Contemporary Mathematics

On the second adjunction mapping and the very ampleness of the triadjoint bundle

Mauro C. Beltrametti and Andrew J. Sommese

In Memoriam: Professor Wei-Liang Chow

Abstract. Let L̂ be a very ample line bundle on a three dimensional projective manifold X̂, i.e., assume that L̂ ≈ i^{∗}O_{ℙ^{N}} (1) for some embedding i : X̂ ↪ ℙ^{N}. In this article, we continue a study, started in previous articles, of the rational map, ψ : X̂ → ℙ^{h−1}, associated to |KX̂ + L̂|, where h := h^{0}(KX̂ +L̂) under the assumption that the Kodaira dimension of KX̂ +L̂ is three. The degree properties of the map ψ when dim ψ(X̂) = 2 are studied, e.g., it is shown that in this case if h ≥ 6, e.g., if X̂ is of nonnegative Kodaira dimension, then the degree of the mapping ψŜ for Ŝ ∈ |L̂| is at most 13, and the degree of the finite part of the Remmert-Stein factorization of ψ is at most 4. If dim ψ(X̂) = 3 then the degree of the mapping ψ is at most 53. Further it is shown that the morphism associated to |3(KX̂ + L̂)| factors as the natural morphism from X̂ to the second reduction of (X̂, L̂) followed by an embedding.

Introduction

For simplicity let X̂ ⊂ ℙ^{N} be a connected three dimensional algebraic submanifold of N-dimensional complex projective space. The n ≥ 4 dimensional analogues of the results we discuss below are easy consequences of the three dimensional results. The very complete theory [3] of the adjunction mapping, i.e., the mapping associated to |KX̂ + 2L̂|, is a powerful tool to study the relations between the curve sections on X̂ and the projective geometry of X̂. The second adjunction mapping, i.e., the mapping associated to |KX̂ + L̂|, is a natural tool to study the relations between surface sections and the projective geometry of X̂. Our knowledge of this mapping is in a much more primitive state than our knowledge of the adjunction mapping. In a series of articles [18, 4, 5] we have been studying different aspects of

1991 Mathematics Subject Classification. 14E35, 14C20, 14J40. The first author thanks the Sonderforschungsbereich 170 of the University of Göttingen for

their support.

The second author thanks the Sonderforschungsbereich 170 of the University of Göttingen, the Alexander von Humboldt Stiftung, and the National Science Foundation (DMS 93-02121) for their support.

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this mapping. We know that the mapping exists [18] except for a very special class of degenerate varieties. Moreover [4] we have some strong lower bounds for the number of sections of |KX̂ + L̂|, e.g., if the Kodaira dimension of X̂ is nonnegative, then h^{0}(KX̂ + L̂) ≥ 5 with equality only if (X̂, L̂) is a quintic hypersurface of ℙ^{4}. In [5] we studied the case when the mapping associated to |KX̂ + L̂| has a one dimensional image. In analogy with surface results of Beauville [1] we showed that in this case the mapping is well behaved, e.g., if the Kodaira dimension of X̂ is nonnegative, and the mapping associated to |KX̂ +L̂| has a one dimensional image, then the mapping is a morphism and, if h := h^{0}(K_{X} + L) ≫ 0 (e.g., h ≥ 21), then the image of the morphism is a curve of genus ≤ 1 with K3 surfaces as fibers.

In this article we study the degree of the mapping associated to |KX̂ + L̂| when its image is two or three dimensional, and we further give a complete description of the mapping associated to |3(KX̂ + L̂)|.

In §1 we introduce background material we use in the form we need it and prove a number of preliminary technical lemmas.

In §2 we prove some general facts about the rational mapping ψ : X̂ → ℙ^{h−1} associated to |KX̂ + L̂|.

In §3 we study the degree properties of the map ψ when dim ψ(X̂) = 2 and the Kodaira dimension of KX̂ + L̂ is three. For example, we show that in this case if h^{0}(KX̂ + L̂) ≥ 6, e.g., if X̂ is of nonnegative Kodaira dimension, then the degree of the mapping ψŜ for Ŝ ∈ |L̂| is at most 13, and the degree of the finite part of the Remmert-Stein factorization of ψ is at most 4.

In §4 we study the degree properties of the map ψ when dim ψ(X̂) = 3. For example, we show that the degree of the mapping ψ is at most 53. In §4 we also show that the morphism associated to |3(KX̂ + L̂)| factors as the natural morphism (described in §1) from X̂ to the second reduction of (X̂, L̂) followed by an embedding.

We will study the mapping associated to |2(KX̂ + L̂)| in a sequel.

1. Background material

We refer to our book [3] for all the results on adjunction theory that we need in this paper and also for a complete list of references to the original sources.

We work over the complex numbers ℂ. Throughout the paper we deal with projective varieties V. We denote by O_{V} the structure sheaf of V and by K_{V} the canonical bundle. For any coherent sheaf F on V, h^{i}(F) denotes the complex dimension of H^{i}(V, F).

Let L be a line bundle on V. The line bundle L is said to be numerically effective (nef, for short) if L · C ≥ 0 for all effective curves C on V. L is said to be big if κ(L) = dim V, where κ(L) denotes the Kodaira dimension of L. If L is nef then this is equivalent to c_{1}(L)^{n} > 0, where c_{1}(L) is the first Chern class of L and n = dim V.

1.1. Notation. The notation used in this paper are standard from algebraic geometry. Let us only fix the following. ≈ (respectively ∼), linear (respectively numerical) equivalence of line bundles; χ(L) = ∑_{i}(−1)^{i}h^{i}(L), the Euler characteristic of a line bundle L; |L|, the complete linear system associated to a line bundle L on a variety V, Γ(L) = H^{0}(L), the space of the global sections of L; e(V), the topological Euler characteristic of V; c_{n}(V), the n-th Chern class of the tangent bundle of V for V smooth, which for V compact equals e(V); κ(V) := κ(K_{V}), the Kodaira dimension, for V smooth; ⌈x⌉, the smallest integer bigger or equal to a rational number x.

We say that a sheaf on a variety is spanned if global sections span it at all points of the variety. Line bundles and divisors are used with little (or no) distinction. Hence we shall freely switch from the multiplicative to the additive notation and vice versa.

1.2. Genus formula. For a line bundle L on a variety V of dimension n the sectional genus, g(L) = g(V, L), of (V, L) is defined by 2g(L) − 2 = (K_{V} + (n − 1)L) · L^{n−1}. Note that if |L| contains n − 1 elements H_{1}, …, H_{n−1} meeting in a reduced irreducible curve C, then g(L) = g(C) = 1 − χ(O_{C}), the arithmetic genus of C.

1.3. Reductions. (see e.g., [3, Chapters 7, 12]). Let (X̂, L̂) be a smooth projective variety of dimension n ≥ 2 polarized with a very ample line bundle L̂. A smooth polarized variety (X, L) is called a (first) reduction of (X̂, L̂) if there is a morphism π : X̂ → X expressing X̂ as the blowing up of X at a finite set of points, B, such that L := (π_{∗}L̂)^{∗∗} is ample and L̂ ≈ π^{∗}L − [π^{−1}(B)] or, equivalently, KX̂ + (n − 1)L̂ ≈ π^{∗}(K_{X} + (n − 1)L).

