MULTIPLIER IDEALS AND ADJOINT LINEAR SYSTEMS ON SURFACES

SEUNGHUN LEE

Abstract. In this note, we generalize the results of Ein and Lazarsfeld on the multiplier ideals to length 2 subschemes. We use them to study the separation of tangent directions for adjoint linear systems on surfaces. We also discuss some results for threefolds and higher dimensional varieties.

Introduction

For the last few years, the following conjecture of Fujita [6] has attracted many mathematicians.

Conjecture 0.1. Let X be a smooth projective variety of dimension n and A be an ample divisor on X. Then K_{X} + tA is base point free if t ≥ n + 1, and K_{X} + tA is very ample if t ≥ n + 2.

However in spite of all the success in base point freeness, it has been less successful in obtaining effective results in the very ampleness beyond surfaces.

On surfaces, above conjecture can be proved using several methods. First, one could apply the vector bundle technique or the Zariski decomposition method (cf. [12], [15], [16], [19], [21]). Here, it is essential to used the Hodge index theorem. Second, one could use the Kawamata-Viehweg vanishing theorem in the spirit of Ein and Lazarsfeld [3]. The first result in this direction was obtained by Maşek [18], where he also treated surfaces with boundary.

In the first part of this note, we develop another approach for the separation of tangent directions, and apply them to adjoint linear systems on surfaces in §3. It is based on the work [4] of Ein and Lazarsfeld. Two key results are Theorem 1.11 and Theorem 1.14. With these, the strategy for the separation of tangent directions become analogous to that of the base point freeness or the separations of distinct points. We explain them below.

Let X be a smooth projective variety and H be an ample divisor on X. Let W be a length 2 subscheme in X supported at a closed point p. In [4], Ein and Lazarsfeld introduced the notion of critical varieties (minimal centers in [10]) in the context of base point freeness, which plays such a crucial role in many papers on base point freeness. This notion is naturally extended to W (see Definition 1.7). With a tie-breaking condition satisfied, it means the following; a Q-Cartier divisor G on X is called critical at W if Z(G), the multiplier ideal scheme for G, contains W but Z((1 − ϵ)G) does not for any 0 < ϵ << 1. We call the support of I_{Z((1−ϵ)G)}/I_{Z(G)} the critical variety of G at W.

Date: February 23, 2000.

Suppose now that we have a Q-Cartier divisor G on X critical at W, and H − G is nef and big. Then Theorem 1.11 implies that if the critical variety is a point, then |K_{X} + H| separate W. Of course, there is no reason that the critical variety should be a point in the first place. However one can make the critical variety smaller using the nice lifting argument of Ein and Lazarsfeld thanks to Theorem 1.14. It especially takes care of the case where the multiplier ideals is non reduced alone the critical variety, which is one of the new phenomenon in the separation of tangent directions. Details are explained in the proof of the following theorem. For the definition of order, see Definition 3.1.

Theorem 0.2 (cf. [4], [20]). Let X be a smooth projective surface. Let W be a subscheme of X with l := length W ≤ 2^{1}. Let A be a nef and big Q-divisor on X. Let Δ = ⌈A⌉ − A and μ = ord_{W} Δ. Suppose that there exist positive numbers β_{1} and β_{2} satisfying the following conditions:

A^{2} > l · β_{2}^{2} ≥ l · (2 − μ)^{2}.

A · C ≥ (length (W ∩ C)) · β_{1} for all curve C in X.

β_{1} ≥ min \{2 − μ, \frac{β_{2}}{β_{2} − 1 + μ}\} .

Then |K_{X} + Δ + A| separates W. Especially, if μ ≥ 2, then A need to be only nef and big.

At least for the base point freeness part, it was known to Ein and Lazarsfeld [4]. The very ample part of the previous theorem is essentially due to Maşek [18], [20].

As one may have noticed, Theorem 0.2 does not implies the numerical bound of Reider [21] for the very ampleness part, which can be though of as a strong version of Conjecture 0.1. In fact, alone the same line we can also prove the following proposition. A similar bound was obtained by Langer [16].

Proposition 0.3. Let X be a smooth projective surface. Let W be a length two subscheme of X consisting of a closed point p in X and a tangent direction v of X at p. Let A be a nef and big Q-Cartier divisor on X. Let Δ = ⌈A⌉ − A and μ_{p} := ord_{p}Δ. Then |K_{X} + Δ + A| separates W in the following two cases.

1. μ_{p} ≥ 3.

1In this note, we only work out up to lenght 2 subscheme. But the method developed in §2 is general enough to deal with any subscheme of finte length and this theorem is expected to be true without the restirction on the length of W.

2. μ_{p} < 3, and there exists positive numbers σ_{1} and σ_{2} satisfying the following three inequalities:

A^{2} > σ_{2}^{2} ≥ (3 − μ_{p})^{2}.  (0.1)
A · C ≥ σ_{1} for all curve C in X through p.  (0.2)
σ_{1} ≥ \frac{2σ_{2}}{σ_{2} − 1 + μ_{p}} .  (0.3)

Up to this point, the proofs essentially depend only on vanishing theorem. However it seems inevitable to rely on some other methods to obtain his bound. So far, all the known method uses Hodge index theorem. In §2, we adopt a simple technique from the graded linear series (cf. [5], [7], [8], [9], [10]), and use them to obtain the bound at the end of §3.

I should mention that one important ingredient in the proofs of the previous results is a multiplicity bound for the critical curves on surfaces. It is easy to obtain in this case because they are divisors. In general, Helmke gave a bound in [8] for reduced W. But there are no known bounds otherwise. Any bound on them with non reduced W will lead to an effective result on very ampleness for arbitrary dimension.

From his bound, one might raise the following Content-Length: 8883 X-Lines: 288 Status: RO

Question 0.4. Let X be a smooth variety of dimension n and let W be a non reduced length 2 subscheme in X. Let p = Supp W. Let G be a Q-Cartier divisor on X critical at W with the critical variety Z of dimension d > 0. Is it true that

mult_{p}Z ≤ \binom{n}{d}?

In his proof, it was crucial that (X, G) is log canonical at p so that the coefficients of the exceptional divisors over p is bounded. But when W is non reduced it is not clear how to bound the exceptional divisors which is not over the tangent direction corresponding to W.

In §4, we consider some partial results in higher dimensional varieties. The next two result are probably well known among the experts. First, for smooth fano threefolds we can verify Conjecture 0.1.

Theorem 0.5. Let X be a smooth projective threefolds with an ample anticanonical divisor −K_{X}. Then | − tK_{X}| is very ample for all t ≥ 4.

One can have a similar result for smooth threefolds X with ample K_{X} instead of −K_{X}. But the bound is weaker than the conjectural one.

Proposition 0.6. Let X be a smooth projective threefolds with an ample canonical divisor K_{X}. Then |tK_{X}| is very ample for all t ≥ 10.

In fact, the proof of Theorem 0.5 can be easily extended to arbitrary smooth projective varieties with an ample (anti)canonical divisor to produce cubic bounds on very ampleness.

In general, we only have the following

Theorem 0.7. Let X be a normal Q-Gorenstein projective variety of dimension n. Let p be a point in X. Let Δ be an effective Q-divisor on X such that (X, Δ) is Kawamata log terminal at p. Let H be an ample Qdivisor on X such that K_{X} + Δ + H is Cartier. We assume that H satisfies the following inequality.

H^{d} · Z > \{∑_{k=1}^{n} \sqrt[k]{2} · k\}^{d}

for all subvariety Z through p, where d = dim Z > 0. Then there exist a length 2 subscheme W of X supported at p, and a divisor D in |K_{X} + Δ + H ⊗ I_{p}| such that W ⊄ D, where I_{p} is the sheaf of maximal ideal at p.

The following is an immediate corollary of this.

Corollary 0.8. Let X be a smooth projective variety of dimension n with an ample divisor H. Then for any closed point p in X, generic elements of |K_{X} + tH ⊗ I_{p}| are smooth at p for t ≥ n^{2} + n + 1.

As for the proof, we again use the multiplier ideals method. But to avoid the difficulties coming from the singularities of X and critical varieties, we use the specialization argument of Anghern and Siu [1]. The difficulty in obtaining a similar bound for the separations of tangent direction lies in that the tangent direction which we want to separate might be contained in the embedded components of the tangent cone of the multiplier ideal scheme at p.

