Sharp Matsusaka-type theorems on surfaces^{∗}

M.C. Beltrametti, A.J. Sommese

Abstract

Let L be an ample line bundle on a smooth surface S. We give sharp lower bounds for tL to be k-jet ample based on the nefvalue of the pair (S, L). We also give sharp lower bounds for tL to be k-spanned.

Introduction

Let L be an ample line bundle on a smooth projective surface S.

In §2 of these notes we restate Fernández Del Busto’s Matsusaka-type theorem from [8] using the notion of nefvalue of the pair (S, L). As a consequence we can use standard adjunction theory results to give sharp and stronger forms of the results in [8].

In §3 we prove a Matsusaka-type theorem for line bundles on surfaces (see Theorem (3.3)). We extend the arguments of Fernández Del Busto [8] to cover the case of k-spanned line bundles, and make some improvements that give better estimates even in the k-jet ample case studied in [8].

In the case of k-spannedness we get a numerical bound for tL to be k-spanned, t positive integer, which is linear in k, while the corresponding bound in [8] for tL to be k-jet ample is quadratic in k (see Theorem (3.3)).

The notions of k-spannedness, k-very ampleness and k-jet ampleness are k-th order embedding notions, which have been extensively studied in the last few years (see (1.2) and also §§8.5, 10.7 of the book [6] for more references on this topics).

In §1 we recall some background material we need.

The second author thanks the National Science Foundation (DMS 93-02121) for their support.

1 Background material

1.1 Notation. We work over the complex field ℂ. By variety we mean an irreducible and reduced projective scheme, V. We denote the structure sheaf by O_{V}.

Basically we use the standard notation from algebraic geometry. Let us only fix the following.

h^{i}(F), the complex dimension of H^{i}(V, F), for a coherent sheaf F on V, i ≥ 0; Γ(L), the space of the global sections of a line bundle, L, on a variety V; we say that L is spanned if it is spanned at all points of V by Γ(L);

∗1991 Mathematics Subject Classification. Primary 14J05, 14E25; Secondary 14C20 .

Keywords and phrases. Surface (smooth complex projective), line bundle, k-spannedness,

k-very ampleness, k-jet ampleness, adjunction theory.

|L|, the complete linear system associated to a line bundle L;

≈ (respectively ∼), the linear (respectively numerical) equivalence of divisors;

V^{[r]} = the Hilbert scheme of all 0-dimensional schemes (Z, O_{Z}) of V with length(O_{Z}) = r. Since we are working in characteristic zero, length(O_{Z}) = h^{0}(O_{Z}).

For a divisor D ∈ Pic(V) ⊗ ℚ we denote

{D}, the integral part and ⌈D⌉ = −{−D}, the rounding up.

Line bundles and divisors are used with little (or no) distinction. We almost always use the additive notation.

1.2 k-th order embeddings. (See [3], [5], [4]) Fix a nonnegative integer k. Let L be a line bundle on a smooth projective variety X of dimension n. We say that L is k-very ample if the restriction map Γ(L) → Γ(O_{Z}(L)) is onto for any 0-cycle (Z, O_{Z}) ∈ X^{[k+1]}.

We say that L is k-spanned if Γ(L) → Γ(O_{Z}(L)) is onto for any curvilinear 0-cycle (Z, O_{Z}) ∈ X^{[k+1]}, i.e., length(O_{Z}) = k + 1 and Z is locally contained in a smooth curve. More explicitly, if Supp(Z) = {x_{1}, …, x_{r}}, x_{1}, …, x_{r} distinct points, then Z is defined by the ideal sheaf J_{Z} such that J_{Z}O_{X,x} is isomorphic to O_{X,x} if x ∉ {x_{1}, …, x_{r}} and J_{Z}O_{X,x_{i}} is generated by (u_{i1}, …, u_{in−1}, u_{in}^{k_{i}}) at x_{i}, where (u_{i1}, …, u_{in}) are local coordinates at x_{i} on X, with ∑_{i=1}^{r} k_{i} = k + 1. We also say in this case that k_{i} is the local length of Z at x_{i}, i = 1, …, r.