Note that there is a one-to-one correspondence between smooth divisors of |L| which contain the set B and smooth divisors of |L̂|.

Except for an explicit list of well understood pairs (X̂, L̂) (see in particular [3, §§7.2, 7.3, 7.4]) we can assume:

a) KX̂ + (n − 1)L̂ is spanned and big, and K_{X} + (n − 1)L is very ample. Note that this reduction, (X, L), is unique up to isomorphism. We will refer to it as the first reduction of (X̂, L̂);

b) K_{X} + (n − 2)L is nef and big, for n ≥ 3.

Note that h^{0}(KX̂ + (n − 2)L̂) = h^{0}(K_{X} + (n − 2)L). Indeed one has (see [3, (7.6.1)])

Γ(aKX̂ + bL̂) ≅ Γ(aK_{X} + bL) (1)

for integers a, b with b ≤ a(n − 1). Hence in particular

H^{0}(KX̂ + L̂) ≅ H^{0}(K_{X} + L).

It thus follows that the rational map ψ̂ associated to |KX̂ + L̂| has the same degree and the same dimension and degree of the image as the rational map ψ associated to |K_{X} + L|.

Note also that κ(KX̂ + (n − 2)L̂) ≥ 0 implies κ(KX̂ + (n − 1)L̂) = n, so that the first reduction (X, L) exists, as well as K_{X} + (n − 2)L is nef (see [3, (7.6.9]).

Since by the above we can assume that K_{X} + (n − 2)L is nef and big, from the Kawamata-Shokurov base point free theorem (see [12], §3) we know that |m(K_{X} + (n − 2)L)|, for m ≫ 0, gives rise to a morphism φ : X → X′ with connected fibers and normal image. Thus there is an ample line bundle K′ on X′ such that K_{X} +(n−2)L ≈ φ^{∗}K′. The pair (X′, K′) is known as the second reduction of (X̂, L̂). The morphism φ is very well behaved (see e.g., [3, §§7.5, 7.6, 7.7 and Chapter 12] for complete results). Furthermore X′ has terminal, 2-Gorenstein (i.e., 2K_{X′} is a line bundle) isolated singularities and K′ ≈ K_{X′} + (n − 2)L′, where L′ := (φ_{∗}L)^{∗∗} is a 2-Cartier divisor such that 2L ≈ φ^{∗}(2L′) − D for some effective Cartier divisor D on X which is φ-exceptional (see [3, (7.5.7)]). For definition and properties of terminal singularities we also refer to [12].

We will use in a crucial way the following theorem [3, (12.2.1)] which gives the structure of the second reduction in the threefolds case.

Theorem 1.4. Let X̂ be a connected smooth threefold with L̂ a very ample line bundle on X̂. Assume that the first reduction (X, L) and the second reduction (X′, K′), φ : X → X′, as in (1.3) exist. Then φ is an isomorphism outside of an algebraic subset Z of X′ such that dim Z ≤ 1. We have:

a) Any 1-dimensional component of Z is a smooth curve, C, such that C ⊂ reg(X′), φ^{−1}(C) = D ∪ {∪D_{i}}_{i∈I} where D and the D_{i}’s are smooth irreducible divisors. The restriction φ_{D} of φ to D is a ℙ^{1}-bundle φ_{D} : D → C and N_{D/X|F} ≅ O_{ℙ^{1}} (−1) for a fiber F of φ_{D}. Furthermore if I is not empty, φ(D_{i}) is a point for each i ∈ I. Let x_{i} = φ(D_{i}), i ∈ I. Then φ^{−1}(x_{i}) = 𝔽_{1}, and for each i ∈ I, D meets 𝔽_{1} transversally along the irreducible curve, E, of 𝔽_{1}, with (E^{2})_{𝔽_{1}} = −1.

b) If x is a 0-dimensional component of Z then φ^{−1}(x) is a reduced divisor, D,

and either b1) D ≅ ℙ^{2}, N_{D/X} ≅ O_{ℙ^{2}} (−2); b2) D ≅ ℙ^{2}, N_{D/X} ≅ O_{ℙ^{2}} (−1); b3) D ≅ 𝔽̃_{2}, N_{D/X} ≅ −G where −2G ≅ K_{D}; b4) D ≅ 𝔽_{0}, N_{D/X} ≅ O_{𝔽_{0}}(−1, −1); or b5) D = D_{1} ∪ D_{2} where D_{1} ≅ 𝔽_{2}, D_{2} ≅ ℙ^{2}, N_{𝔽_{2}/X} ≈ −E − f, N_{ℙ^{2}/X} ≅

O_{ℙ^{2}} (−2) and D_{1}, D_{2} meet transversally in a smooth rational curve which is the irreducible curve, E, of 𝔽_{2} with (E^{2})_{D_{1}} = −2 and is a line on ℙ^{2}.

Remark 1.5. Let X̂ be a smooth n-fold with L̂ a very ample line bundle. Assume that the first reduction (X, L) of (X̂, L̂) exists. Further assume that K_{X} + (n − 2)L is nef. Then 2(K_{X} + (n − 2)L) is spanned by global sections.

Let p : X → Y be the morphism with connected fibers and normal image associated to |m(K_{X} + (n − 2)L|, m ≫ 0. Then K_{X} + (n − 2)L ≈ p^{∗}L for some ample line bundle L on Y.

If dim Y = 3 the result is proved in [18, Note added in proof].

Assume that dim Y = 2. Then (X, L) is a quadric fibration under p and from [6, §8] we know that Y is smooth and L ≈ K_{Y} + M for some ample line bundle M on Y. Thus 2L ≈ K_{Y} + L + M.

We claim that (L + M)^{2} ≥ 5. Indeed, if not,

(L + M)^{2} = L^{2} + 2L · M + M^{2} = 4,

whence L^{2} = L · M = M^{2} = 1. Therefore L ∼ M by the Hodge index theorem. Then K_{Y} ≈ O_{Y}. This contradicts the parity condition from genus formula (1.2).

Thus by Reider’s theorem either 2L ≈ K_{Y} + (L + M) is spanned, and hence 2(K_{X} + (n − 2)L) is spanned, or there exists an effective curve C on Y such that

(L + M) · C ≤ C · C < \frac{(L + M) · C}{2} < 1.

This contradicts the ampleness of L and M.

Assume that dim Y = 1. Then (X, L) is a Del Pezzo fibration under p and exactly the same argument as in [18, (0.5.2)] applies to say that 2L is spanned.

1.6. Pluridegrees. Let (X̂, L̂), (X, L) be as in (1.3) with n = 3. Define the pluridegrees, for j = 0, 1, 2, 3, by

d̂_{j} := (KX̂ + X̂)^{j} · L̂^{3−j} and d_{j} := (K_{X} + L)^{j} · L^{3−j}.

If γ denotes the number of points blown up under π : X̂ → X, then because KX̂ + L̂ ≅ K_{X} + L + ∑_{i} E_{i}, the invariants d̂_{j}, d_{j} are related by

d̂_{0} = d_{0} − γ ; d̂_{1} = d_{1} + γ ; d̂_{2} = d_{2} − γ ; d̂_{3} = d_{3} + γ.

