Z, O_{Z}) ⊂ Y be a zero-dimensional subscheme of length k + 1, defined by the ideal I_{Z}, and supported on the points y_{1}, …, y_{r}. Let k_{i} = length(O_{Z,y_{i}}). Given an element

g = (g_{1}, …, g_{r}) ∈ ⨁_{i=1}^{r} H^{0}(π^{∗}L ⊗ O_{Z,y_{i}}) = H^{0}(π^{∗}L ⊗ O_{Z}) ,

we will construct a section s ∈ H^{0}(Y, π^{∗}L) whose image in H^{0}(π^{∗}L ⊗ O_{Z}) is g.

The strategy is different to that in the proof of Theorem 2.1. Roughly speaking, for q ≥ 0, we will approximate s by building a sequence of affine subspaces

… ⊂ V_{2}^{q} ⊂ V_{1}^{q} ⊂ V_{0}^{q} ⊂ H^{0}(X, L − qM)

consisting of sections s_{q,i} such that the length of the subscheme of Z where ∑ t^{q}π^{∗}s_{q,i} mod I_{Z} agrees with g increases with i reaching length Z in the last step. First of all we take care of subschemes of Z supported in the ramification locus of π. Starting with a fiber containing most of the points y_{i} we deal then with subschemes of Z supported in regular fibers of π. Each step consists now of fixing sections in V_{i}^{q} for q appropriately big and imposing new conditions on sections in V_{i}^{q} for lower q thus defining subspaces V_{i+1}^{q}. Positivity assumptions of the theorem assure that we never run out of sections. Let us now turn to the details.

After reordering y_{1}, …, y_{r} we may assume that

y_{1}, …, y_{n_{1}−1} ∈ R
y_{n_{1}}, …, y_{n_{2}−1} ∈ π^{−1}(x_{1})
⋮
y_{n_{m}}, …, y_{n_{m+1}−1} = y_{r} ∈ π^{−1}(x_{m}) .

for distinct points x_{1}, …, x_{n} ∈ X \ B. Letting l_{i} = n_{i+1} − n_{i}, we may also assume that l_{1} ≤ … ≤ l_{m} and k_{n_{i}} = max(k_{n_{i}}, …, k_{n_{i+1}−1}). We will use the abbreviations I_{i} = I_{Z,y_{i}} for i = 1, …, r and K_{i} = k_{n_{1}} + … + k_{n_{i+1}−1} for i = 1, …, m.

Identifying the local rings of the points in the same fiber by means of π, we consider the ideals

J_{i} =_{def} I_{n_{i}+1} ∩ … ∩ I_{n_{i+1}−1} ⊂ O_{Y,y_{n_{i}}}.

and for i = 1, …, m we denote by

ρ_{i} : O_{Y,y_{n_{i}}}/(I_{n_{i}} ∩ J_{i}) ⟶ O_{Y,y_{n_{i}}}/J_{i}

and

η_{i} : O_{Y,y_{n_{i}}}/(I_{n_{i}} ∩ J_{i}) ⟶ O_{Y,y_{n_{i}}}/I_{n_{i}}

the quotient maps. Applying Lemma 3.2 we get the inequalities

length(O_{X,x_{i}}/J_{i}) ≤ τ(K_{i}, l_{i})  (4)

for all i ∈ {1, …, m} such that l_{i} ≥ 2.

Let y_{i} be one of the points y_{1}, …, y_{n_{1}−1} on the ramification divisor R and let W be the restriction of Z to y_{i}. Choosing coordinates u_{1}, …, u_{n} at the point y_{i} as in Case 3 of the proof of Theorem 2.1, we have a decomposition

I_{W_{i}} = ⨁_{q=0}^{d−1} u^{q}π^{−1}𝔞_{i,q} ,

where 𝔞_{i,q} are ideals in O_{X,π(y_{i})}, hence we get an isomorphism

O_{W_{i}} ≅ ⨁_{q=0}^{d−1} O_{X}/𝔞_{i,q}  (5)

Note that 𝔞_{i,q} ⊂ 𝔞_{i,q+1} for q = 0, …, d − 2, giving a cofiltration

O_{X}/𝔞_{i,0} ↠ O_{X}/𝔞_{i,1} ↠ … ↠ O_{X}/𝔞_{i,d−1} → 0 .