Let x_{1}, …, x_{r} be r distinct points on X. Let 𝔪_{i} be the maximal ideal sheaves of the points x_{i}, i = 1, …, r. Consider the 0-cycle Z = k_{1}x_{1} +⋯+k_{r}x_{r}, ∑_{i=1}^{r} k_{i} = k+1, defined by the ideal sheaf J_{Z} such that J_{Z}O_{X,x} is isomorphic to O_{X,x} if x ∉ {x_{1}, …, x_{r}} and J_{Z}O_{X,x_{i}} is isomorphic to the k_{i}-th power of the maximal ideal 𝔪_{i}O_{X,x_{i}} of the local ring O_{X,x_{i}}, i = 1, …, r. We say that L is k-jet ample if the restriction map Γ(L) → Γ(O_{Z}(L)) is onto for any such 0-cycle (Z, O_{Z}).

Note that for k = 0, the conditions L is 0-very ample, L is 0-spanned, L is 0-jet ample are all equivalent to L being spanned by Γ(L). Furthermore for k = 1, the conditions L is 1-very ample, L is 1-spanned, L is 1-jet ample are all equivalent to L being very ample. Note also that k-jet ampleness implies k-very ampleness (see [5, Prop. 2.2]) and, of course, k-very ampleness implies k-spannedness (in fact, the notions of k-very ampleness and k-spannedness coincide for k ≤ 2).

Let S be a smooth projective surface and let B be a big divisor on S, i.e., the Kodaira dimension κ(B) = 2. For n ≫ 0, let nB = M_{n} + F_{n} be the decomposition of nB in its moving part M_{n} and its fixed part F_{n}. In particular M_{n} is nef and big and h^{0}(M_{n}) = h^{0}(nB). One has the following result (see [8], [11]).

Lemma 1.3 (Ein-Lazarsfeld-Nakamaye) Notation as above. Assume that for n ≫ 0 there

is a constant β > 0 with h^{0}(nB) ≥ \frac{n^{2}}{2} β + O(n). Then, for n ≫ 0,

M_{n}^{2} ≥ n^{2}β + O(n).

Furthermore, let N be any nef and big divisor on S. Then

0 ≤ \frac{1}{n}F_{n} · N ≤ B · N − \sqrt{β}\sqrt{N^{2}}.

By a real divisor we mean an element of Pic(S) ⊗_{ℤ} ℝ. By a nef real divisor R we mean and element R ∈ Pic(S) ⊗_{ℤ} ℝ such that R · C ≥ 0 for all effective curves C on S. We will also use the following consequence of the Riemann-Roch theorem (see [8, Lemma 1]).

Lemma 1.4 Let E be a Cartier divisor on a smooth projective surface S. Assume that as an element of Pic(S) ⊗_{ℤ} ℝ, we have E = D − R where R is a nef real divisor and D is a positive real multiple of an ample Cartier divisor A. If D^{2} − 2D · R > 0, then for all n ≫ 0 we have

h^{0}(nE) ≥ \frac{n^{2}}{2} (D^{2} − 2D · R) + O(n).

Proof. Note that we have A · E = A · D − A · R > A · R ≥ 0. Thus we conclude that h^{2}(nE) = h^{0}(K_{S} − nE) = 0 for all n ≫ 0. Therefore from the Riemann-Roch theorem we have for all n ≫ 0 that

h^{0}(nE) ≥ E^{2} \frac{n^{2}}{2} + O(n).

Now note that E^{2} = D · (D − 2R) + R^{2} ≥ D · (D − 2R). Q.E.D.

Definition 1.5 Let A be an ample Cartier divisor on a smooth surface S. The nefvalue of the pair (S, A) is be the real number τ defined by

τ = inf{r ∈ ℚ, K_{S} + rA is nef}.

2 A sharp Matsusaka-type theorem for k-jet ampleness

The following is a simple consequence of Fernández Del Busto’s theorem, but by acknowledging directly the adjunction theory [6], we can use known adjunction theory results to obtain considerably sharper lower bounds for k-jet ampleness.

Theorem 2.1 Let A be an ample divisor on a nonsingular projective surface S. Let τ be the nefvalue of the pair (S, A), or any real value greater than the nefvalue of the pair (S, A). If

t + τ > \frac{1}{2} (\frac{(A · (K_{S} + τA) + 1)^{2}}{A^{2}} + k^{2} + 4k + 6)

then tA is k-jet ample.