Notations. We will work over the field ℂ of complex numbers. The following is the list of the notations that might be used differently by other people.

• = equivalence as Weil divisor.

• ∼_{lin}^{Q} Q-linear equivalence.

• ⌊·⌋ round-down(integral part).

• ⌈·⌉ round-up.

• {·} fractional part.

• T_{p}X the tangent space of X at p.

• ℙ(T_{p}X) the projectivised tangent space of X at p.

• C_{p}X the tangent cone of X at p.

• ℙ(C_{p}X) the projectivised tangent cone of X at p.

1. Multiplier Ideals

Here we collect some properties of multiplier ideals which are needed later in the proofs. The main reference for this section is [2], [4]. We also extend some of the results there to subschemes of finite length. All the proofs can be found there except Proposition 1.10 and Theorem 1.11, 1.14. But we give the proofs for the convenience of readers.

The following vanishing theorem of Kawamata and Viehweg is our basic tool.

Theorem 1.1 (Theorem 1-2-3 in [11]). Let X be a smooth variety and let f : X ⟶ S be a proper morphism onto a variety S. Let D be a f-nef and f-big Q-divisor whose support is in normal crossing. Then

R^{i}f_{∗}O_{X}(K_{X} + ⌈D⌉) = 0

for all i > 0.

Lemma 1.2 (Lemma 1-3-2 in [11]). Let f : Y ⟶ X be a proper birational morphism from a nonsingular variety Y onto a variety X. Let P be an effective f-exceptional Cartier divisor and let D be a (−f)-nef Q-Cartier divisor on Y. Then

f_{∗}O_{P} (P + ⌊D⌋) = 0.

1.1. Critical Varieties. The notations introduced here, we keep them throughout §1.

Let X be a normal Q-Gorenstein variety, i.e. K_{X} is Q-Cartier. Let W be a 0-dimensional subscheme in X. Let G be an effective Q-Cartier divisor on X. Let f : Y ⟶ X be an embedded resolution of G. Define b_{i} and g_{i} by

K_{Y} − f^{∗}K_{X} = ∑ b_{i}F_{i}  and  f^{∗}G = ∑ g_{i}F_{i},

where F_{i}’s are the distinct irreducible smooth divisors in simple normal crossing. Let

∑⌈(b_{i} − g_{i})⌉F_{i} = P − N,

where P and N are effective divisors with no common components. We observe that P is f-exceptional. Thus by Lemma 1.2

f_{∗}O_{P} (P − N) = f_{∗}O_{P} (P) = 0.

Then

f_{∗}O_{Y} (P − N) = f_{∗}O_{Y} (−N) ⊂ O_{X}.

Definition 1.3. f_{∗}O_{Y} (P − N) is called the multiplier ideal of G. Let Z(G) be the scheme defined by this ideal and we will note the multiplier ideal by I_{Z(G)}. One can check that I_{Z(G)} is independent of the choice of the embedded resolutions.

Proposition 1.4. 1. R^{1}f_{∗}O_{Y} (−N) = 0 and R^{i}f_{∗}O_{Y} (P − N) = 0 for all

i > 0.

2. Assume that X is projective and let H be an Cartier divisor on X such that H −K_{X} −G is nef and big. Then H^{i}(X, O_{X}(H)) = 0 for all i > 0.

Proof. (1) follows from Theorem 1.1 and Lemma 1.2. For (2), we note that

H^{i}(Y, O_{Y} (P − N) ⊗ f^{∗}O_{X}(H)) = 0

for all i > 0. Thus it follows from the Serre’s spectral sequence and (1).

Definition 1.5. We say that (G, f), or simply G, satisfies tiebreaking if b_{i} − g_{i} ≠ b_{j} − g_{j} for all i ≠ j.

Remark 1.6. If G is ample, for arbitrary small 0 < ϵ << 1, one can always find a Q-Cartier divisor G′ ∼_{lin}^{Q} (1+ϵ)G together with an embedded resolution of G′ so that G′ satisfies tiebreaking.

Definition 1.7. 1. We say that G is critical at W if it satisfies tiebreak-

ing, W ⊂ Z(G), but W ⊄ Z((1 − ϵ)G) for any ϵ > 0.

2. Suppose G is critical at W. Then there is a unique F_{i} such that

⌈K_{Y} − f^{∗}(K_{X} + G)⌉ = ⌈K_{Y} − f^{∗}(K_{X} + (1 − ϵ)G)⌉ − F_{i}

for a sufficiently small 0 < ϵ << 1. We call F_{i} the critical component of G and f(F_{i}) the critical variety of G at W.

3. Let l ∈ ℕ and let p be a closed point in X. We say that G is l-critical at p if

(a) G is critical at a length l subscheme of X whose support is p.

(b) For all λ < 1, Z(λG) does not contain any length l subschemes in

X supported at p.

4. Suppose G is l-critical at p. Then G is critical at some W with the length l and Supp W = p. We call the critical component(resp. variety) of G at W as the l-critical component(resp. variety) of G at p. By 1.(a) they are well defined.

Proposition 1.8. We assume that W is a closed point in X. If G is critical at p with the critical variety Z, then Z is normal at p.

Proof. Let F be the critical component. Since it is local, we may assume that N = F.

From Proposition 1.4,

0 ⟶ I_{Z} ⟶ O_{X} ⟶ f_{∗}O_{F} ⟶ 0.

Thus Z is normal at p.

Content-Length: 435 X-Lines: 16 Status: RO

Proposition 1.9. Let H be an irreducible normal Cartier divisor on X intersecting G properly and let G_{H} = G|_{H}. Then,

I_{Z(G_{H})} ⊂ Img (I_{Z(G)} ⟶ O_{H}.

Proof. We may assume that f : Y ⟶ X is also an embedded resolution of H. Let V be the proper transform of H in Y.

From Proposition 1.4, we have

f_{∗}O_{Y} (K_{Y} + V − ⌊f^{∗}(K_{X} + G + H)⌋) → f_{∗}O_{V} (K_{V} − ⌊f^{∗}(K_{H} + G_{H})⌋) → 0.

The first term is a subsheaf of I_{Z(G)}. The second term is I_{Z(G_{H})}. Thus we have the inclusion.

Content-Length: 866 X-Lines: 28 Status: RO The following proposition is implicit in [4], [14].

Proposition 1.10. Assume that W is a length 2 subscheme of X supported at a closed point p. Let G be critical at W with the critical variety Z of dimension d > 0 and the critical component F. Suppose that the following two conditions hold.

1. Z is smooth at p, and f|_{F} is smooth over a neighborhood of p.

2. F ∩ f^{−1}(p) is not contained in P.

Let \overline{ϕ} ∈ I_{W/Z}^{kd+1} for some integer k > 0. Then for any lifting ϕ of \overline{ϕ},

W ⊂ Z(D + (1 − ϵ)G),

where 0 < ϵ << 1 and D = \frac{1}{k}div(ϕ).

Proof. Since it is a local problem, we may assume that P − N = −F by assumption 2.

From the surjection I_{W/X}^{kd+1} ⟶ I_{W/Z}^{kd+1} ⟶ 0, we can pick a local lifting

ϕ ∈ I_{W/X}^{kd+1} of \overline{ϕ}. Let D = \frac{1}{k}div(ϕ). From Theorem 1.14, it is enough to show that W ⊂ Z(D +(1−ϵ)G). Content-Length: 4558 X-Lines: 152 Status: RO

Let C_{1} be an irreducible fiber of f over p. Let g_{1} : Y_{1} ⟶ Y be the blowing up at C_{1} with exceptional divisor E_{1}. Then there is a unique subvariety C_{2} of dim C_{2} = n − 1 − d in E_{1} corresponding to W. Let g_{2} : Y_{2} ⟶ Y_{1} be the blowing up at C_{2} with the exceptional divisor E_{2}. We set g = g_{1} ∘ g_{2}. Note that

1. K_{Y_{2}} − g^{∗}K_{Y} = dg_{2}^{∗}E_{1} + dE_{2}.

2. g^{∗}f^{∗}D ≥ g_{∗}^{−1}f^{∗}D + dg_{2}^{∗}E_{1} + dE_{2}.