We put d̂:= d̂_{0} , d := d_{0}. If K_{X} +L is nef, by the generalized Hodge index theorem (see e.g., [3, (2.5.1), (13.1)], [9, (1.2)]) one has

d_{1}^{2} ≥ dd_{2} and d_{2}^{2} ≥ d_{1}d_{3} (2)

and the parity Lemma (13.1.1) of [3] says that

d ≡ d_{1} mod(2) ; d_{2} ≡ d_{3} mod(2). (3)

Moreover if K_{X} + L is nef and big the numbers d_{j} are positive.

We need the following versions of the double point formula (for the proofs and full details see [4, (0.5)] and also [3, (8.2), (13.1)]).

Proposition 1.7. Let (X̂, L̂) be a smooth projective 3-fold, polarized with a very ample line bundle, L̂. Assume κ(KX̂ + L̂) ≥ 2. Let (X, L) and π : X̂ → X be the first reduction and the first reduction map respectively. Let d̂_{j}, d_{j}, 0 ≤ j ≤ 3, be the pluridegrees of (X̂, L̂) and (X, L) respectively. Let γ be the number of points blown up by π. Let Ŝ be a smooth element in |L̂|. Then

44h^{0}(KX̂ + L̂) + 58χ(OŜ) + 2h^{0}(KX̂) + 4 ≥ 12d_{2} + 17d_{1} + d_{3} + (20 − d̂)d̂+ 5γ.

There is the following slightly stronger version of the usual double point formula (1.7).

Proposition 1.8. Let (X̂, L̂) be a smooth projective 3-fold, polarized with a very ample line bundle, L̂. Let (X, L) and π : X̂ → X be the first reduction and the first reduction map respectively. Let d̂_{j}, d_{j}, 0 ≤ j ≤ 3, be the pluridegrees of (X̂, L̂) and (X, L) respectively. Let γ be the number of points blown up by π. Let Ŝ be a smooth element in |L̂|. Let q := h^{1}(O_{X}). Then

44h^{0}(KX̂ + L̂) + 60χ(O_{S}) + 2h^{0}(KX̂) + 2q − 2 ≥ 13d_{2} + 17d_{1} + d_{3} + (20 − d̂)d̂+ 5γ.

Proof. Let S be a smooth surface in |L| corresponding to Ŝ. Let p_{g} := h^{0}(K_{X}), p_{g}(S) = h^{0}(K_{S}). The proof is the same as in [4, (0.5.2)]. The only difference is to leave the q term and not using the Noether inequality K_{S}^{2} ≥ 2p_{g}(S)− 4 in relation (0.5.2.5) of [4]. Thus, instead of that relation we find,

e(X̂) ≤ −4h^{0}(KX̂ + L̂) + 2p_{g} + 24χ(O_{S}) − 2d_{2} + 2q − 2 + 2γ.

Substituting in (0.5.2.1) of [4] we get the result.

The following inequalities follow from the log version of the usual Yau inequality given in [20, §5] (see [4, (0.6)] or [3, (13.1.7), (13.1.8)] for full details).

Proposition 1.9. (Tsuji’s inequality) Let (X̂, L̂) be a smooth projective 3-fold, polarized with a very ample line bundle, L̂. Assume that κ(KX̂ +L̂) ≥ 0. Let (X, L) be the first reduction of (X̂, L̂). Let S be a smooth element of |L|. Then we have

(K_{X} + L)^{3} + \frac{8}{3}K_{S} · L_{S} ≤ 32(2h^{0}(K_{X} + L) − χ(O_{S})),

or

h^{0}(K_{X} + L) ≥ \frac{d_{3}}{64} + \frac{d_{1}}{24} + \frac{χ(O_{S})}{2}.

1.10. Castelnuovo’s bound. Let C be a reduced, irreducible projective curve. Assume that ψ : C → ℙ^{N} is a generically one-to-one morphism, and that ψ(C) does not lie in any hyperplane. Let d denote the degree of ψ(C) in ℙ^{N}. Let g(C) be the arithmetic genus of C. Then Castelnuovo’s bound (see e.g., [8, Theorem 3.7]) reads

g(C) ≤ Castel(d, N) := [\frac{d − 2}{N − 1}] (d − N − ([\frac{d − 2}{N − 1}] − 1) \frac{N − 1}{2}) ,

where [x] means the greatest integer ≤ x.

1.11. Threefolds of log-general type. Let (X̂, L̂), (X, L) be as in (1.3) with n = 3 and let d̂_{j}, d_{j}, j = 0, 1, 2, 3, be the pluridegrees as in (1.6). We say that (X̂, L̂) is of log-general type if K_{X} + L is nef and big. Hence in particular the numbers d_{j} are positive in this case.

Assume (X̂, L̂) of log-general type. Let Ŝ be a smooth element of |L̂| and S the corresponding smooth surface in |L|. Since K_{X} + L is nef and big the canonical bundle K_{S} of S is nef and big (see e.g., [3, (7.6.10)]), so that S is a minimal surface of general type. Hence we have

d_{2} = K_{S} · K_{S} < 9χ(O_{S}). (4)

Indeed the Miyaoka inequality yields d_{2} ≤ 9χ(O_{S}). Note that the equality cannot happen. Otherwise S is a ball quotient and hence a K(Π, 1), which would contradict [16, (1.3)]. Throughout the paper we will often use (4) in the form

\frac{χ(O_{S})}{2} ≥ \frac{d_{2} + 1}{18}. (5)

Note that since χ(O_{S}) is an integer we have the stronger integrality relation

χ(O_{S}) ≥ ⌈\frac{d_{2} + 1}{9}⌉ . (6)

Note that if (X̂, L̂) is of log-general type then the restriction, φ_{S}, of the second reduction map φ to S coincides with the m-canonical map associated to |mK_{S}|, m ≥ 2. This follows from the exact sequence

0 → K_{X} + (m − 1)(K_{X} + L) → m(K_{X} + L) → mK_{S} → 0,

by noting that K_{X} + L is nef and big. Therefore the image S′ := φ_{S}(S) is the canonical model of S.

Assume that κ(X) ≥ 0. Then from [17, (1.5), (3.1)] (see also [3, (13.1.3)]), we know that

d_{3} ≥ d_{2} ≥ d_{1} ≥ d. (7)

Note that if κ(X) ≥ 0, then (X̂, L̂) is of log-general type. Indeed, if h^{0}(tK_{X}) > 0 for some positive integer t, then t(K_{X} + L) gives a birational embedding, given on a Zariski open set by sections of Γ(L), and thus κ(K_{X} + L) = 3.

The following result is not optimal, but it is more than enough for our purposes.

Lemma 1.12. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Then either |L̂| embeds X̂ in ℙ^{4} or

1. d ≥ d̂= L̂^{3} ≥ 8;

2. d_{1} ≥ 6, d_{2} ≥ 3, and d_{3} ≥ 1.

Proof. We can assume that Γ(L̂) embeds X̂ in ℙ^{N} with N ≥ 5. Since a smooth Ŝ ∈ |L| is of general type we have by a result of Castelnuovo (see [3, Theorem (8.1.1)], [14, (0.6)]) that d̂ > 2(N − 1 − 2) + 2 ≥ 6. Thus d ≥ d̂ ≥ 7. If we show that d̂≠ 7 we are done with showing the degree lower bound. To see this note that N = 5 since otherwise d̂> 2(N − 1 − 2) + 2 ≥ 8. Thus by Castelnuovo’s bound we conclude that g(C) ≤ 6 for any smooth curve section of X̂ ⊂ ℙ^{5}. Thus we conclude that d_{1} ≤ d̂_{1} ≤ 3. But then d_{2} ≤ 1 by the Hodge index relations (2). Since K_{X} + L is nef and big we know that d_{3} ≥ 1 and thus by the Hodge index relations we conclude that d_{1} ≤ 1. This implies by the Hodge index relations the absurdity that d = 1.