Proof. This follows from [8, Theorem*, page 520]. We need only show that

\frac{1}{2} (\frac{(A · (K_{S} + τA) + 1)^{2}}{A^{2}} + k^{2} + 4k + 6 − 2τ) > \frac{1}{2} (\frac{(A · K_{S} + 1)^{2}}{A^{2}} + k^{2} + 4k + 6 − K_{S}^{2})

or equivalently that

(A · (K_{S} + τA) + 1)^{2} > (A · K_{S} + 1)^{2} + 2τA^{2} − A^{2}K_{S}^{2}.

Expanding and simplifying, we see that this is equivalent to A^{2}(K_{S} + τA)^{2} ≥ 0, which is the case since A is ample and K_{S} + τA is nef. Q.E.D.

Corollary 2.2 Let A be an ample divisor on a nonsingular projective surface S. Let τ be the nefvalue of the pair (S, A), or any real value greater than the nefvalue of the pair (S, A). If

t + τ > \frac{1}{2} (\frac{(A · (K_{S} + τA) + 1)^{2}}{A^{2}} + 11)

then tA is very ample.

Adjunction theory gives much information on pairs in terms of their nefvalues. Using the classification of Lanteri and Palleschi [10] (see also [12, Corollary 3.4.2]), we see that either τ ≤ 2 or (S, A) is (ℙ^{2}, O_{ℙ^{2}}(1)), in which case the estimate is clear.

Corollary 2.3 Let A be an ample divisor on a nonsingular projective surface. If

t > \frac{1}{2} (\frac{(A · (K_{S} + 2A) + 1)^{2}}{A^{2}} + k^{2} + 4k + 2)

then tA is k-jet ample.

Corollary 2.4 Let A be an ample divisor on a nonsingular projective surface. If

t > \frac{1}{2} (\frac{(A · (K_{S} + 2A) + 1)^{2}}{A^{2}} + 7)

then tA is very ample.

Using again the classification of Lanteri and Palleschi [10] (see also [12, Corollary 3.4.2]), we see that either τ ≤ 1 or S is ℙ^{2}, or a smooth quadric, in which case the estimate is clear, or (S, A) is a scroll.

Corollary 2.5 Let A be an ample divisor on a nonsingular projective surface, and assume that (S, A) is not a scroll (neither ℙ^{2} nor a quadric). If

t > \frac{1}{2} (\frac{(A · (K_{S} + A) + 1)^{2}}{A^{2}} + k^{2} + 4k + 4)

then tA is k-jet ample.

Corollary 2.6 Let A be an ample divisor on a nonsingular projective surface, and assume that (S, A) is not a scroll. If

t > \frac{1}{2} (\frac{(A · (K_{S} + A) + 1)^{2}}{A^{2}} + 9)

then tA is very ample.

Remark 2.7 The above result is sharp, i.e, the assumption that (S, A) is not a scroll is essential. Let S = ℙ(E ⊕ L) where E is a degree 1 line bundle on a smooth genus g curve C and L is a degree δ line bundle on C. Let A denote the tautological line bundle on S. If the corollary was true for scrolls then by restricting to the section corresponding to the quotient morphism E ⊕ L → E → 0 we would have that g equals the sectional genus

g(A) = \frac{(K_{S} + A) · A}{2} + 1 and hence that tE is very ample for

t > \frac{1}{2} (\frac{(2g − 1)^{2}}{2δ} + 9) .

By choosing δ ≫ 0 we would obtain the absurdity that given any degree 1 line bundle E on an arbitrary curve C, we would have 5E is very ample.

Remark 2.8 It is a theorem of the authors [5] that if a line bundle A is 1-jet ample, then kA is k-jet ample. Asymptotically this implies that if A is ample, then there is a t_{0} > 0 such that kt_{0}A is k-jet ample. It is therefore natural to ask the following question.

Question 2.9 Let t(k) be the minimum k such that t(k)A is k-jet ample. Then compute

the lim inf and the lim sup of \frac{t(k)}{k} as k → ∞.

3 A Matsusaka-type theorem for k-spannedness

In what follows (Z, O_{Z}) will denote a curvilinear 0-cycle of length k + 1 on a smooth projective surface S, and L an ample line bundle on S.