3. g^{∗}F = g_{∗}^{−1}F + g_{2}^{∗}E_{1} + E_{2} and (f ∘ g)_{∗}O_{Y_{2}}(−g_{2}^{∗}E_{1} − E_{2}) ⊂ I_{W}. Thus

K_{Y_{2}} − (f ∘ g)^{∗}(K_{X} + (1 − ϵ)G + D) = K_{Y_{2}/Y} + g^{∗}(K_{Y/X} − f^{∗}((1 − ϵ)G + D)) ≤ −g_{2}^{∗}E_{1} − E_{2},

for 0 < ϵ << 1 and the assertion follows.

1.2. Two key theorems. In this section we give the proofs of the theorems mentioned in the introduction. The next theorem appears in several papers including [1], [2], [3], [8], [9], [10], [13] in the context of freeness. The following version is due to Ein.

Theorem 1.11. We assume that X is projective. Let H be a Cartier divisor on X such that H − K_{X} − G is nef and big. We assume that I_{Z(G)} ⊂ I_{W}. Let N = N_{0} + N_{+} be a decomposition of N such that for all F_{i} ⊂ Supp N_{0}, F_{i} ⊂ f^{−1}W or F_{i} ∩ f^{−1}W = ∅. Let Z_{+} be the scheme defined by the ideal sheaf f_{∗}O_{Y} (P − N_{+}), i.e. I_{Z_{+}} = f_{∗}O_{Y} (P − N_{+}). Then for any Σ with W ∩ Z_{+} ⊂ Σ ⊂ W, there is D ∈ |H| such that Σ = D ∩ W, i.e. we have the following surjection:

H^{0}(X, I_{Z_{+}} ⊗ O_{X}(H)) ⟶ I_{W ∩Z_{+}/W} ⟶ 0.  (1.1)

Proof. Let J = I_{Z_{+}}/I_{Z(G)} ⊂ O_{Z(G)}. By Proposition 1.4,

H^{0}(X, I_{Z_{+}} ⊗ O_{X}(H)) → H^{0}(Z(G), J ⊗ O_{Z(G)}(H)) → 0.  (1.2)

By the construction, J is trivial in a punctured neighborhood of W. So we can write J = J_{1} ⊕ J_{2} so that Supp J_{1} ⊂ W and Supp J_{2} ∩ W = ∅. Thus

H^{0}(J ⊗ O_{Z(G)}(H)) ≅ J_{1} ⊕ H^{0}(Z(G), J_{2} ⊗ O_{Z(G)}(H)).  (1.3)

Since I_{Z(G)} ⊂ I_{W}, we have

J_{1} ⟶ I_{W ∩Z_{+}/W} ⟶ 0.  (1.4)

Now (1.1) follows from (1.2), (1.3), and (1.4).

Corollary 1.12. Let X, H, and G be as in Theorem 1.11 and we assume further that G is critical at W.

1. Then there is D ∈ |H| such that length W = length (D ∩ W) + 1.

2. We assume that W = {p, v}, where v ∈ ℙT_{p}X.

(a) If the subscheme of Z(G) which consists of positive dimensional components does not contains W, then |H| separates v at p.

(b) If the critical variety of G is a point, then |H| separates v at p.

Remark 1.13. Corollary 1.12.1 can be used to separates any subschemes of finite length inductively.

The next Theorem is essentially due to Ein and Lazarsfeld.

Theorem 1.14. Suppose G is critical at W with the critical variety Z. Let B be a non-zero effective Q-Cartier divisor on Z. Let D_{1} and D_{2} be two liftings of B, i.e. B = D_{i}|_{Z} for i = 1, 2. If

W ⊂ Z((1 − s)D_{1} + (1 − t)G)

for all sufficiently small s and t, then

W ⊂ Z((1 − s)D_{2} + (1 − t)G)

for all sufficiently small s and t.

Proof. We may assume that f : Y ⟶ X is also embedded resolution of D_{1} and D_{2}. Let F be the critical component. Then taking t sufficiently small, we may assume that

K_{Y} − ⌊f^{∗}(K_{X} + G)⌋ = K_{Y} − ⌊f^{∗}(K_{X} + (1 − t)G)⌋ − F.

Let P_{i} − N_{i} := K_{Y} − ⌊f^{∗}(K_{X} + (1 − t)G + (1 − s)D_{i})⌋ for i = 1, 2. Then taking s small and generic, we may assume further that

K_{Y} − ⌊f^{∗}(K_{X} + G + (1 − s)D_{i})⌋ = P_{i} − N_{i} − F.

Since D_{1} and D_{2} are the liftings of the same non-zero effective divisor B on Z, F ⊄ Supp f^{∗}D_{i} for i = 1, 2, and f^{∗}D_{1}|_{F} = f^{∗}D_{2}|_{F}. We note that ord_{F} N_{1} = ord_{F} N_{2}. Then since f is an embedded resolution for G, D_{1}, and D_{2},

P_{1} − N_{1} − (ord_{F} N_{1})F |_{F} = P_{2} − N_{2} − (ord_{F} N_{2})F |_{F} .

Thus

O_{F} (P_{1} − N_{1}) = O_{F} (P_{2} − N_{2}).  (1.5)

From Proposition 1.4 we have the following commutative diagram.

0⟶  I_{Z(G)}  ⟶  O_{X}  ⟶ O_{Z(G)} ⟶0
↑  ↑  ↑
0⟶ I_{Z(G+(1−s)D_{i})} ⟶ I_{Z((1−t)G+(1−s)D_{i})} ⟶ J_{i} ⟶0

where J_{i} = f_{∗}O_{F} (P_{i} − N_{i}). From (1.5) J_{1} = J_{2}. So the theorem follows from W ⊂ Z(G).

Content-Length: 11053 X-Lines: 363 Status: RO

Definition 1.15. Suppose G is critical at W with the critical variety Z. Let B be an effective Q-Cartier divisor on Z. An effective Q-Cartier divisor D is said to be a nice lifting of B with respect to G if

D|_{Z} = B  and  Supp J ⊂ Z

where J is defined by

0 → I_{Z((1−t)G+D)} → I_{Z((1−t)G)} → J → 0.

for 0 < t << 1

Proposition 1.16. Suppose G is critical at W with the critical variety Z. Let B be an effective Q-Cartier divisor on Z. Let A be a Q-ample Cartier divisor on X such that B ∼_{lin}^{Q} A|_{Z}. Then there is a nice lifting D of B with respect to G such that D ∼_{lin}^{Q} A.

Proof. We fix m >> 0 so that O_{X}(mA) ⊗ I_{Z} is globally generated and

H^{1}(X, O_{X}(mA) ⊗ I_{Z}) = 0.

Let

ϕ : H^{0}(X, O_{X}(mA)) → H^{0}(Z, O_{Z}(mA)) → 0.

Let Λ ⊂ |mA| be the sublinear system consisting of those divisors D′ such that Z ⊂ D′ or D′|_{Z} = mB. Since O_{X}(mA) ⊗ I_{Z} is globally generated, ℙ(ϕ^{−1}(0)) is base point free on X − Z. Thus Λ is also base point free on X − Z. Now the generic elements of \frac{1}{m}Λ give nice liftings.

2. Graded Linear Series

Here we extend a property of graded linear series in [4] to length two subscheme. The idea comes from [7]. For more details on the graded linear series we refer to [4], [5].

Let X be a projective variety of dimension n and H be a nef and big Q-Cartier divisor on X. Let W be a length 2 subscheme of X. We also fix positive dimensional subvariety Z through W. Let β be a nonnegative real number. We consider the following graded linear series.

Ω^{β} = Ω^{β}(W, H) = ⨁_{k=0}^{∞} Ω_{k}^{βk}

where

Ω_{k}^{kβ} = {s ∈ H^{0}(X, O_{X}(kH))|s ∈ I_{W}^{⌈βk⌉} for 1 ≤ i ≤ h}

if kH is Cartier and Ω_{k}^{kβ} = 0 otherwise. We define the degree of Ω^{β} as

deg(Ω^{β}) = lim sup_{k} \{\frac{n!}{k^{n}} dim Ω_{k}^{βk}\} .