The inequalities d_{1} ≥ 6 and d_{2} ≥ 3 follow from the Hodge index relations and parity conditions (3).

The following result of the authors and M. Schneider [2, Lemma (3.3)] will be useful.

Lemma 1.13. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Then 6h^{0}(K_{X} + L) − d_{2} + d_{3} − 4χ(O_{S}) = 2h^{0}(2K_{X} + L). Furthermore h^{0}(2K_{X} + L) = 0 if either 2d_{3} < d_{2}, or 2d_{2} < d_{1}, or 2d_{1} < d.

Proof. The result [2, Lemma (3.3)] is stated for threefolds in ℙ^{5}, but the argument for the first assertion works with no change in general. The statements about h^{0}(2K_{X} + L) = 0 all follow by noting that 2K_{X} + L = 2(K_{X} + L) − L and that K_{X} + L and L are nef and big.

The following will be useful.

Lemma 1.14. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Assume that h^{0}(L̂) ≥ 6 (which follows from |KX̂ + L̂| not giving a birational mapping). Then

1. if d_{3} ≥ 9 it follows that d ≥ 9, d_{1} ≥ 9, d_{2} ≥ 9;

2. if d_{3} ≥ 18 it follows that d ≥ 10, d_{1} ≥ 14, d_{2} ≥ 16;

3. if d_{3} ≥ 81 it follows that d ≥ 11, d_{1} ≥ 23, d_{2} ≥ 44;

4. if d_{2} ≥ 9 it follows that d ≥ 9, d_{1} ≥ 9;

5. if d_{2} ≥ 19 it follows that d ≥ 10, d_{1} ≥ 14;

6. if d_{2} ≥ 41 it follows that d ≥ 11, d_{1} ≥ 23.

Proof. All of these are proved the same way. For example if d_{3} ≥ 81 then using the fact that d ≥ 8, d_{1} ≥ 6, d_{2} ≥ 3 we see by using the Hodge index inequalities dd_{2} ≤ d_{1}^{1}, d_{1}d_{3} ≤ d_{2}^{2} repeatedly and taking account of parity (3) that d_{1} ≥ 18. Using Castelnuovo’s inequality (1.10) we see that for degree at most 8 the genus g of a curve section of X̂ is bounded by 9. In this case we see by the genus formula that d + d_{1} ≤ 16. This implies d ≥ 9. But in this case Castelnuovo’s inequality implies that g ≤ 12 and hence d + d_{1} ≤ 22. So d ≥ 10. Using this, the Hodge index inequalities repeatedly and taking account of parity we conclude that d_{1} ≥ 22. Again using Castelnuovo’s inequality we get the contradiction that d_{1} ≤ 20. Thus d ≥ 11. Use d ≥ 11 and the Hodge inequalities again to prove the inequalities d_{1} ≥ 23, d_{2} ≥ 44.

We also need the following general fact.

Lemma 1.15. Let F be a rank r generically spanned vector bundle on an irreducible and reduced projective variety X such that c_{1}(det(F)) = 0. Then F is trivial, i.e., F ≅ ⊕^{r}O_{X}.

Proof. Choose r sections s_{j} ∈ Γ(X, F) that span F at the generic point. Then s := s_{1} ∧ ⋯ ∧ s_{r} is a non trivial section of det F which is not zero at the generic point. Let D = [s] be the divisor associated to s. Since c_{1}(det F) = 0 we see that D is empty, and hence S is nowhere zero. Thus the s_{j}’s are nowhere zero and linearly independent at every point. Therefore they generate F and F is trivial.

For the reader’s convenience we also include the main result that we use from [4] and which allows us to assume h := h^{0}(K_{X} +L) ≥ 6 in the nonnegative Kodaira dimension case.

Theorem 1.16. Let L̂ be a very ample line bundle on a projective manifold X̂ of dimension 3. If the Kodaira dimension of KX̂ + L̂ is 3 then h^{0}(KX̂ + L̂) ≥ 2. If the Kodaira dimension of X̂ is nonnegative, then h^{0}(KX̂ + L̂) ≥ 5 with equality only if (X̂, L̂) is a degree 5 hypersurface of ℙ^{4}.

Corollary 1.17. Let L̂ be a very ample line bundle on a projective manifold X̂ of dimension 3. If the Kodaira dimension of KX̂ + L̂ is 3 then p_{g}(Ŝ) ≥ 2 for any smooth Ŝ ∈ |L̂|.

Proof. By Theorem (1.16) we know that h^{0}(KX̂ + L̂) = h^{0}(K_{X} + L) ≥ 2. Then, since L̂ is very ample, we conclude from [4, (0.11)] that

p_{g}(S) = p_{g}(Ŝ) = h^{0}(KŜ) = h^{0}(K_{X̂|Ŝ} + L̂Ŝ) ≥ 2.

2. Notation and general set up

Let (X̂, L̂) be a smooth threefold polarized with a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type, which implies that the first reduction (X, L), π : X̂ → X, of (X̂, L̂) exists and furthermore K_{X} + L is nef and big. Let Ŝ be a general element of |L̂| and let S be the corresponding smooth surface in |L|. Recall that the restriction πŜ : Ŝ → S maps Ŝ onto its minimal model, S, and S is of general type. Recall that h^{0}(KX̂ + L̂) = h^{0}(K_{X} + L) by (1).

Let ψ : X → ℙ^{n} be the rational map associated to |K_{X} + L|. Let Σ := ψ(X). Since from [4, (1.2)] we know that h^{0}(K_{X} + L) ≥ 2, we have dim Σ ≥ 1. In this paper we deal with the case when dim Σ = 2, 3 (see [5] for the case dim Σ = 1). Let h := h^{0}(K_{X} + L). Then Σ ⊂ ℙ^{h−1}, so that

(8) deg Σ ≥ h − dim Σ.

2.1. General construction. With the notation as above, let Γ ⊂ X × Σ be the graph of ψ. Let ρ : \overline{X} → Γ be a smooth resolution of Γ. Let σ : \overline{X} → X be the composition of ρ with the product projection X × Σ → X, and let \overline{ψ} : \overline{X} → Σ be the composition of ρ with the product projection X × Σ → Σ. Let \overline{ψ} = s ∘ \overline{r} be the Remmert-Stein factorization of \overline{ψ}, where \overline{r} is a morphism with connected fibers and s is a finite morphism. Thus we have a commutative diagram

\begin{matrix}\overline{X} − σ^{−1}(E) & ⟶^{\overline{r}} & C \\ σ ↓ & ↘^{\overline{ψ}} & ↓ s \\ X − E & ⟶^{ψ} & Σ \end{matrix} (9)

where E is the indeterminacy locus of ψ. Then there exists a rational map r := \overline{r} ∘ σ^{−1} : X → C such that ψ = s ∘ r. We say that s ∘ r is the Remmert-Stein factorization of the rational map ψ. Note also that the general fiber, F, of r is irreducible.