3.1 Blowing up of curvilinear 0-cycles. Here we only carry out from [2, §2] the case when Supp(Z) is a single point z. The discussion in the case Supp(Z) = {z_{1}, …, z_{r}}, 1 ≤ r ≤ k + 1, is a straightforward modification of it.

Let π_{α} : S_{α} → S_{α−1} be the blowing up of S_{α−1} at a point p_{α−1}, E_{α} = π_{α}^{−1}(p_{α−1}) the exceptional divisors, α = 1, …, k + 1, \overline{E}_{α} the proper transforms of E_{α} under π_{k+1} ∘ ⋯ ∘ π_{α+1}, α = 1, …, k, and \overline{E}_{k+1} = E_{k+1}. We have

\overline{E}_{α} · \overline{E}_{β} = 1 if |α − β| = 1; \overline{E}_{α} · \overline{E}_{β} = 0 if |α − β| > 1;

and

\overline{E}_{α} · \overline{E}_{α} = −2 if α ≠ k + 1; \overline{E}_{k+1} · \overline{E}_{k+1} = −1.

Let \overline{S} = S_{k+1}, S_{0} = S, p_{0} = z and denote by π : \overline{S} → S the composition of the π_{α}’s. Let J_{Z} be the ideal sheaf of Z in S. By a suitable choice of the points p_{α}’s it is easy to check that

π^{∗}J_{Z} ≈ −\overline{E}_{1} − 2\overline{E}_{2} − ⋯ − (k + 1)\overline{E}_{k+1},

and

K\overline{S} ≈ π^{∗}K_{S} + \overline{E}_{1} + 2\overline{E}_{2} + ⋯ + (k + 1)\overline{E}_{k+1}.

In the general case, let k_{i} be the length of Z at x_{i}, i = 1, …, t. That is J_{Z}O_{S,x_{i}} is generated by (u_{i}, v_{i}^{k_{i}}), where u_{i}, v_{i} are local coordinates at x_{i} on S, i = 1, …, t. Then the exceptional divisor of the blowing up along Z is \overline{E} = ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} α\overline{E}_{α}^{(i)}, where the \overline{E}_{α}^{(i)}’s are the proper transforms of the exceptional divisors E_{α}^{(i)}’s constructed as above and corresponding to each point z_{i}, i = 1, …, r.

We need the following improvement of [8, Lemma 3]. We will use the notation β̃(t) to keep the analogy with [8] clear.

Lemma 3.2 Let t, d, d_{1}, σ be integers and let x be a real number. Assume that d ≥ 1, t ≥ 0, and σ ≥ 4 and define β̃(t) := (t + x) ((t − x)d − 2d_{1}). If

t > \frac{1}{2} (\frac{(d_{1} + xd + 1)^{2}}{d} + σ − 2x)

then

1. β̃(t) > σ; and

2. td − d_{1} − \sqrt{d (β̃(t) − σ)} < 1.

Proof. Note that using the hypothesized lower bound we have

t + x > σ/2 ≥ 2.  (1)

Assume that β̃(t) ≤ σ, i.e., that (t+x) ((t − x)d − 2d_{1}) ≤ σ. Using the hypothesized lower bound for t we have:

(t + x) (\frac{1}{2} [(d_{1} + xd + 1)^{2} + dσ − 2dx] − xd − 2d_{1}) ≤ σ.

Noting that

\frac{1}{2} [(d_{1} + xd + 1)^{2} + dσ − 2dx] − xd − 2d_{1} = \frac{1}{2} [(−d_{1} − xd + 1)^{2} + dσ]

we have

\frac{(t + x)dσ}{2} ≤ (t + x) (\frac{1}{2} [(−d_{1} − xd + 1)^{2} + dσ]) ≤ σ.

Thus we must have d(t + x) ≤ 2, which contradicts equation (1). Therefore we have that β̃(t) > σ.

Now assume that the second inequality we want to prove is false, i.e., that td − d_{1} −

\sqrt{d (β̃(t) − σ)} ≥ 1. We then have

td − d_{1} − 1 ≥ \sqrt{d (β̃(t) − σ)}.

Squaring both sides and substituting for β̃(t) we have

(td − d_{1} − 1)^{2} ≥ d ((t + x) ((t − x)d − 2d_{1}) − σ) .