Let s_{k} be a general section of Ω_{k}^{βk}. We define

M(β) = M(β, W, Z) = lim inf_{k} \{\frac{1}{k} mult_{Z}(s_{k})\} .

Proposition 2.1. M(β)’s are lower convex on ℝ_{≥0} for 1 ≤ i ≤ h.

Proof. First we show that M(x) is lower convex on ℚ_{≥0}. Let β_{1}, β_{2} and λ ∈ ℚ_{>0} with 0 < λ < 1. Let k_{1} and k_{2} be two sufficiently large and divisible integers such that λ = \frac{k_{1}}{k_{1}+k_{2}} . Let s_{j}, j = 1, 2, be generic section in

s_{j} ∈ H^{0}(X, O_{X}(k_{j}H) ⊗ ⨂_{i=1}^{h} I_{i}^{⌈β_{j}k_{j}⌉})

such that

M(β_{j}) = \frac{mult_{Z}s_{j}}{k_{j}} + δ_{j}

for some δ_{1}, δ_{2} << 1. Since

⌈β_{1}k_{1}⌉ + ⌈β_{2}k_{2}⌉ ≥ ⌈(λβ_{1} + (1 − λ)β_{2})(k_{1} + k_{2})⌉,

we have

M(λβ_{1} + (1 − λ)β_{2}) ≤ \frac{mult_{Z}(s_{1}s_{2})}{k_{1} + k_{2}} = λ\frac{mult_{Z}s_{1}}{k_{1}} + (1 − λ)\frac{mult_{Z}s_{2}}{k_{2}} .

Thus M(x) is lower convex on ℚ_{≥0}. Now this implies that M(x) is continuous on ℚ_{≥0} and, since M(x) is increasing function, M(x) is continuous on ℝ_{≥0}. Thus M(x) is lower convex on ℝ_{≥0}.

The idea of the following proposition is due to Fujita [7]. Kawamata [10] and Helmke [8] also have similar result. Here we state it for only surfaces, which suit our purpose. But it can be easily extended to arbitrary dimensions.

Proposition 2.2. We assume that X is a surface and that Z is a curve. Let β, γ in ℝ_{≥0} such that deg(Ω^{x}) > 0 for all β ≤ x < γ.

1. Assume that W = {p, q}, two distinct points. Let m_{p} = ord_{p}Z and m_{q} = ord_{q}Z. Then

deg(Ω^{γ}) ≥ deg(Ω^{β}) − ∫_{β}^{γ} 2(t − m_{p}M(t))dt − ∫_{β}^{γ} 2(t − m_{q}M(t))dt.

2. Assume that W = {p, v} where v ∈ ℙ(T_{p}X). Let m = 2ord_{W} Z−ord_{p}Z. Then

deg(Ω^{γ}) ≥ deg(Ω^{β}) − 2 ∫_{β}^{γ} 2(t − mM(t))dt.

Proof. The proofs are similar, so we only prove (2). Let f : X_{p} → X be the blowing up at p with the exceptional divisor E and let g : X_{v} → X_{p} be the blowing up at v with exceptional divisor F_{v}. Let Z_{v} be the proper transform of Z in X_{v}. Let F_{p} be the proper transform of E, F := F_{p} + 2F_{v}, and let h = f ∘ g.

Let k be sufficiently large and divisible positive integer. For βk ≤ j ≤ γk, we have the following exact sequences.

0 ⟶ H^{0}(X_{v}, O_{X_{v}}(h^{∗}kL − (j + 1)F))

⟶ H^{0}(X_{v}, O_{X_{v}}(h^{∗}kL − jF)) ⟶^{g_{k_{j}}} H^{0}(F, O_{F} (−jF))

and

0 ⟶ O_{F_{p}}(−jF − 2F_{v}) ⟶ O_{F} (−jF) ⟶ O_{2F_{v}}(−jF) ⟶ 0.

Since H^{0}(F_{p}, O_{F_{p}}(−jF − 2F_{v})) = 0,

H^{0}(F, O_{F} (−jF)) ⊂ H^{0}(2F_{v}, O_{2F_{v}}(−jF)).

Z appears with multiplicity kM(\frac{j}{k}) in the base locus of |Ω_{k}^{j}|. Thus from Proposition 2.1 and m = F_{v} · Z_{v},

Img (g_{k,j}) ⊂ H^{0}(2F_{v}, O_{2F_{v}}(−jF − ⌈mkM(\frac{j}{k})⌉))

⊂ H^{0}(2F_{v}, O_{2F_{v}}(−jF − ⌈j \frac{m}{β}M(β)⌉)).

We set

α := ⌈j \frac{m}{β}M(β)⌉.

Now from the the following exact sequence,

0 ⟶ O_{F_{v}}(−jF − F_{v} − α) ⟶ O_{2F_{v}}(−jF − α) ⟶ O_{F_{v}}(−jF − α) ⟶ 0. we obtain

dim Img (g_{k,j}) ≤ 2 · (j − ⌈j \frac{m}{β}M(β)⌉) + 1.

Thus

deg(Ω^{γ}) ≥ deg(Ω^{β}) − 2 (1 − \frac{m}{β}M(β)) (γ^{2} − β^{2}).

So by taking Riemann sum, we obtain the inequality.

3. On Surfaces

In this section, first we will state and give proofs of three propositions, Proposition 3.2, 3.3, and 3.5, which together imply Theorem 0.2. The arguments are essentially the same and we could give one proof that works for all three of them. But, it is our intention to emphasize the handling of the separation of tangent directions and to show new phenomenon. So with the risk of boring the reader, we separate them into three propositions. One can see the similarities among the proofs as well as the differences. As for the separation of tangent directions (Proposition 3.5), they were very much inspired by Maşek [18], [20] and we follows his argument very closely.

In the second part of this section, we will give the proof of Proposition 0.3 and the numerical bound of Reider using graded linear series.

First we clarify what we mean by order of Q-divisors.

Definition 3.1. Let X be a variety. Let G be a Q-Cartier divisor on X.

1. Let p be a closed point of X. We define the order of G at p, ord_{p}G, as the coefficient of the exceptional divisor of the blowing up of X at p in the total transform of G.

2. Let W be a length two subscheme of X consists of two distinct closed points p and q in X. We define the order of G at W, ord_{W} G, as the minimum between ord_{p}G and ord_{q}G.

3. Let W be a non reduced length two subscheme of X supported at p, i.e. W = {p, v} for some v ∈ ℙ(T_{p}X). Let h : U ⟶ X be the blowing up of X at p with the exceptional divisor E and let g : V ⟶ U be the blowing up of U at v with the exceptional divisor F. We define the order of G at W, ord_{W} G, as the coefficient of g^{∗}E + F.

3.1. Proof of Theorem 0.2. First we deal with the case of freeness (cf. [4], [20]).

Proposition 3.2. Let X be a smooth projective surface. Let A be a nef and big Q-divisor on X. Let Δ = ⌈A⌉ − A and let μ := ord_{p}Δ. Then |K_{X} + Δ + A| is free at p in the following two cases.

1. μ ≥ 2.

2. μ < 2, and there are positive constants β_{1}, β_{2} satisfying the following three inequalities.

A^{2} > β_{2}^{2} ≥ (2 − μ)^{2}.  (3.1)
A · C ≥ β_{1} for all curve C through p.  (3.2)
β_{1} ≥ min \{2 − μ, \frac{β_{2}}{β_{2} − 1 + μ}\} .  (3.3)

Proof. (1) From the assumption, we already have I_{Z(Δ)} ⊂ I_{p}. Furthermore the connected component of Z(Δ) at p has dimension 0. Thus it follows from Theorem 1.11.

(2) From (3.1) and Riemann-Roch theorem, we can create a Q-Cartier divisor D on X such that

η := ord_{p}D > β_{2} ≥ 2 − μ.  (3.4)

Then by tie-breaking technique, we can create a Q-divisor G ∼_{lin}^{Q} λA such that Δ + G is critical at p with the critical variety Z, and λ < 1. We note that essentially, G = λD.

If the critical variety is a point, then Theorem 1.11 immediately implies that |K_{X} + Δ + A| is free at p. So we assume that Z is a curve. We note that Z is smooth at p by Proposition 1.8.