This implies length(O_{X}/𝔞_{i,q}) ≤ ⌊\frac{length(O_{W_{i}})}{q+1}⌋. Under the isomorphism (5) the element g_{i} corresponds to a d-tuple (g_{i,0}, …, g_{i,d−1}) for i = 1, …, n_{1} − 1. Since L − qM is σ(k, d, q)- very ample, the subspace

V_{0}^{q} = {s ∈ H^{0}(X, L − qM) | s mod 𝔞_{i,q} = g_{i,q} for i = 1, …, n_{1} − 1}

is at least σ_{0}(k, d, q)-very ample on X \ R, where

σ_{0}(k, d, q) = σ(k, d, q) − ⌊\frac{length(O_{W_{i}})}{q + 1}⌋

for q = 0, …, min(k, d − 1). For any section s of the form

s = π^{∗}s_{0} + tπ^{∗}s_{1} + … + t^{min(k,d−1)}π^{∗}s_{min(k,d−1)}

with s_{q} ∈ V_{0}^{q} we then have

s mod I_{i} = g_{i} in O_{Y,y_{i}}

for i = 1, …, n_{1} − 1.

From now on we proceed in m steps. The recursive procedure works so that the i-th step takes care of what we generate in the (m − i + 1)-st fiber.

Step 1. For all q ≥ l_{m} such that V_{0}^{q} ≠ H^{0}(X, L − qM) we fix a section s_{q} ∈ H^{0}(X, L − qM). For i = 1, …, l_{m} we denote by h_{i} the image

(φ^{β(n_{m},n_{m}+i−1)})^{∗}(∑_{q≤l_{m}} t^{q}π^{∗}(s_{q})) mod I_{n_{m}} ∩ … ∩ I_{n_{m+1}−1}

in the local ring O_{Y,y_{nm}}. We choose preimages ĝ_{i} of g_{n_{m}+i−1} in O_{Y,y_{nm}}/I_{n_{m}} ∩…∩I_{n_{m+1}−1}. For q ≤ l_{m} − 1 we let V_{1}^{q} be the subspace of V_{0}^{q} of sections s_{q} satisfying the following conditions in O_{Y,y_{nm}}:

t^{l_{m}−1}π^{∗}s_{l_{m}−1} mod J_{m} = α_{l_{m}−1,1}ρ_{m}(ĝ_{1} − h_{1}) + … + α_{l_{m}−1,l_{m}}ρ_{m}(ĝ_{l_{m}} − h_{l_{m}})
⋮
tπ^{∗}s_{1} mod J_{m} = α_{1,1}ρ_{m}(ĝ_{1} − h_{1}) + … + α_{1,l_{m}}ρ_{m}(ĝ_{l_{m}} − h_{l_{m}})
π^{∗}s_{0} mod I_{n_{m}} ∩ J_{m} = α_{0,1}(ĝ_{1} − h_{1}) + … + α_{0,l_{m}}(ĝ_{l_{m}} − h_{l_{m}}) ,

where α_{i,j} are solutions of the Vandermonde type systems of linear equations

α_{0,i} + … + α_{l_{m}−1,i} = δ_{1i}
α_{0,i} + ε^{β(n_{m},n_{m}+1)}α_{1,i} + … + ε^{β(n_{m},n_{m}+1)}α_{l_{m}−1,i} = δ_{2i}
⋮
α_{0,i} + ε^{(l_{m}−1)β(n_{m},n_{m+1}−1)}α_{1,i} + … + ε^{(l_{m}−1)β(n_{m},n_{m+1}−1)}α_{l_{m}−1,i} = δ_{l_{m}i} .

We observe that (4) implies that V_{1}^{q} remains at least σ_{1}(k, d, q)-very ample, where for q ≥ 1

σ_{1}(k, d, q) = σ_{0}(k, d, q) − τ(K_{m}, l_{m})

and for q = 0

σ_{1}(k, d, 0) = σ_{0}(k, d, 0) − ∑_{i=n_{m}}^{n_{m+1}−1} K_{i} = ∑_{i=1}^{n_{m}−1} k_{i}.