Simplifying and dividing by 2d we have

t ≤ \frac{1}{2} (\frac{(d_{1} + xd + 1)^{2}}{d} + σ − 2x)

which contradicts the hypothesized lower bound for t. Q.E.D.

We can prove now an effective version of Matsusaka’s Big Theorem in case of kspannedness. The proof of the theorem below is due to Fernández Del Busto. The argument we give is a modification of his proof. The difference is that we deal with curvilinear 0-cycles of given length. This allows us to obtain a linear bound in k instead of a quadratic bound as in the corresponding result for k-jet ampleness proved in [8] (compare with [8, §1] and Corollary (3.4) below).

Theorem 3.3 Let L be an ample line bundle on a nonsingular projective surface S. Let τ be the nefvalue of the pair (S, L) or any real value grater than the nefvalue of the pair (S, L).If

t + τ > \frac{1}{2} (\frac{(L · (K_{S} + τL) + 1)^{2}}{L^{2}} ) + 2k + 3,

then tL is k-spanned.

Proof. Through the proof we will use Lemma (3.2) with d := L^{2}, d_{1} := L · K_{S} and x = τ. Let (Z, O_{Z}) be a curvilinear 0-cycle on S of length k + 1, with Supp(Z) = {z_{1}, …, z_{r}}, and let k_{1}, …, k_{r} be r positive integers such that ∑_{i=1}^{r} k_{i} = k + 1, with k_{i} the local length of Z at z_{i}, i = 1, …, r, as in (1.2). Let J_{Z} be the ideal sheaf of Z in S. Let π be the blowing up of S along the 0-cycle Z. Then to show that tL is k-spanned, it suffices to show that, with the same notation as in (3.1), and by using Leray’s spectral sequence,

H^{1}(S, tL ⊗ J_{Z}) ≅ H^{1}(\overline{S}, π^{∗}(tL) − ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} α\overline{E}_{α}^{(i)}) = (0).  (2)

Since K_{S} + τL is nef, we can write tL − K_{S}, t a positive integer, as the difference of an ample real divisor and a nef real divisor

tL − K_{S} = (t + τ)L − (K_{S} + τL).  (3)

Consider the divisor

B := π^{∗}(tL − K_{S}) − 2 ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} α\overline{E}_{α}^{(i)}.

For a positive integer n, consider the exact sequence

0 → nB → π^{∗}(n(tL − K_{S})) → Q → 0,

where Q denotes the quotient sheaf. Since

h^{0}(Q) = ∑_{i=1}^{r} h^{0}(O_{S}/(x_{i}, y_{i}^{k_{i}})^{2n}) = ∑_{i=1}^{r} ℓ(O_{S}/(x_{i}, y_{i}^{k_{i}})^{2n}),

we have that

h^{0}(nB) ≥ h^{0}(n(tL − K_{S})) − ∑_{i=1}^{r} ℓ(O_{S}/(x_{i}, y_{i}^{k_{i}})^{2n}),  (4)

where (x_{i}, y_{i}) are local coordinates at the points z_{i}, i = 1, …, r. Now, let σ := 4(k + 1) and let β̃(t) be as in (3.2). Let us assume

t > \frac{1}{2} (\frac{(L · (K_{S} + τL) + 1)^{2}}{L^{2}} + 4(k + 1) − 2τ) .  (5)

Thus by Lemma (3.2) we conclude that

β̃(t) − 4(k + 1) > 0.  (6)

Therefore, recalling (3), Lemma (1.4) applies (with D = (t + τ)L, R = K_{S} + τL, so that β̃(t) = D · (D − 2R)) to say that, for n ≫ 0,

h^{0}(n(tL − K_{S})) ≥ \frac{n^{2}}{2} β̃(t) + O(n).

On the other hand,

ℓ(O_{S}/(x_{i}, y_{i}^{k_{i}})^{2n}) = k_{i} \frac{2n(2n + 1)}{2} = k_{i}n(2n + 1), i = 1, …, r.