Again from Riemann-Roch theorem and (3.2), we can create a Q-divisor B ∼_{lin}^{Q} H|_{Z} on Z with ord_{p}B > β_{1} − ϵ for some 0 < ϵ << 1. Since we have surjections

I_{p/X}^{k} ⟶ I_{p/Z}^{k} ⟶ 0

for all k ∈ ℕ, we can find a local lifting D^{loc} of B with ord_{p}D^{loc} > β_{1} − ϵ. Let D′ be a nice lifting of B with respect to Δ + G.

From Theorem 1.14, there exist 0 < λ′ such that

p ⊂ Z(Δ + (1 − δ)G + λ′D′)

and

λ′(β_{1} − ϵ) ≤ 2 − μ − ord_{p}(1 − δ)G

for some 0 < δ << 1. Content-Length: 2698 X-Lines: 90 Status: RO

We note that from (3.4), \frac{2−μ−λη}{1−λ} is a decreasing function of λ. Also note that since the critical variety is a divisor, μ + λη ≥ 1. Thus

\frac{2 − μ − λη}{1 − λ} < min \{2 − μ, \frac{β_{2}}{β_{2} − 1 + μ}\} .

Taking ϵ and δ sufficiently small, λ′ + λ < 1 by (3.3). Since the critical variety of Δ + (1 − δ)G + λ′D′ is a point, by Theorem 1.11 |K_{X} + Δ + A| is free at p.

Next we consider the separation of distinct points (cf [18], [20]).

Proposition 3.3. Let X be a smooth projective surface. Let W be a length two subscheme consisting of two distinct closed points p and q of X. Let A be a nef and big Q-divisor on X. Let Δ = ⌈A⌉ − A and μ := ord_{W} Δ. Then |K_{X} + Δ + A| separates p and q in the following two cases.

1. μ ≥ 2.

2. μ < 2, and there are positive constants β_{1}, β_{2} satisfying the following three inequalities.

A^{2} > 2β_{2}^{2} ≥ 2(2 − μ)^{2}.  (3.5)
A · C ≥ (length (W ∩ C)) · β_{1} for all curve C in X.  (3.6)
β_{1} ≥ min \{2 − μ, \frac{β_{2}}{β_{2} − 1 + μ}\} .  (3.7)

Proof. (1) From the assumption, we already have I_{Z(Δ)} ⊂ I_{W}. Furthermore the connected component of Z(Δ) at p has dimension 0. Thus it follows from Theorem 1.11.

(2) Since the length of W is two, from (3.5) and Riemann-Roch theorem, we can create a Q-Cartier divisor D on X such that

η := ord_{W} D > β_{2} ≥ 2 − μ.  (3.8)

Then by tie-breaking technique, we can create a Q-divisor G ∼_{lin}^{Q} λA such that Δ + G is critical at W with the critical variety Z, and λ < 1. We note that essentially, G = λD.

If the critical variety is a point, then Theorem 1.11 immediately implies that |K_{X} + Δ + A| separates W. So we assume that Z is a curve. We note that Z is smooth at W ∩ Z by Proposition 1.8.

Again from Riemann-Roch theorem and (3.6), we can create a Q-divisor B ∼_{lin}^{Q} H|_{Z} on Z with ord_{W ∩Z}B > β_{1} − ϵ for some 0 < ϵ << 1. Since we have surjections

I_{W ∩Z/X}^{k} ⟶ I_{W ∩Z/Z}^{k} ⟶ 0  (3.9)

for all k ∈ ℕ, we can find a local lifting D^{loc} of B with

ord_{W ∩Z}D^{loc} > β_{1} − ϵ.  (3.10)

Let D′ be a nice lifting of B with respect to Δ + G.

From Theorem 1.14, there exist 0 < λ′ such that

W ⊂ Z(Δ + (1 − δ)G + λ′D′)

and

λ′(β_{1} − ϵ) ≤ 2 − μ − ord_{W} (1 − δ)G,  (3.11)

for some 0 < δ << 1. Also since the critical variety is a divisor, by examining the coefficient of the exceptional divisor of the blowing up at Z∩W we obtain

λ′(β_{1} − ϵ) ≤ 1.  (3.12)

Content-Length: 4234 X-Lines: 135 Status: RO We note that from (3.8), \frac{2−μ−λη}{1−λ} is a decreasing function of λ. Thus

\frac{2 − μ − λη}{1 − λ} < 2 − μ.

Also note that

min \{\frac{2 − μ − λη}{1 − λ} , \frac{1}{1 − λ}\} ≤ \frac{η}{η − 1 + μ}.

Thus

min \{\frac{2 − μ − λη}{1 − λ} , \frac{1}{1 − λ}\} < min \{2 − μ, \frac{β_{2}}{β_{2} − 1 + μ}\} .  (3.13)

Taking ϵ and δ sufficiently small, we have λ′ + λ < 1 by (3.7), (3.11), (3.12), and (3.13). Since the critical variety of Δ + (1 − δ)G + λ′D′ is a point, by Theorem 1.11 |K_{X} + Δ + A| separates W.

Remark 3.4. Proposition 4.2 in [20] can also be proved in a similar manner.

Now we consider the separation of tangent directions (cf. [18], [20]).

Proposition 3.5. Let X be a smooth projective surface. Let W be a length two subscheme of X consisting of a closed point p in X and a tangent direction v of X at p. Let A be a nef and big Q-Cartier divisor on X. Let Δ = ⌈A⌉ − A and μ = ord_{W} Δ. Then |K_{X} + Δ + A| separates v at p in the following two cases.

1. μ ≥ 2.

2. μ < 2, and there are positive constants β_{1}, β_{2} satisfying the following three inequalities.

A^{2} > 2β_{2}^{2} ≥ 2(2 − μ)^{2}.  (3.14)
A · C ≥ (length (W ∩ C)) · β_{1} for all curve C in X.  (3.15)
β_{1} ≥ min \{2 − μ, \frac{β_{2}}{β_{2} − 1 + μ}\} .  (3.16)

Proof. (1) From the assumption, we already have I_{Z(Δ)} ⊂ I_{W}. Furthermore the connected component of Z(Δ) at p has dimension 0. Thus it follows from Theorem 1.11.

(2) Since the length of W is two, from Riemann-Roch theorem and (3.14), we can create a Q-Cartier divisor D on X such that

η := ord_{W} D > 2 − μ.  (3.17)

Then by tie-breaking technique, we can create a Q-divisor G ∼_{lin}^{Q} λA such that Δ + G is critical at W with the critical variety Z, and λ < 1. We note that essentially, G = λD.

If the critical variety is a point, then again Theorem 1.11 immediately implies that |K_{X} + Δ + A| separates W. So we assume that Z is a curve. Here Z may be singular at p. But from tie-breaking, we may assume that

mult_{p}⌊Δ + G⌋ ≤ 2,

thus mult_{p}Z ≤ 2. Below, we separate into two cases.

(2.1) First, assume that mult_{p}Z = 1. Near p, one of the following cases should occur.

⌊Δ + G⌋ = Z and v ∉ ℙ(T_{p}Z)  (3.18)
⌊Δ + G⌋ = Z and v ∈ ℙ(T_{p}Z)  (3.19)
⌊Δ + G⌋ = Z + Z′, Z ≠ Z′, and v ∈ ℙ(T_{p}Z)  (3.20)
⌊Δ + G⌋ = Z + Z′, Z ≠ Z′, and v ∉ ℙ(T_{p}Z)  (3.21)
⌊Δ + G⌋ = 2Z and v ∉ ℙ(T_{p}Z)  (3.22)

where Z′ is another curve through p. As for (3.18), since I_{Z}(Δ + G) ⊂ I_{W} and O_{X}(−⌊Δ + G⌋) ⊄ I_{W}, again Theorem 1.11 implies that |K_{X} + Δ + A| separates W. For the remaining cases, we can argue all together as follows. The argument here is similar to the one in the previous proposition.

As before, from (3.15) and the Riemann-Roch theorem, we can create a

Q-divisor B ∼_{lin}^{Q} A|_{Z} on Z so that ord_{W ∩Z}B > β_{1} − ϵ for some 0 < ϵ << 1.