Let s be of the form π^{∗}s_{0} + … + t^{l}π^{∗}s_{l}, where s_{q} ∈ V_{1}^{q}, for q ≤ l_{m} − 1. Then in the local ring O_{Y,y_{nm+i−1}} we have

s mod I_{n_{m}+i−1} = (φ^{β(n_{m},n_{m}+i−1)})^{∗}s mod I_{n_{m}}
= π^{∗}s_{0} + ε^{β(n_{m},n_{m}+i−1)}tπ^{∗}s_{1} + … + ε^{(l_{m}−1)β(n_{m},n_{m}+i−1)}t^{l_{m}−1}π^{∗}s_{l_{m}−1} +

+(φ^{β(n_{m},n_{m}+i−1)})^{∗}(∑_{q≥l_{m}} s_{q}) mod I_{1} =

= α_{0,1}η_{m}(ĝ_{1} − h_{1}) + … + α_{0,l_{m}}η_{m}(ĝ_{l_{m}} − h_{l_{m}}) + … +

+ε^{β(n_{m},n_{m}+i−1)} (α_{1,1}η_{m}(ĝ_{1} − h_{1}) + … + α_{1,l_{m}}η_{m}(ĝ_{l_{m}} − h_{l_{m}})) +

⋮ +ε^{(l_{m}−1)β(n_{m},n_{m}+i−1)}(α_{l_{m}−1,1}η_{m}(ĝ_{1} − h_{1}) + … + α_{l_{m}−1,l_{m}}η_{m}(ĝ_{l_{m}} − h_{l_{m}}) + +η_{m}(h_{i}) =

= (α_{0,1} + ε^{β(n_{m},n_{m}+i−1)}α_{1,1} + … + ε^{(l_{m}−1)β(n_{m},n_{m}+i−1)}α_{l_{m}−1,1})η_{m}(ĝ_{1} − h_{1}) +

⋮ +(α_{0,l_{m}} + ε^{β(n_{m},n_{m}+i−1)}α_{1,l_{m}} + … + ε^{(l_{m}−1)β(n_{m},n_{m}+i−1)}α_{l_{m}−1,l_{m}})η_{m}(ĝ_{l_{m}} − h_{l_{m}}) + +η_{m}(h_{i})

= g_{n_{m}+i−1}

Step i. Proceeding as in Step 1, we fix sections s_{q} ∈ V_{i−1}^{q} for l_{m−i+1} ≤ q < l_{m−i+2} and we construct subspaces

V_{i}^{q} ⊂ V_{i−1}^{q}

for q < l_{m−i+1}, such that for all s_{q} ∈ V_{i}^{q} one has

∑_{q=0}^{l} t^{q}π^{∗}s_{q} mod I_{j} = g_{j}

for all n_{m−i+1} ≤ j < n_{m−i+2}. Then V_{i}^{q} is still σ_{i}(k, d, q)-very ample, where by (4)

σ_{i}(k, d, q) = σ_{i−1}(k, d, q) − τ(K_{i}, l_{i}) = σ_{0}(k, d, q) − ∑_{ν=i+1}^{m} τ(K_{ν}, l_{ν})

for q ≥ 1 and

σ_{i}(k, d, 0) = σ_{0}(k, d, 0) − K_{i} = ∑_{ν=1}^{n_{m−i+1}−1} k_{ν}.

After m − 1 steps, by Lemma 3.3, the space V_{m−1}^{q} will still be σ_{m−1}(k, d, q)-very ample with

σ_{m−1}(k, d, q) ≥ τ(K_{1}, l_{1})

for 1 ≤ q < l_{1} and

σ_{m−1}(k, d, 0) ≥ K_{1}.

This guarantees that we can find sections s_{1}, …, s_{l_{1}} such that

s =_{def} ∑_{q=0}^{l} t^{q}π^{∗}s_{q}

satisfies

s mod I_{Z} = g.

This completes the proof of the theorem.

As in section 2 we get immediately the same result for unbranched coverings.

Theorem 3.4 Let X be a smooth projective variety. Let M be a line bundle on X such that O_{X}(dM) ≅ O_{X} and let π : Y ⟶ X be the cyclic covering of degree d defined by M. Let L be a line bundle on X and k a non-negative integer. If L − qM is σ(k, d, q)-very ample for q = 0, …, min(k, d − 1) then π^{∗}L is k-very ample.