Thus we infer from (4) that, for n ≫ 0,

h^{0}(nB) ≥ \frac{n^{2}}{2} β̃(t) − n(2n + 1) ∑_{i=1}^{r} k_{i} + O(n)

= \frac{n^{2}}{2} β̃(t) − n(2n + 1)(k + 1) + O(n)
= \frac{n^{2}}{2} (β̃(t) − 4(k + 1)) + O(n)

Thus, recalling (6), we conclude that B is big.

Let nB = M_{n} + F_{n} be the decomposition of nB in its moving part M_{n} and its fixed part F_{n}. Then by Lemma (1.3) we have

\frac{1}{n}F_{n} · π^{∗}L ≤ B · π^{∗}L − \sqrt{\overline{β(t)} − 4(k + 1)}\sqrt{(π^{∗}L)^{2}}
= (tL − K_{S}) · L − \sqrt{\overline{β(t)} − 4(k + 1)}\sqrt{L^{2}}.

Therefore, by Lemma (3.2),

\frac{1}{n}F_{n} · π^{∗}L < 1

and hence the irreducible components of the divisor \{\frac{1}{n}F_{n}\} are exceptional, that is

\{\frac{1}{n}F_{n}\} = ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} η_{α}\overline{E}_{α}^{(i)}, η_{α} ≥ 0.  (7)

Fix n ≫ 0. By Bertini’s theorem we can choose a general divisor D in |nB| such that if D = M_{n} + F_{n}, with F_{n} the fixed part of nB, then M_{n} is reduced. Consider the divisor

D := π^{∗}((t + 1)L − K_{S}) − 2 ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} α\overline{E}_{α}^{(i)} − \frac{1}{n}D.

Since nD = nπ^{∗}L, the divisor D is numerically equivalent to π^{∗}L and hence D is nef and

big. Since M_{n} is reduced, one has \{\frac{M_{n}}{n} \} = 0, so that \{\frac{D}{n} \} = \{\frac{F_{n}}{n} \}, and therefore

⌈\frac{−D}{n} ⌉ = − \{\frac{D}{n} \} = − ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} η_{α}\overline{E}_{α}^{(i)}.

By using (7), the Kawamata-Viehweg vanishing theorem thus implies the vanishing of the higher cohomology groups of the divisor

F := K\overline{S} + ⌈π^{∗}((t + 1)L − K_{S}) − 2 ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} α\overline{E}_{α}^{(i)} − \frac{D}{n} ⌉  (8)

= (t + 1)π^{∗}L − ∑_{i=1}^{r} ∑_{α=1}^{k_{i}}(α + η_{α})\overline{E}_{α}^{(i)}.  (9)

In particular H^{1}(\overline{S}, F) = (0). This implies that

H^{1}(\overline{S}, (t + 1)π^{∗}L − ∑_{i=1}^{r} ∑_{α=1}^{k_{i}} α\overline{E}_{α}^{(i)} = (0)

and hence that (t + 1)L is k-spanned. Recalling the assumption (5), we are done. Q.E.D.

For k = 1, the result gives a slightly sharper result for very ampleness than the result of the last section (compare also with [7, (13.10)]).

Corollary 3.4 Let L be an ample line bundle on a nonsingular projective surface S, then

tL is very ample if t + τ > \frac{(L · (K_{S} + τL) + 1)^{2}}{2L^{2}} + 5.

Proof. Note that τ ≤ 3 (see e.g., [10]). Then use (3.3) with k = 1. Q.E.D.

In particular using adjunction theory as before (to obtain Corollaries (2.4), (2.6)) we have the following.

Corollary 3.5 Let L be an ample line bundle on a nonsingular projective surface S. Then

1. tL is very ample if t > \frac{(L · (K_{S} + 2L) + 1)^{2}}{2L^{2}} + 3;

2. If (X, L) is not a scroll, then tL is very ample if t > \frac{(L · (K_{S} + L) + 1)^{2}}{2L^{2}} + 4.

Remark 3.6 (The Del Pezzo surface case (cf., [2, (2.6)] and [5, (5.2)]) Let (S, L) be a Del Pezzo surface, i.e., L = −K_{S} is ample and τ = 1 is the nefvalue. The result (2.6) of [2] gives in the worst case that −tK_{S} is k-spanned if t ≥ k + 2. Theorem (3.3) gives

• Assume t > \frac{1}{2} (\frac{1}{K_{S}^{2}} + 4k + 4), i.e., t ≥ 2k + 3. Then −tK_{S} is k-spanned.