Since we have surjections

I_{W/X}^{k} ⟶ I_{W ∩Z/Z}^{k} ⟶ 0  (3.23)

for all k ∈ ℕ, we can find a local lifting D^{loc} of B with

ord_{W} D^{loc} > β_{1} − ϵ.  (3.24)

We note the difference between (3.9),(3.10) and (3.23), (3.24). Let D′ be a nice lifting of B with respect to Δ + G.

From Theorem 1.14, there exist 0 < λ′ such that Content-Length: 5646 X-Lines: 184 Status: RO

λ′(β_{1} − ϵ) ≤ 2 − μ − (1 − δ)ord_{W} G

and

W ⊂ Z(Δ + (1 − δ)G + λ′D′)

for some 0 < δ << 1.

We note that from (3.17), \frac{2−μ−λη}{1−λ} is a decreasing function of λ. Also note that μ + λη ≥ 1 since Z contains W for (3.19) and (3.20), and mult_{p}⌊Δ + G⌋ = 2 for the rest of the cases. Thus

\frac{2 − μ − λη}{1 − λ} < min \{2 − μ, \frac{β_{2}}{β_{2} − 1 + μ}\} .

Taking ϵ and δ sufficiently small, λ′ + λ < 1 by (3.16). Since the critical variety of Δ + (1 − δ)G + λ′D′ is a point, by Theorem 1.11 |K_{X} + Δ + A| separates W.

So far, we have shown the handling of all the possible cases in general except when Z is singular at p. For surfaces, it is bounded by 2 as we have already mentioned. Unfortunately, even for surfaces it is already too big to argue as (2.1). So we take an alternative and restrict the linear system to the proper transform of Z in the blowing up of X at p using Lemma 3.6.(1), and apply Lemma 3.6.(2). The reason to restrict to the proper transform of Z in the blowing of X at p rather than in an embedded resolution of G is as follows.

Let f_{1} : V_{1} ⟶ X and f_{2} : V_{2} ⟶ X be two birational morphisms from smooth surfaces V_{1} and V_{2} such that f_{2} = f_{1} ∘ g for some g : V_{2} ⟶ V_{1}. Then

f_{2}_{∗}O_{V_{2}}(K_{V_{2}} − ⌊f_{2}^{∗}(K_{X} + G)⌋) ⊂ f_{1}_{∗}O_{V_{1}}(K_{V_{1}} − ⌊f_{1}^{∗}(K_{X} + G)⌋).

So, even though we do need to take advantage of the fact that (X, G) is not log canonical at p, to use Lemma 3.6.(2) most effectively we need to take the smallest resolution whose corresponding ideal sheaf on X for (1 − ϵ)G, 0 < ϵ << 1, is non trivial at p. In the following situation, the blowing up of X at p turned out to be the right one for us (cf. [18])

(2.2) Assume that mult_{p}Z = 2. Here we need the following well-known

Lemma 3.6. 1. Let S be a smooth projective surface and let M be a nef and big Q-Cartier divisor on M. Then H^{i}(S, O_{S}(K_{S} + ⌈M⌉)) = 0 for all i > 0.

2. Let C be an irreducible Gorenstein curve and B be a Cartier divisor on C. Then O_{C}(K_{C} + B) is globally generated if degB > 1.

Proof. (1) is Lemma 1.1 in [3]. For (2), we note that from the Serre’s duality and the assumption H^{1}(C, I_{p} ⊗ O_{C}(K_{C} + B)) = Hom(I_{p}, O_{C}(−B)) = 0 for all closed point p in C.

Now let f : Y ⟶ X be the blowing up of X at p with the exceptional divisor E. Let F be the proper transform of Z in Y. Put H = K_{X} + Δ + A. We have the following two exact sequences

0⟶ O_{Y} (−E − F + f^{∗}H) ⟶ O_{Y} (−E + f^{∗}H) ⟶ O_{F} (−E + f^{∗}H) ⟶0

and

0⟶ O_{X}(−Z + H) ⟶ I_{p} ⊗ O_{X}(H) ⟶ I_{p/Z} ⊗ O_{Z}(H) ⟶0,

where the 0-th cohomology groups of the first two columns are isomorphic. Note that ⌈K_{Y} − f^{∗}(K_{X} + G)⌉ = −F − E and ⌈−G⌉ = Z, Then since

H^{1}(Y, O_{Y} (−E − F + f^{∗}H)) = H^{1}(X, O_{X}(−Z + H)) = 0

by Lemma 3.6.(1), the third column also induces an isomorphism between 0-the cohomology groups. Thus to separates v at p, it is enough to show that O_{F} (−E − F + f^{∗}H) is globally generated. Since ⌈−f^{∗}(G − Z)⌉ = 0, we have

O_{F} (−E − F + f^{∗}H) = O_{F} (K_{F} + ⌈f^{∗}(Δ + A − G)⌉)
= O_{F} (K_{F} + f^{∗}(Δ + A − Z))
= O_{F} (K_{F} + f^{∗}⌈Δ + A − G⌉).

Thus, from Lemma 3.6.(2) it is enough to show

Z · ⌈(Δ + A − G)⌉ > 1.  (3.25)

But

Z · ⌈(Δ + A − λD)⌉
=Z · ((1 − λ)A) + Z · (Δ + {λD})
≥(1 − λ)2β_{1} + 2 · ord_{p}(Δ + {λD})
≥(1 − λ)2β_{1} + 2 · ord_{W} (Δ + {λD})
≥(1 − λ)2β_{1} + 2 · λη + 2 · μ − 3 if v ∈ ℙ(C_{p}Z), or
≥(1 − λ)2β_{1} + 2 · λη + 2 · μ − 2 if v ∉ ℙ(C_{p}Z).

So (3.25) in turn follows from

β_{1} > \frac{2 − μ − λη}{1 − λ}

by taking ϵ and δ small enough. But λη + μ ≥ 1 since mult_{p}Z = 2. Thus as before,

\frac{2 − μ − λη}{1 − λ} < min \{2 − μ, \frac{β_{2}}{(β_{2} − 1 + μ)}\} .

3.2. Reider’s Bound. The theorem of Reider [21] implies the following

Proposition 3.7. Let X be a smooth projective surface. Let A be a Cartier divisor on X. Then K_{X} + A is very ample if A^{2} ≥ 10 and A · C ≥ 3 for all curves C in X.

Proposition 0.3 implies the part of the separation of tangent directions, which we prove below.

Proof of Proposition 0.3. If μ_{p} ≥ 3, then it follows from Theorem 1.11. Actually, it generates 1-jet. So we assume that μ_{p} < 3. We set μ = ord_{W} Δ.

From Riemann-Roch theorem and (0.1), we can create a Q-Cartier divisor D on X such that

η_{p} := ord_{p}D > 3 − μ_{p}.  (3.26)

We set η := ord_{W} D. Then by tie-breaking technique, we can create a Qdivisor G ∼_{lin}^{Q} λA such that Δ + G is critical at W with the critical variety Z, and λ < 1. We note that essentially, G = λD. Content-Length: 156 X-Lines: 4 Status: RO

If the critical variety is a point, then again Theorem 1.11 immediately implies that |K_{X} + Δ + A| separates W. So we assume that Z is a curve.

From tie-breaking, we may assume that

mult_{p}⌊Δ + G⌋ ≤ 2,

thus mult_{p}Z ≤ 2. Below, we separate into three cases. Content-Length: 55 X-Lines: 1 Status: RO

(2.1) First, assume that mult_{p}Z = 1 and v ∈ T_{p}Z.

From (0.2) and the Riemann-Roch theorem, we can create a Q-divisor B ∼_{lin}^{Q} A|_{Z} on Z so that 2 · ord_{W} B > σ_{1} − ϵ for some 0 < ϵ << 1. Since we have surjections

I_{W/X}^{k} ⟶ I_{W/Z}^{k} ⟶ 0

for all k ∈ ℕ, we can find a local lifting D^{loc} of B with

2 · ord_{W} D^{loc} > σ_{1} − ϵ.  (3.27)

Let D′ be a nice lifting of B with respect to Δ + G. From Theorem 1.14, there exist 0 < λ′ such that

λ′(σ_{1} − ϵ) ≤ 4 − 2 · μ − 2 · (1 − δ)ord_{W} G

and

W ⊂ Z(Δ + (1 − δ)G + λ′D′)

for some 0 < δ << 1. From the assumption, μ_{p} + λη_{p} ≥ 1 and 2μ + 2λη − (μ_{p} + λη_{p}) ≥ 1. Thus

\frac{4 − 2 · μ − 2 · λη}{1 − λ} ≤ \frac{3 − μ_{p} − λη_{p}}{1 − λ} < \frac{2σ_{2}}{σ_{2} − 1 + μ_{p}} .