Remark 3.5 In order to convey some feeling for the numbers σ(k, d, q) appearing in the hypotheses of Theorem 3.1, we include a table for d = 15 and 0 ≤ k ≤ 15. (The k-th column lists the values σ(k, 15, q) for 1 ≤ q ≤ k.) For instance π^{∗}L is 2-very ample if L is so and both L − M and L − 2M are globally generated, and it is 4-very ample if L is so and L − M and L − 2M are very ample and L − 3M and L − 4M are globally generated.

k  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
L − 1M  0 0 1 1 2 2 3 4 4 5 6 6 7 8 9
L − 2M  0 0 1 2 2 3 4 4 5 6 6 7 8 9
L − 3M  0 0 1 2 3 3 4 5 6 6 7 8 9
L − 4M  0 0 1 2 3 4 4 5 6 7 8 8
L − 5M  0 0 1 2 3 4 5 5 6 7 8
L − 6M  0 0 1 2 3 4 5 6 6 7
L − 7M  0 0 1 2 3 4 5 6 7
L − 8M  0 0 1 2 3 4 5 6
L − 9M  0 0 1 2 3 4 5
L − 10M  0 0 1 2 3 4
L − 11M  0 0 1 2 3
L − 12M  0 0 1 2
L − 13M  0 0 1
L − 14M  0 0

4 Examples and applications

Theorems 2.1 and 3.1 state, roughly speaking, that under suitable assumptions on the bundles L − qM the positivity of a pullback π^{∗}L is at least as high as the positivity of L. It is then natural to ask whether the converse statement is also true. The following example shows that this is not the case:

Example 4.1 Let (X, L) be a principally polarized abelian variety. A d-torsion point M ∈ Pic^{0}(X) give rise to a cyclic covering π : Y ⟶ X such that (Y, π^{∗}L) is an abelian variety of type (1, …, 1, d). If d is sufficiently large, then π^{∗}L will be very ample. (One knows by work of Debarre-Hulek-Spandaw [11] that it is enough to take d > 2^{g}.) But L is not even globally generated. In the surface case, where the generation of jets and the k−very ampleness is well understood (see [2]), one can even give explicit values d_{k} (resp. d′_{k}) for every k > 0 such that π^{∗}L is k-jet ample (resp. k−very ample) whenever d ≥ d_{k} (resp. d ≥ d′_{k}).

The previous example shows that the positivity can very well increase after taking a pullback. In general, however, it does not need to increase at all, as the following example shows:

Example 4.2 Consider a product of elliptic curves X = E_{1} × … × E_{n} and the product polarization

L = O_{X} (∑_{i=1}^{n} pr_{i}^{∗}(0)) ,

where pr_{i} is the i-th projection. The choice of a d-torsion point on E_{n} determines a covering Ẽ_{n} ⟶ E_{n} which in turn induces a cyclic covering Y = E_{1}×…×E_{n−1}×Ẽ_{n} ⟶ X such that

π^{∗}L = O_{Y} (∑_{i=1}^{n−1} pr_{i}^{∗}(0) + d · pr_{n}^{∗}(0)) .

The line bundles O_{X}((k + 2)L) and O_{Y} ((k + 2)π^{∗}L) are then k-jet ample (by [1]), but neither of them is ℓ-jet ample (or even ℓ-very ample) for any ℓ > k, since the restricted bundles O_{X}((k+2)L)|E_{n} and O_{Y} ((k+2)π^{∗}L)|Ẽ_{n} are only of degree k+2 (cf. [8, Proposition 2.1]).

Even if L is positive, the positivity of π^{∗}L may drop if the assumptions on the line bundles L − qM are not satisfied. The following example illustrates their importance.

Example 4.3 Let X ≅ IP^{n} and let r and d be integers greater or equal 2. Let M = O_{IP}^{n}(r) and let B be a smooth member of |dM|. Let π : Y ⟶ X be the cyclic covering defined by B and let L = O_{IP}^{n}((d − 1)r). Then L is d−jet ample but π^{∗}L fails to be that positive. The reason is that L−qM fails to be (d−q)−jet ample for q = d−1. Indeed, as in the proof of the case 3 in Theorem 2.1 let y ∈ R be a point on the ramification divisor R ⊂ Y and u_{1}, …, u_{n} be local coordinates such that R is locally defined by u_{1} = 0. From the decomposition

H^{0}(π^{∗}L) ≅ H^{0}(O_{IP}^{n}((d − 1)r)) ⊕ u_{1} · H^{0}(O_{IP}^{n}((d − 2)r)) ⊕ … ⊕ u_{1}^{d−1} · H^{0}(O_{IP}^{n})

we infer that no section of π^{∗}L generates a jet of the form u_{1}^{d−1}u_{i} for i = 2, …, n. In fact Theorem 2.1 implies that π^{∗}L is (d − 1)− jet ample.