Thus we have a commutative diagram

\begin{matrix}X & ⟶^{ψ} & Σ \\ σ ↑ & ↘^{r} & ↑ s \\ \overline{X} & ⟶^{\overline{r}} & Y. \end{matrix}

We will denote K := K_{X} +L. Then r is the meromorphic map given by the complete linear system |K|. It is a general fact, see, e.g., [11, Chapter II, 7.17.3], that the smooth resolution ρ : \overline{X} → Γ can be chosen such that there exists a spanned line bundle \overline{K} on \overline{X} such that

\overline{K} ≈ σ^{∗}K − E, (10)

for some effective divisor E on \overline{X} with σ(E) set theoretically equal to the base locus of |K|, and Γ(K) ≅ Γ(\overline{K}) under the map given by the inclusion σ_{∗}\overline{K} ⊂ K. Note that \overline{r} is the morphism given by |\overline{K}| and \overline{K} = \overline{r}^{∗}H for some ample and spanned line bundle H on Y.

If dim Σ = 2, let 𝔣 be a general fiber of ψ and let f be a general fiber of r. Then note that

#s(L · f) = L · 𝔣, (11)

where #s denotes the degree of s.

The following simple lemma is useful.

Lemma 2.2. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Assume further that |K_{X} + L| gives a rational map, ψ. If h^{0}(K_{X}) > 0 then ψ is a birational map.

Proof. Indeed if h^{0}(K_{X}) > 0 a section of Γ(K_{X}) gives an embedding Γ(L) ↪ Γ(K_{X} + L) and hence, since L is very ample off a finite set of points, |K_{X} + L| defines a birational map, ψ, defined on a Zariski open set by sections of L.

We also need the following general fact.

Lemma 2.3. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that κ(X) ≥ 0. Assume further that |K_{X} + L| gives a rational map, ψ, having a 2-dimensional image. Let ψ = s ∘ r be the Remmert-Stein factorization of ψ as in (2.1). Assume that the base locus B of |K_{X} +L| is of codimension at least two. Then ψ is not a morphism and the general fiber f of r meets B.

Proof. First note that B is not empty. To see this note that since K_{X} + L is big, it would follow from B being empty that K_{X} + L is the pullback of a line bundle from the lower dimensional image under ψ.

To see that ψ is not a morphism choose a small complex neighborhood U of x ∈ B such that (K_{X} + L)_{U} is isomorphic to the trivial bundle and set Z := B ∩ U. Using this trivialization a basis of H^{0}(K_{X} + L) restricts to a set of holomorphic sections f_{0}, …, f_{N} which only have Z as common zeros on U and such that the map [f_{0}, …, f_{N}] has a two dimensional image. For ψ_{U} = [f_{0}, …, f_{N}] to be a morphism at x it is necessary that there is a holomorphic function g on U and holomorphic functions h_{0}, …, h_{N} on U such that f_{i} = gh_{i} for all i and h_{k}(x) ≠ 0 for some k. If this happened then we would have all of the f_{i} zero on the nontrivial zero set of g. This contradicts the codimension at least two hypothesis since Z contains the zero set of g.

With the same notation as in (2.1), let K = K_{X} +L, σ^{∗}K ≈ \overline{r}^{∗}H +E, where σ : \overline{X} → X, \overline{r} : \overline{X} → Y and H is an ample and spanned line bundle on Y. Let D_{1}, D_{2} be general members of |\overline{r}^{∗}H|. Since H is spanned we have that D_{1} ∩D_{2} ∼ (H ·H)\overline{f}, where \overline{f} is a general fiber of \overline{r}. From (8) and (1.16) we conclude that H · H ≥ 3. If the general fiber f of r does not meet B, it thus follows that σ(D_{1}) ∩ σ(D_{2}) is disconnected on X, since otherwise D_{1} ∩ D_{2} would consist of only one fiber.

Since the line bundle L := O_{X}(σ(D_{1})) on X agrees with K_{X} + L off the set B whose codimension is at least two, it follows that K_{X} + L ≈ L.

Look at the Koszul complex

0 → −2L → −L ⊕ −L → O_{X} → O_{σ(D_{1})∩σ(D_{2})} → 0.

Since L is nef and big h^{i}(−2L) = h^{i}(−L) = 0 for i < 3. On the other hand h^{0}(O_{X}) = 1 and h^{0}(O_{σ(D_{1})∩σ(D_{2})}) ≥ 2 by disconnectedness. This leads to a contradiction. Thus we conclude that the general fiber f of r meets B.

3. The second adjunction mapping: the case of a 2-dimensional image

We follow the notation in §2. In this section we assume that the rational map ψ : X → ℙ^{n} associated to |K_{X} + L| as an image Σ := ψ(X) of dimension 2. Let 𝔣 be a general fiber of ψ.

We have the following general bounds.

Lemma 3.1. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Assume further that |K_{X} + L| gives a rational map, ψ, having a 2dimensional image. Let 𝔣 be a general fiber of ψ and let K = K_{X} + L. Let h := h^{0}(K_{X} + L). Then

1. d_{1} = K · L · L > 0;

2. d_{2} = L · K · K ≥ L · 𝔣(h − 2);

3. d_{3} = K · K · K ≥ K · 𝔣(h − 2) ≥ h − 2.

Proof. Notation as in §2. Since K is nef and big and L is ample one has d_{1} > 0. Write H := \overline{K}. Then σ^{∗}K ≈ H + E by (10). Compute

d_{2} = σ^{∗}K · σ^{∗}K · σ^{∗}L = σ^{∗}K · (H + E) · σ^{∗}L
= (H + E) · H · σ^{∗}L + σ^{∗}K · E · σ^{∗}L
= H · H · σ^{∗}L + E · H · σ^{∗}L + σ^{∗}K · E · σ^{∗}L.

Clearly σ^{∗}K · E · σ^{∗}L ≥ 0. Also, H · H · σ^{∗}L = (L · 𝔣) deg Σ ≥ L · 𝔣(h − 2) by (8). Noting that L and H are spanned and E is effective we have E · H · σ^{∗}L ≥ 0, and thus 2) is proved.

To show 3), compute

d_{3} = (σ^{∗}K)^{3} = (σ^{∗}K)^{2} · (H + E)
≥ (σ^{∗}K)^{2} · H = (H + E) · H · σ^{∗}K ≥ H · H · σ^{∗}K,

where the two inequalities follow from the fact that H, E are effective and σ^{∗}K is nef. By using (8) and the fact that K · 𝔣 ≥ K · f > 0 since K is nef and big and f moves in a family covering X (see also [3, (2.1.2)]), we have H · H · σ^{∗}K = (K · 𝔣) deg Σ ≥ K · 𝔣(h − 2) ≥ h − 2, so we are done.

We can prove now the main result of this section.

Theorem 3.2. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Assume further that |K_{X} + L| gives a rational map, ψ, having a 2dimensional image. Let 𝔣 be a general fiber of ψ and let K = K_{X} + L. Let h := h^{0}(K_{X} + L).