Thus λ + λ′ < 1 and |K_{X} + Δ + A| separates W. Content-Length: 819 X-Lines: 27 Status: RO

(2.2) Second, assume that mult_{p}Z = 1 but v ∉ T_{p}Z. In this case, we may assume that

ord_{p}⌊Δ + G⌋ = 2.  (3.28)

Otherwise, for the same reason as the case (3.18), Theorem 1.11 would imply that |K_{X} + Δ + A| separates W.

Now from (0.2) and the Riemann-Roch theorem, we can create a Q-divisor B ∼_{lin}^{Q} A|_{Z} on Z so that ord_{p}B > σ_{1} − ϵ for some 0 < ϵ << 1. Since we have surjections

I_{W/X}^{k} ⟶ I_{W ∩Z/Z}^{k} ⟶ 0

for all k ∈ ℕ, we can find a local lifting D^{loc} of B with

ord_{W} D^{loc} > (σ_{1} − ϵ).  (3.29)

Note the difference between (3.27) and (3.29). Let D′ be a nice lifting of B with respect to Δ + G.

From Theorem 1.14, there exist 0 < λ′ such that

2 · λ′(σ_{1} − ϵ) ≤ 4 − 2 · μ − 2 · (1 − δ)ord_{W} G

and

W ⊂ Z(Δ + (1 − δ)G + λ′D′)

for some 0 < δ << 1. Since μ_{p} + λη_{p} ≥ 2,

\frac{4 − 2 · μ − 2 · λη}{1 − λ} ≤ \frac{4 − μ_{p} − λη_{p}}{1 − λ} < \frac{2σ_{2}}{σ_{2} − 2 + μ_{p}} .

But from (0.1),

\frac{2σ_{2}}{σ_{2} − 2 + μ_{p}} ≤ \frac{4σ_{2}}{σ_{2} − 1 + μ_{p}} .

Thus λ + λ′ < 1 by (0.3). Content-Length: 6181 X-Lines: 202 Status: RO

(2.3) Assume that mult_{p}Z = 2. The proof is identical to that of case (2.2) in Proposition 3.5, except the parts following inequality (3.25). We replace them with the followings.

Z · ⌈(Δ + A − λD)⌉
=Z · ((1 − λ)A) + Z · (Δ + {λD})
≥(1 − λ)σ_{1} + ord_{p}(Δ + {λD})
≥(1 − λ)σ_{1} + λη_{p} + μ_{p} − 2.

So (3.25) follows from

σ_{1} > \frac{3 − μ_{p} − λη_{p}}{1 − λ}

taking ϵ and δ small enough. But λη_{p} + μ_{p} ≥ 1, (in fact ≥ 2). Thus as before,

\frac{3 − μ_{p} − λη_{p}}{1 − λ} < \frac{2σ_{2}}{σ_{2} − 1 + μ_{p}} .

Next we use Proposition 2.2 to separates distinct points in Proposition 3.7. Most of the proof is overlapping with that of Proposition 3.3. So we only sketch them.

Sketch of Proof. Here we keep the same notation introduced in Proposition 3.3. The only case that need to be considered is when the critical variety Z is a curve and G is critical at p and also at q. Here we need to show that the following inequality holds

2 − λη < (1 − λ)2.  (3.30)

One can easily check that if ord_{W} D > 3, then (3.30) holds. So from now on, we will show that in fact we can create such D. We may assume that

η − 2 << 1.

Let Ω^{x} = Ω^{x}(W, A) and M(x) = M(x, W, Z). Since M(x) is a convex function of x, \frac{M(η)}{η} ≥ \frac{M(λη)}{λη}. Thus

M(η) ≥ \frac{1}{λ} ≥ \frac{η}{2}.

Let s_{γ} be the left derivative of M(x) at γ > 2. Let s = lim_{γ→2} s_{γ} and T (x) = s(x − 2) + 1. Again, since M(x) is convex, we may assume that T (x) ≤ M(x).

Suppose T (x) = M(x), for all x ≥ 2 with deg Ω^{x} > 0. We may assume that η − 2 << 1 so that T (η) = s(η − 2) + 1 < η − 1. Then, since \frac{1}{λ} = M(η) = T (η),

\frac{2 − λη}{1 − λ} = \frac{2M(η) − η}{M(η) − 1} = \frac{2T (η) − η}{T (η) − 1} < 1.

Thus (3.30) holds and |K_{X} + A| separates p and q.

Now we assume that T (x) < M(x) for some x ≥ 2 with deg Ω^{x} > 0. We have deg Ω^{x} ≥ 10 − 2x^{2}. Let us fix r > 2 and let β_{0} = 2 − \frac{1}{s}. Let

f(s) = 10 − 2 · β_{0}^{2} − 2 · ∫_{β_{0}}^{r} 2(x − T (x))dx.

Then simple calculation shows that f(s) achieves the minimum g(r) = −6+ 8r − 2r^{2}. g(r) > 0 for 2 ≤ r < 3. Since M(x) > T (x) for some x ≥ 2 with deg Ω^{x} > 0, by Proposition 2.2, g(3) ≥ 0 already implies that deg Ω^{{3+ϵ}} > 0 for a sufficiently small ϵ > 0. Thus deg Ω^{3} > 0.

4. On Higher Dimensions

Proof of Theorem 0.5. First, | − nK_{X}| induces morphism for n ≥ 2 and | − nK_{X}| separates two distinct points for n ≥ 4 (cf. Corollary 2.6, 2.10 in [17]).

Now let p be a closed point in X and v ∈ ℙ(T_{p}X). We choose a generic D from |−2K_{X} ⊗I_{p}|. We separates into two cases according to the singularities of D at p.

(1) Suppose that D is singular at p. Here we argue with the multiplier ideals. Let D_{1}, D_{2} be two generic elements of | − 2K_{X}| through p. Then by assumption, mult_{p}D_{i} ≥ 2 for i = 1, 2. Furthermore, since | − 2K_{X}| induces a finite morphism, we may assume that D_{1} and D_{2} are smooth and intersects properly in a punctured neighborhood of p. We pick a Q-divisor

Δ ∼_{lin}^{Q} ϵ · (−K_{X}) with 0 < ϵ << 1. Then

I_{Z((1−δ)(Δ+D_{1}+D_{2}))} ⊂ I_{p}^{2}

and Z((1−d)(Δ+D_{1}+D_{2})) has no positive dimensional components through p for sufficiently small 0 < δ << 1. Thus by Theorem 1.11, | − tK_{X}| is very ample.

(2) Now assume that D is smooth at p. If v ∉ ℙ(T_{p}D), then | − tK_{X}| separates {p, v} by the remark given in the beginning of the proof and the assumption t ≥ 4. So we assume that v ∈ ℙ(T_{p}X).

We note that D is connected since H^{1}(X, O_{X}(−2K_{X})) = 0, and K_{D} = −K_{X}|_{D} and −tK_{X}|_{D} = tK_{D}. We may assume that D is smooth. In fact, D is normal by the assumption. Thus (if necessary) we can work on the minimal resolution of D instead of D. Then the pull back of −K_{X}|_{D} is no longer ample but it is still nef and big and the bound of the intersection numbers with the curves through p is unchanged.

Now we apply Proposition 3.7 to (D, tK_{D}). Since K_{D}^{2} ≥ 2 and t ≥ 4, |tK_{D}| separate v at p. Then since H^{1}(X, O_{X}(−(2 + t)K_{X})) = 0, | − tK_{X}| also separates v at p.

Remark 4.1. The argument in (1) works with D being smooth at p. Only then the bound is worse. This way, one can obtain effective bounds on smooth varieties with (anti)canonical ample divisors.

The proof of Proposition 0.6 is similar, so we leave it to the interesting readers.

Proof of Theorem 0.7. We begin with the following simple lemma.