4 An asymptotic result for k-very ampleness

In this section we show an asymptotic result for k-very ampleness and a few consequences of it.

Theorem 4.1 Let L be an ample line bundle on a smooth surface S. Let k_{0} be an integer such that A := (k_{0} − 1)L − K_{S} is nef and big and such that A · L ≥ 2. Then for t ≥ k + k_{0}, tL is k-very ample and therefore k-spanned.

Proof. Note that

((k + k_{0})L − K_{S})^{2} = ((k + 1)L + A)^{2}  (10)
≥ (k + 1)^{2} + 4(k + 1) + 1 ≥ k^{2} + 6k + 6.

Using the main theorem of [4] it follows that tL = K_{S} + (tL − K_{S}) is k-very ample if tL − K_{S} is nef and big; (tL − K_{S})^{2} ≥ 4k + 5; and there exist no effective divisors C on S with

(tL − K_{S}) · C − k − 1 ≤ C^{2} < \frac{(tL − K_{S}) · C}{2} < k + 1.  (11)

For t ≥ k+k_{0} we have tL−K_{S} = (t−k_{0} +1)L+A is ample and (tL−K_{S})^{2} ≥ k^{2} +6k+6 > 4k + 5. Assume that there was an effective C on S with

(tL − K_{S}) · C − k − 1 ≤ C^{2} < \frac{(tL − K_{S}) · C}{2} < k + 1.

Note that

(tL − K_{S}) · C ≥ k + 2.  (12)

To see this note that (tL − K_{S}) · C = (t − k_{0} + 1)L · C + A · C ≥ (k + 1)L · C + A · C. Thus (tL − K_{S}) · C ≥ k + 1. If (tL − K_{S}) · C < k + 2 we have that A · C = 0. Since C is effective and A is nef and big we conclude by the Hodge index theorem that C^{2} < 0. But this contradicts 0 ≤ (tL − K_{S}) · C − k − 1 ≤ C^{2}. Thus (tL − K_{S}) · C ≥ k + 2. Therefore C^{2} ≥ 1 by (11). Note that C^{2} ≥ 1 also implies that A · C ≥ 1.

From (k + 1)L · C + A · C ≤ (tL − K_{S}) · C ≤ 2k + 1 we conclude that L · C = 1. Thus, since C^{2} ≥ 1, we conclude that

L^{2} = C^{2} = 1  (13)

and therefore by the Hodge index theorem L ∼ C. Thus A · C = A · L ≥ 2.

Therefore (tL − K_{S}) · C ≥ (t − k_{0} + 1)L · C + A · C ≥ k + 1 + A · C ≥ k + 3. Thus we have the contradiction C^{2} ≥ (tL − K_{S}) · C − k − 1 ≥ 2 to equation (13). Q.E.D.

Corollary 4.2 Assume that A is a nef and big line bundle on a smooth surface S such that K_{S} + A is ample. Then (k + 2)(K_{S} + A) is k-very ample and therefore k-spanned.

Proof. We take L := K_{S} + A and k_{0} = 2 in Theorem (4.1). Indeed A = (k_{0} − 1)(K_{S} + A) − K_{S} is nef and big. Also A · L = A · (K_{S} + A) is positive since both bundles are nef and big, and is even by the usual parity relation. Q.E.D.

Corollary 4.3 Let L be an ample line bundle on a smooth surface S. There exists an integer k_{0} > 0 such that for t ≥ k + k_{0}, tL is k-very ample and therefore k-spanned.

Proof. Choose a k_{0} > 0 such that A := (k_{0} − 1)L − K_{S} is ample and A · L ≥ 2. Now apply the above theorem. Q.E.D.

In the case of surfaces with ample canonical bundle we have the following (compare with [1, (5.4)]).

Corollary 4.4 Assume that L is an ample line bundle on a smooth surface S and K_{S} ∼ rL for some r > 0. Then tL is k-very ample and therefore k-spanned for t ≥ k + r + 3. If further L^{2} ≥ 2, then tL is k-very ample and therefore k-spanned for t ≥ k + r + 2.