1. If h ≥ 6, e.g., if κ(X) ≥ 0, then L · 𝔣 ≤ 13;

2. If h = 5, then L · 𝔣 ≤ 14;

3. If h = 4, then L · 𝔣 ≤ 17;

4. If h = 3 then L · 𝔣 ≤ 24.

Proof. We can assume without loss of generality that h^{0}(K_{X}) = 0 since otherwise ψ has a 3-dimensional image (see (2.2)).

Assume that L·𝔣 ≥ 14. Using relation (5) and Lemma (3.1), 2) we have for any

smooth S ∈ |L| that χ(O_{S}) ≥ \frac{d_{2} + 1}{9} ≥ \frac{14(h − 2) + 1}{9} and thus by Lemma (1.13) that

d_{3} + 6h ≥ 4\frac{14(h − 2) + 1}{9} + 14(h − 2),

i.e., that d_{3} ≥ \frac{128}{9}h−40. Therefore from Tsuji’s inequality (1.9) we obtain d_{1} ≤ 51. If h ≥ 15 then d_{2} ≥ 13 · 14 = 182. Using Castelnuovo’s inequality and the Hodge inequality dd_{2} ≤ d_{1}^{2} as in Lemma (1.14) we obtain d ≥ 14 and d_{1} ≥ 52.

If h = 14 then proceeding the same way we obtain d_{2} ≥ 168 and d_{1} ≥ 50. Thus

using Tsuji’s inequality (1.9) and Lemma (1.13) we obtain χ(O_{S}) ≤ \frac{509}{27}, and thus χ(O_{S}) ≤ 18. Then Miyaoka’s inequality (4) contradicts d_{2} ≥ 168.

If h = 13 then proceeding the same way we obtain d_{2} ≥ 154 and d_{1} ≥ 45. Thus

using Tsuji’s inequality (1.9) and Lemma (1.13) we obtain χ(O_{S}) ≤ \frac{53}{3}, and thus χ(O_{S}) ≤ 17. Again (4) contradicts d_{2} ≥ 154.

If h = 12 then proceeding the same way we obtain d_{2} ≥ 140 and d_{1} ≥ 43.

Thus using Tsuji’s inequality (1.9) and Lemma (1.13) we obtain χ(O_{S}) ≤ \frac{439}{27}, and thus χ(O_{S}) ≤ 16. Using the double point formula (1.7) we find that d ≥ 55 and the Hodge inequality d_{1}^{2} ≥ dd_{2}, together with parity condition (3), gives d_{1} ≥ 89 contradicting d_{1} ≤ 51.

This same reasoning takes care of the cases h = 7 to 11.

If h = 6 then proceeding the same way we obtain d_{2} ≥ 56 and d_{1} ≥ 25. From Lemma (1.13) we get d_{3} ≥ 4χ(O_{S})+20. Using Tsuji’s inequality with these bounds for d_{1} and d_{3} we find χ(O_{S}) ≤ 8. Using the double point formula (1.7) and the Hodge inequalities gives d ≥ 39 and d_{1} ≥ 47. Using d_{2} ≥ 56 and the Miyaoka inequality we obtain χ(O_{S}) ≥ 7. Then Lemma (1.13) yields d_{3} ≥ 48. This gives a contradiction to Tsuji’s inequality

6 ≥ \frac{48}{64} + \frac{47}{24} + \frac{7}{2} > 6.

Assuming that L · 𝔣 ≥ 15 the same reasoning leads to a contradiction in the case h = 5.

Now assume that h = 4 and L · 𝔣 ≥ 18. From d_{2} ≥ 18(h − 2) ≥ 36 and relation (5) we conclude that χ(O_{S}) ≥ 5. From Lemma (1.13) we conclude that d_{3} ≥ 36−24+20 = 32. From d_{2} ≥ 36, the inequality d ≥ 10 from Lemma (1.14) and the Hodge inequality dd_{2} ≤ d_{1}^{2}, together with the parity condition, we conclude that d_{1} ≥ 20. From Tsuji’s inequality we conclude that χ(O_{S}) ≤ 5. Using the double point formula (1.7) with h = 4, χ(O_{S}) ≤ 5 we conclude that d ≥ 28 and d_{1} ≥ 37. Using this we conclude from Tsuji’s inequality the contradiction that χ(O_{S}) ≤ 4.

Assuming that L·𝔣 ≥ 25 the same reasoning used in the above paragraphs leads to a contradiction in the case h = 3.

Let ψ = s∘r be the Remmert-Stein factorization of ψ as in §2. In the remaining part of this paragraph we will give a bound for the genus of a general (connected) fiber of r and for the degree of the finite morphism s.

First, we need the following general bound for the degree of a general fiber f of r.

Proposition 3.3. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Assume further that |K_{X} + L| gives a rational map, ψ, having a 2-dimensional image. Let ψ = s ∘ r be the Remmert-Stein factorization of ψ and let f be a general fiber of the connected part r. Let B be the base locus of |K_{X} + L|. Then

1. L · f ≥ 3;
2. L · f ≥ 4 if κ(X) ≥ 0 and dim B = 0;
3. L · f ≥ 4 if κ(X) = 3.

Proof. Let us prove that L · f ≥ 3. Assume L · f ≤ 2. Then f ≅ ℙ^{1}. Since f moves in a h^{0}(N_{f/X})-dimensional family, the normal bundle N_{f/X} of f in X is generically spanned and hence it is spanned since f ≅ ℙ^{1}. Therefore det N_{f/X} ≅ O_{ℙ^{1}}(a) for some integer a ≥ 0 and the adjunction formula gives

K_{X|f} ≈ K_{f} − det(N_{f/X}) ≈ O_{ℙ^{1}}(−2 − a).

Thus (K_{X} + L) · f ≤ −1 if L · f = 1 and (K_{X} + L) · f = 0 if L · f = 2. The first case contradicts the nefness assumption of K_{X} + L, the latter case contradicts the bigness assumption of K_{X} + L (see e.g., [3, (2.1.2)]).

To show 3), assume L · f = 3. Then Castelnuovo’s bound gives g(f) ≤ 1. Let f̃ be a desingularization of f. Hence g(f̃) ≤ g(f) ≤ 1, so that either f ≅ ℙ^{1} or f is an elliptic curve. In both the cases the additivity of the Kodaira dimension, κ(X) ≤ 2 + κ(f) ≤ 2, contradicts the assumption κ(X) = 3.

To show 2), assume L · f = 3. Since κ(X) ≥ 0, the same argument as above, by using the additivity of the Kodaira dimension, shows that the only possibility is g(f) = 1 and f is a smooth elliptic curve. Since f moves in a h^{0}(N_{f/X})-dimensional family, the normal bundle N_{f/X} of f in X is generically spanned and therefore deg(det N_{f/X}) ≥ 0.

Assume deg(det N_{f/X}) > 0. Then from K_{f} ≈ K_{X|f} + det(N_{f/X}) we conclude that K_{X} · f < 0. Since κ(X) ≥ 0 this contradicts the fact that f moves.

So we can assume deg(det N_{f/X}) = 0 and therefore Lemma (1.15) implies that N_{f/X} is trivial. Since dim B = 0, Lemma (2.3) applies to say that the general fiber f meets B. Then there exists a pencil of fibers f through a point x ∈ B. By deformation theory this implies that deg(det N_{f/X}) > 0, contradicting the present assumption.