Lemma 4.2. With the same notations as above, we assume further that there is a Q-divisor G in X satisfying the following conditions:

1. H − G is nef and big.

2. Δ + G is 2-critical at p, and the 2-critical variety is a point. Then for generic D in |K_{X} +Δ+H ⊗I_{p}|, W ⊄ D where W is the connected component of Z(Δ + G) at p.

Proof. From the assumptions, we have the following exact sequence.

0 ⟶ O_{X}(K_{X} + Δ + H) ⊗ I_{Z(Δ+G)} ⟶ O_{X}(K_{X} + Δ + H) ⊗ I_{Z(Δ+(1−ϵ)G)} ⟶ J ⟶ 0,

where the connected component of Z(Δ + (1 − ϵ)G) at p is p itself, and J is supported at p for 0 < ϵ << 1. Thus H^{0}(X, J) ≠ 0. Content-Length: 0 X-Lines: 0 Status: RO

From Proposition 1.4, we also have the exact sequence in 0-th cohomology groups derived from the above exact sequence. Now we assume the contrary, i.e. every section of H^{0}(X, O_{X}(K_{X} + Δ + H) ⊗ I_{p}) is singular at p. Then since W has length 2 by the assumption 2,

H^{0}(X, O_{X}(K_{X} + Δ + H) ⊗ I_{Z(Δ+G)})

= H^{0}(X, O_{X}(K_{X} + Δ + H) ⊗ I_{Z(Δ+(1−ϵ)G)}),

and H^{0}(X, J) = 0. So it is a contradiction.

Content-Length: 473 X-Lines: 19 Status: RO

Below we will create a Q-divisor G satisfying the two conditions in the previous lemma.

Let q(k), k = 1, ⋯ n, be a sequence of rational numbers such that

H^{d} · Z > {∑_{k=1}^{n} q(k)}^{d}

for all subvariety Z through p where d = dim Z, and q(k) > \sqrt[k]{2} · k for all k = 1, ⋯, n. We set

A := \frac{1}{∑_{k=1}^{n} q(k)} H.

Let W_{0} be a length 2 subscheme of X supported at p and we put d_{0} := n. (1-st step) From the Riemann-Roch theorem we can create a Q-divisor D_{1} ∼_{lin}^{Q} q(d_{0})A

such that ord_{W_{0}}D_{1} > n. Then by tie-breaking technique, we can create a Q-Cartier divisor

G_{1} ∼_{lin}^{Q} λ_{1}q(d_{0})A

for some λ_{1} < 1 so that Δ + G_{1} is 2-critical at p with the 2-critical variety Z_{1}. We note that essentially G_{1} = λ_{1}D_{1}. Content-Length: 6673 X-Lines: 253 Status: RO

If Z_{1} is a point, then Lemma 4.2 finishes the proof. So we assume that d_{1} := dim Z_{1} > 0.

((i+1)-th step) Suppose that we have created a Q-Cartier divisor

G_{i} ∼_{lin}^{Q} λ_{i}(∑_{k=1}^{i} q(d_{k−1}))A

with λ_{i} < 1, such that Δ + G_{i} is 2-critical at p with the 2-critical variety Z_{i} at p with d_{i} := dim Z_{i} > 0.

Let T be a curve on Z_{i} obtained by generic hyperplanes through p. Now from the Riemann-Roch theorem, we can create a family of Q-divisors {B_{c}}_{c∈T −p} on Z_{i} such that

B_{c} ∼_{lin}^{Q} q(d_{i})A|_{Z_{i}}

and

ord_{W_{c}}B_{c} > d_{i}

where W_{c} is the length two subscheme of T supported at c.

Let D_{c}^{loc} be a local lifting of B_{c} such that ord_{W_{c}}D_{c}^{loc} > d_{i}. Then from Proposition 1.10,

W_{c} ⊂ Z(Δ + (1 − δ)G_{i} + D_{c}^{loc})  (4.1)

for sufficiently small 0 < δ << 1. Now we take the limit (cf. Theorem 6.7.1 [14])

B_{0} := limit_{c→0}B_{c}

and let D_{i+1} be a nice lifting of B_{0} with respect to Δ + G_{i}. Then from Proposition 1.9, Theorem 1.14 and (4.1), Z(Δ + (1 − δ)G_{i} + D_{i+1}) contains a length two subscheme. Thus we can find a Q-divisor

G_{i+1} ∼_{lin}^{Q} λ_{i+1}(∑_{k=1}^{i+1} q(d_{k−1}))A

for some λ_{i+1} < 1 so that Δ + G_{i+1} is 2-critical at p.

Acknowledgment. Parts of this paper is contained in author’s thesis at University of Illinois at Chicago under the guidance of Prof. Lawrence Ein. He would like to thank Prof. Robert Lazarsfeld and Prof. Vladimir Maşek for their generous support during my Ph.D program. He would like to thank Prof. Takao Fujita, Prof. Stefan Helmke, Prof. Yujiro Kawamata for their valuable advise. While preparing this paper, the author has enjoyed stay at Korea Institute for Advanced Study, Max-Planck-Institut für Mathematik at Bonn, RIMS, Tokyo Institute of Technology, University of Tokyo, and Waseda University He would like to thank those institutes for their warm hospitality. He would also like to thank the organizers of the wonderful workshop held in Bedlewo Poland September 1999, where he had opportunities to discuss about this paper with other mathematicians. Finally, he would like to thank Prof. Lawrence Ein for his constant interest and encouragement.

References

1. U. Angehrn and Y.-T. Siu, Effective Freeness and Point Separation for Adjoint Bundles Inv. Math, 122 (1995), pp. 291–308

2. L. Ein, Multiplier ideals, Vanishing theorem and Applications Proc. Sympos. Pure. Math. Part1. 62, Amer. Math. Soc., Providence, RI, (1997) pp.203–219

3. L. Ein and R. Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefold J. Amer. Math. Soc, 6 (1993), pp.875–903

4. L. Ein and R. Lazarsfeld, multiplier ideals manuscript

5. L. Ein, R. Lazarsfeld, and M. Nakamaye, Zero-estimate, intersection theory, and a theorem of Demailly Higher-dimensional complex varieties, de Gruyter, Berkin, (1996), pp.183–207

6. T. Fujita, Birational geometry of algebraic varieties: Open problem the 23rd International Symposium of the Division of Mathematics of the Taniguchi Foundation, Katata, (1988), pp.42–45

7. T. Fujita Remarks on Ein-Lazarsfeld criterion of spannedness of adjoint bundles of polarized threefold alg-geom eprint 9311013

8. S. Helmke On Fujita’s conjecture Duke Math. J, 88 (1997), pp.201–216

9. S. Helmke On global generation of adjoint linear systems Math. Ann. 313 (1999), pp.635–652

10. Y. Kawamata On Fujita’s freeness conjecture for 3-folds and 4-folds Math. Ann. 308 (1997), pp.491–505 11. Y. Kawamata, K. Matsuda, K. Matsuki Introduction to the minimal model problem

Adv. Stud. Pure. Math., 10, North-Holland, Amsterdam-New York, (1987) pp.283– 360

12. T. Kawachi and V. Maşek Reider-type theorems on normal surfaces J. Alg. geom. 7 (1998), pp.239–249

13. J. Kollár Effective base-point freeness Math. Ann., 296 (1993), pp.595–605

14. J. Kollár Singularities of Pairs Proc. Sympos. Pure. Math. Part1. 62, Amer. Math. Soc., Providence, RI, (1997) pp.203–219

15. A. Langer Adjoint linear systems on normal surfaces J. Algebraic Goem. 8 (1999), pp.41–66

16. A. Langer Adjoint linear systems on normal log surfaces preprint

17. S. Lee Remarks on the pluricanonical and the adjoint linear systems on projetive threefolds Comm. in Alg., 27 (1999), pp.4459-4476.

18. V. Maşek Thesis, UCLA (1993)

19. V. Maşek Kawachi’s invariant for normal surface singularities Int. J. Math. 9 (1998), pp.623–640

20. V. Maşek Very ampleness of adjoint linear systems on smooth surfaces with boundary Nagoya Math. J. 153 (1999), pp.1–29

21. I. Reider Vector bundles of rank 2 and linear systems on algebraic surfaces Ann. of Math. 127 (1988), pp.309–316