Proof. First assume that L^{2} ≥ 2. We can take k_{0} = r + 2 in Theorem (4.1) since (k_{0} − 1)L − K_{S} ∼ (k_{0} − r − 1)L is ample and ((k_{0} − 1)L − K_{S}) · L ≥ 2 in this case. If L^{2} = 1, then take k_{0} = r + 3. Q.E.D.

Taking r = 1 we have the following result.

Corollary 4.5 Assume that K_{S} is ample on a smooth surface S. Then tK_{S} is k-very ample and therefore k-spanned for t ≥ k + 4. If further K_{S}^{2} ≥ 2, then tK_{S} is k-very ample and therefore k-spanned for t ≥ k + 3.

Remark 4.6 To write down an explicit k_{0} in Theorem (4.1) is straightforward. First use Fernández Del Busto’s result to choose an explicit k′ such that k′L is very ample. Now by the usual formula for the doublepoint divisor [9] we have that (k^{′2}L^{2} − 4)L − K_{S} is spanned. Thus (k^{′2}L^{2} − 2)L − K_{S} is ample and satisfies

L · ((k^{′2}L^{2} − 2)L − K_{S}) = L · (((k^{′2}L^{2} − 4)L − K_{S}) + 2L) ≥ 2L^{2} ≥ 2.

Thus we can take k_{0} = k^{′2}L^{2} − 1.

It would be interesting to know what the best choice of k_{0} is in Theorem (4.1). as k → ∞, i.e., letting t(k) be the minimal integer such that t(k)L is k-very ample, find the the lim inf and the lim sup of t(k) − k as k → ∞.

References

[1] M.C. Beltrametti, S. Di Rocco and A. Lanteri, “On higher order embeddings and nconnectedness,” preprint.

[2] M.C. Beltrametti, P. Francia and A.J. Sommese, “On Reider’s method and higher order embeddings,” preprint Max-Planck-Institüt, 7, (1987). Abridged version by same title, Duke Math. J. 58 (1989), 425–439.

[3] M.C. Beltrametti and A.J. Sommese, “On k-spannedness for projective surfaces,” Algebraic Geometry, Proceedings L’Aquila, 1988, Lecture Notes in Math., 1417 (1990), 24–51, Springer-Verlag.

[4] M.C. Beltrametti and A.J. Sommese, “Zero cycles and k-th order embeddings of smooth projective surfaces,” Projective Surfaces and their Classification, Proceedings Cortona, 1988, Symposia Mathematica, INDAM, 32, (1992), 33–48, Academic Press.

[5] M.C. Beltrametti and A.J. Sommese, “On k-jet ampleness,” Complex Analysis and Geometry, (ed. by V. Ancona and A. Silva), The University Series in Mathematics, Plenum Press, (1993), 355–376.

[6] M.C. Beltrametti and A.J. Sommese, The Adjunction Theory of Complex Projective Varieties, Expositions in Mathematics, 16, W. de Gruyter, 1995.

[7] J.-P. Demailly, “L^{2} vanishing theorems for positive line bundles and adjunction theory,” CIME Session Transcendental Methods in Algebraic Geometry, Cetraro, Italy, July 1994, Prépublication de l’Institut Fourier, Grenoble, n. 288 (1994).

[8] G. Fernández Del Busto, “A Matsusaka-type theorem for surfaces,” J. Algebraic Geometry 5 (1996), 513–520.

[9] S.L. Kleiman, “The enumerative theory of singularities,” in Real and Complex Singularities, Oslo 1976, ed. by P. Holme, 297–396, Alphen aan den Rijn, Sijthoff and Noordhoff, Rockville, Maryland, 1977.

[10] A. Lanteri and M. Palleschi, “About the adjunction process for polarized algebraic surfaces,” J. Reine Angew. Math. 352 (1984), 15–23.

[11] R. Lazarsfeld, “Lectures on linear series,” Park City/IAS Math. Series, vol. 3 (to appear).

[12] A.J. Sommese, “The birational theory of hyperplane sections of projective threefolds,” unpublished 1981 preprint.

Mauro C. Beltrametti Dipartimento di Matematica Via Dodecaneso 35 I-16146 Genova,Italy beltrame@dima.unige.it

Andrew J. Sommese Department of Mathematics Notre Dame, Indiana, 46556, U.S.A sommese.1@nd.edu http://www.nd.edu/∼sommese/index.html