Remark 4.4 More generally, in the case of branched coverings, using the decomposition (1) one can show as in the previous example that if L − qM fails to generate (k − q)−jets at the points of the branched locus, π^{∗}L fails to be k−jet ample.

Now, we want to show that the assumptions of Theorem 3.1 cannot be weakened.

Example 4.5 (Geiser involution) Let B ⊂ X ≅ IP^{2} be a smooth plane quartic and let π : Y ⟶ X be the double covering branched over B. Then Y is the Del Pezzo surface of degree 2 and −K_{Y} ≅ π^{∗}O_{IP}^{2}(1), (see [6, 10.2.4]). Theorem 3.1 states that −kK_{Y} is k−very ample for k ≥ 2. This reproves a result of [14]. Moreover, in [14] it was shown that −kK_{Y} is not (k + 1)−very ample. This implies that our theorem is sharp.

Finally, we show how our results can be applied in certain situations to study higher order embeddings of blown up varieties.

Example 4.6 (Bertini involution) Let S be the Del Pezzo surface of degree 1. Then | − K_{S}| has a base point x and | − 2K_{S}| is globally generated and defines a 2 : 1 mapping p : S ⟶ Q onto a quadric cone Q ⊂ IP^{3}. It is known (cf. [6, 10.4.3]) that p is branched over a smooth curve and the vertex v = p(x) of Q. Let τ : IF_{2} ⟶ Q be the blowing up of Q at v with the exceptional divisor D and let σ : X ⟶ S be the blowing up of S at x with the exceptional divisor E. Then the following diagram

\begin{matrix}X & ⟶^{π} & IF_{2} \\ ↓_{σ} & & ↓ τ \\ S & ⟶^{p} & Q \end{matrix}

is commutative and π is a double covering branched over a smooth divisor homologous to 2(2D + 3f), where f denotes the pull-back of the ruling on Q by τ. It is interesting to investigate the positivity of the line bundles of the form −sK_{X} − tE, for s, t > 0. Notice that for s, t ≤ 0 the line bundle is not even ample. We observe that −sK_{X} − tE = sσ^{∗}(−K_{S}) − (s + t)E but since the jet ampleness of −sK_{S} is not known for we cannot apply the results from [10] directly. On the other hand −sK_{X} − tE = sπ^{∗}f − tπ^{−1}(D). This suggests to look for the positivity of the line bundles coming from IF_{2} via π. We recall that a line bundle L = aD + bf is k−jet ample on IF_{2} (equivalently k−very ample, cf. e.g. [15], [8]) if and only if L.D = −2a + b ≥ k and L.f = a ≥ k. Then Theorem 2.1 implies that π^{∗}L = −bK_{X} − 2aE is k−jet ample if a ≥ k + 1 and b ≥ 3k and Theorem 3.1 asserts that π^{∗}L is k−very ample for a ≥ ⌊\frac{k−1}{2}⌋ + 2 and b ≥ 3 ⌊\frac{k−1}{2}⌋ + 3.

References

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[14] Di Rocco S.: k-very ample line bundles on Del Pezzo surfaces. Math. Nachr, 179 (1996), 47–56.

[15] Di Rocco S.: Generation of k-jets on toric varieties. Math Z. (to appear)

[16] Geramita, A. V., Gimigliano G., and Harbourne B.: Projectively normal but superabundant embeddings of rational surfaces in projective space. J. of Algebra 169, 791-804, (1994).

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Thomas Bauer, Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraße 1\frac{1}{2}, D-91054 Erlangen, Germany

E-mail: bauerth@mi.uni-erlangen.de

Sandra Di Rocco, KTH, Department of Mathematics, S-100 44 Stockholm, Sweden

E-mail: sandra@math.kth.se

Current address: Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, D-53225 Bonn, Germany Tomasz Szemberg, Instytut Matematyczny PAN, c/o Instytut Matematyki, Uniwersytet Jagielloński, Reymonta 4, PL-30-059 Kraków, Poland

E-mail: szemberg@im.uj.edu.pl