Proposition 3.4. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Assume further that |K_{X} + L| gives a rational map, ψ, having a 2-dimensional image. Let ψ = s ∘ r be the Remmert-Stein factorization of ψ and let f be a general fiber of the connected part r. Let h := h^{0}(K_{X} + L). Then

1. g(f) ≤ 66 if h ≥ 6, e.g., if κ(X) ≥ 0;

2. g(f) ≤ 78 if h = 5;

3. g(f) ≤ 120 if h = 4;

4. g(f) ≤ 253 if h = 3;

5. g(f) ≤ 26 if ψ is a morphism and h ≫ 0 (e.g., h ≥ 77).

Proof. By Theorem (3.2) and (11) we know that L·f ≤ 13 if h ≥ 6, L·f ≤ 14 if h = 5, L · f ≤ 17 if h = 4 and L · f ≤ 24 if h = 3. Since the maximum genus g(f) is taken when f is a plane curve, the genus formula gives the desired bounds in 1), 2), 3) and 4).

Assume now that ψ is a morphism, so that K_{X|f} ≅ K_{f} and hence, by using the lower bound L · f ≥ 3 from (3.3),

K · f = K_{f} + L · f ≥ 2g(f) + 1.

Thus Lemma (3.1) gives d_{2} ≥ L · f(h − 2) ≥ 3(h − 2) and d_{3} ≥ K · f(h − 2) ≥ (2g(f) + 1)(h − 2). Therefore Tsuji’s inequality (1.9) and Miyaoka’s inequality (5) yield

h ≥ \frac{(2g(f) + 1)(h − 2)}{64} + \frac{d_{1}}{24} + \frac{h − 2}{6} + \frac{1}{18}. (12)

Assume g(f) ≥ 27. Then (12) leads to

h ≥ \frac{55}{64}(h − 2) + \frac{h − 2}{6} + \frac{1}{18},

which implies h ≤ 76.

We give now some bound for the degree of the finite morphism s of the Remmert-Stein factorization of the morphism ψ = s ∘ r as in §2.

Proposition 3.5. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L) be the first reduction of (X̂, L̂). Assume further that |K_{X} + L| gives a rational map, ψ, having a 2-dimensional image. Let B be the base locus of |K_{X} + L|. Let ψ = s ∘ r be the Remmert-Stein factorization of ψ and let f be a general fiber of the connected part r. Let #s be the degree of the finite morphism s. Let h := h^{0}(K_{X} + L). Then

1. #s ≤ \frac{13}{L · f} (and hence #s ≤ 4) if h ≥ 6, e.g., if κ(X) ≥ 0;

2. #s ≤ 3 if κ(X) ≥ 0 and dim B = 0 ;

3. #s ≤ 3 if κ(X) = 3.

Proof. Let 𝔣 be a general fiber of ψ. Recall that #s ≤ \frac{13}{L · f} by (11). Assume

h^{0}(K_{X} + L) ≥ 6. Thus Theorem (3.2) implies that L · 𝔣 ≤ 13, so we conclude that

#s ≤ \frac{13}{L · f}. Hence in particular #s ≤ 4 since L · f ≥ 3 by (3.3), 1). This shows

1). By (3.3), 2) we get 2). If κ(X) = 3, (3.3), 3) yields L · f ≥ 4 and hence the

argument above gives #s ≤ \frac{14}{L · f} ≤ 3.

Remark 3.6. Notation as in Proposition (3.5). Assume κ(X) ≥ 0. Then if #s = 4, the fiber f is a smooth elliptic cubic. Indeed, by looking over the proof of (3.5), wee see that #s = 4 implies L · f = 3. Then g(f) = 1 by Castelnuovo’s bound. If f is singular, let f̃ be a desingularization of f. Then g(f̃) = 0, so that f̃ ≅ ℙ^{1} and hence we have the contradiction κ(X) ≤ κ(f) + dim ψ(X) < 0. Thus f is smooth and we are done.

Assume κ(X) = 3. Then if #s = 3, the fiber f is a plane quartic. Indeed #s = 3 implies L · f = 4. If f lies in ℙ^{N} with N ≥ 3, Castelnuovo’s bound yields g(f) = 1 and hence we have the contradiction κ(X) ≤ κ(f) + dim ψ(X) ≤ 2. Thus f lies in ℙ^{2}.

We conclude this section by recalling an example from [13] which shows that the rational map ψ can have a 0-dimensional base locus.

Example 3.7. From Fujita’s lists of Del Pezzo 3-folds (see [10, (8.11)]) we know that there exists a polarized threefold (X, M), M ample line bundle on X, such that K_{X} ≈ −2M and M^{3} = 1. In [13] it is shown that L := 3M is very ample, K_{X} + L ≈ M is not spanned and has a single point as base locus. Furthermore h^{0}(K_{X} + L) = 3, so that |K_{X} + L| gives a birational map ψ : X → ℙ^{2}. The invariants are d_{3} = 1, d_{2} = 3, d_{1} = 9, d = 27.

4. The second adjunction mapping: the case of a 3-dimensional image

Notation as in §2. In this section we assume that the rational map ψ : X → ℙ^{n} associated to |K_{X} + L| as an image Σ := ψ(X) of dimension 3. Our aim is to give a bound for the degree t := #ψ of ψ. From Lemma (2.2) we can assume

h^{0}(K_{X}) = 0. (13)

Let h := h^{0}(K_{X} + L). Then we have

d_{3} ≥ t(h − 3). (14)

Indeed, Σ ⊂ ℙ^{h−1} and therefore deg Σ ≥ h − dim Σ = h − 3, so that

d_{3} = (K_{X} + L)^{3} ≥ t deg ψ(X) ≥ t(h − 3).

Let S be a general smooth member of |L|. We claim that if t ≥ 5 then

d_{2} = K_{S}^{2} ≥ 3h − 7. (15)

By Beauville’s generalization of an inequality of Castelnuovo [1] we know that if the rational map given by |K_{S}| is generically one-to-one then

d_{2} ≥ 3h^{2}(O_{S}) − 7 ≥ 3h − 7,

where the last inequality follows from (13). Thus we can assume that |K_{S}| does not give a generically one-to-one map. Therefore, since L_{S} is very ample outside of a finite set of points, h^{0}(K_{S} −L_{S}) = 0. Thus since h^{0}(K_{X}) = 0, the map H^{0}(K_{X} + L) → H^{0}(K_{S|C}) is injective for any smooth C ∈ |L_{S}|. Then h ≤ h^{0}(K_{S|C}). We conclude from Clifford’s theorem that

h^{0}(K_{S|C}) ≤ \frac{deg(K_{S|C})}{2} + 1 = \frac{d_{1}}{2} + 1

with equality implying that C is hyperelliptic. But since K_{X} + 2L is very ample by the main result of [19] and its restriction to C is K_{C} we conclude that C is not hyperelliptic and thus that d_{1} ≥ 2h − 1. By this inequality, inequality (14), and the Hodge index theorem we conclude that

d_{2}^{2} ≥ d_{3}d_{1} ≥ t(h − 3)(2h − 1).

Noting that h ≥ 4 it is a straightforward check that for t ≥ 5 we have t(h − 3)(2h − 1) ≥ (3h − 7)^{2}, whence inequality (15).

Using this and Miyaoka’s inequality (5) we conclude that for t ≥ 5

χ(O_{S}) ≥ \frac{h − 2}{3}. (16)

Let us now give a bound for the degree of ψ.

Proposition 4.1. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that |KX̂ + L̂| gives a rational map, ψ̂, having a 3dimensional image. Let h := h^{0}(KX̂ + L̂) and let t := #ψ̂ be the degree of ψ̂. Then t ≤ 53.

Proof. Let (X, L) be the first reduction of (X̂, L̂). Let ψ be the rational map associated to |K_{X} + L|. By recalling the discussion following (1), we can without loss of generality work with X, L, ψ.

Since the result is trivial for hypersurfaces in ℙ^{4}, we can assume without loss of generality that h^{0}(L̂) ≥ 6. Thus we have that a smooth Ŝ ∈ |L̂| is a general type surface embedded in ℙ^{m} with m ≥ 4 and its image on X is a smooth minimal model surface of general type S.

Now assume that t ≥ 54. Thus d_{3} ≥ 54(h − 3) by (14). Using this and the inequality (16) in Tsuji’s inequality (1.9) we conclude that

h + 4d_{1} ≤ 275. (17)

Since ψ has a three dimensional image, it follows that h ≥ 4. We use the inequality (17) and a bootstrap procedure to deal with the cases h ≥ 5 and then turn with more a detailed argument to deal with the case h = 4.

If h ≥ 5 then d_{3} ≥ 108. By Lemma (1.14) we conclude that d ≥ 11. Using the Hodge inequalities and parity conditions repeatedly we conclude that d_{1} ≥ 25 and d_{2} ≥ 52. From Miyaoka’s inequality (5) we conclude that χ(O_{S}) ≥ 6. From Tsuji’s inequality we obtain

h ≥ \frac{54(h − 3)}{64} + \frac{25}{24} + \frac{6}{2}.

Simplified this gives h > 9. If h ≥ 10 then d_{3} ≥ 378 and by using the Hodge inequalities and the reasoning of Lemma (1.14) we conclude that d ≥ 13, d_{1} ≥ 41, and d_{2} ≥ 125. From d_{2} ≥ 125 we conclude that χ(O_{S}) ≥ 14. From Tsuji’s inequality we obtain

h ≥ \frac{54(h − 3)}{64} + \frac{41}{24} + \frac{14}{2}.

Simplified this gives h > 39. Then (14) gives d_{3} ≥ 1998 and from the Hodge inequalities and parity conditions as in Lemma (1.14) we conclude that d_{1} ≥ 80. This contradicts the inequality (17).

We now do the case with h = 4. From the inequality (14) we conclude that d_{3} ≥ 54. By Lemma (1.14) we conclude that d ≥ 10. Using the usual Hodge index relations and parity conditions repeatedly we conclude that

d ≥ 10, d_{1} ≥ 18, d_{2} ≥ 32, d_{3} ≥ 54.

By Miyaoka’s inequality we have χ(O_{S}) ≥ 4. Then Tsuji’s inequality (1.9) gives

4 = h ≥ \frac{54}{64} + \frac{18}{24} + \frac{χ(O_{S})}{2},

which implies χ(O_{S}) = 4. Using the double point formula (1.7) with h = χ(O_{S}) = 4 with the Hodge inequalities and parity conditions repeatedly we conclude that

d ≥ 45, d_{1} ≥ 49, d_{2} ≥ 52, d_{3} ≥ 54.

In particular we have the contradiction χ(O_{S}) ≥ 6 from Miyaoka’s inequality.

Remark 4.2. It can be shown by the same reasoning that t ≤ 32 for h ≤ 17. Unfortunately the lack of lower bound for d_{1} involving h linearly prevents us from showing this for all h.

Remark 4.3. It is worth emphasizing that if h^{0}(K_{X}) > 0 then t = 1 (see (2.2)). Further note that if p_{g}(Σ) > 0 where by p_{g}(Σ) we mean h^{3}(O\overline{Σ}) for \overline{Σ}, a desingularization of Σ := dim ψ(X), then h^{0}(K_{X}) > 0 and t = 1.

Theorem 4.4. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Assume further that |KX̂ + L̂| gives a rational map, ψ̂, having a 3-dimensional image. Let h := h^{0}(KX̂ + L̂) and let t := #ψ̂ be the degree of ψ̂. Then t ≤ 31 if q(X) = 0.

Proof. Let (X, L) be the first reduction of (X̂, L̂). As in the last proposition it suffices to work with X, L, and the rational map ψ associated to |K_{X} + L|.

We assume that t ≥ 32 and deduce a contradiction. Since q(X) = 0 we have q(S) = 0 and using the exact sequence

0 → K_{X} → K_{X} + L → K_{S} → 0

we get

χ(O_{S}) ≥ h + 1. (18)

Then Tsuji’s inequality (1.9), (14) and (18) yield

h ≥ \frac{32(h − 3)}{64} + \frac{h + 1}{2} + \frac{d_{1}}{24},

or

d_{1} ≤ 24.

We claim that d_{1} ≤ 24 implies that h ≤ 6. Indeed if h ≥ 7 we have d_{3} ≥ 128 by (14). Using Lemma (1.14) we see that d ≥ 11, d_{1} ≥ 23, d_{2} ≥ 44. Using the Hodge inequalities (2) and parity conditions (3) repeatedly we have the contradiction d_{1} ≥ 27.

If h = 6 then d_{3} ≥ 96 and Lemma (1.14) implies that d ≥ 11, d_{1} ≥ 23, and d_{2} ≥ 44. One more application of the Hodge inequalities yields d_{2} ≥ 48. From Tsuji’s inequality using d_{1} ≥ 23 and d_{3} ≥ 96 we conclude that χ(O_{S}) ≤ 7. From the double point formula (1.7) with χ(O_{S}) ≤ 7 and h = 6 combined with the Hodge inequalities we conclude that d ≥ 50 and the contradiction d_{1} ≥ 64.

If h = 5 then d_{3} ≥ 64 and Lemma (1.14) implies that d ≥ 10, d_{1} ≥ 14, d_{2} ≥ 16. Using the Hodge inequalities and parity conditions repeatedly yields that d ≥ 10, d_{1} ≥ 20, and d_{2} ≥ 36. From Tsuji’s inequality using d_{1} ≥ 20 and d_{3} ≥ 64 we conclude that χ(O_{S}) ≤ 6. From the double point formula (1.7) with χ(O_{S}) ≤ 6 and h = 5 combined with the Hodge inequalities we conclude that d ≥ 45 and the contradiction d_{1} ≥ 52.

Finally if h = 4 and d_{3} ≥ 32 then Lemma (1.14) implies that d ≥ 10, d_{1} ≥ 14, d_{2} ≥ 16. This and the Hodge inequalities, taking account of parity conditions, yields that d ≥ 10, d_{1} ≥ 16, and d_{2} ≥ 23. From Tsuji’s inequality using d_{1} ≥ 16 and d_{3} ≥ 32 we conclude that χ(O_{S}) ≤ 5. From the double point formula (1.7) with χ(O_{S}) ≤ 5 and h = 4 combined with the Hodge inequalities we conclude that d ≥ 37 and the contradiction d_{1} ≥ 37.

5. On embeddings of the second reduction

Notation as in §2. Consider the second reduction (X′, L′), φ : X → X′, L′ := (φ_{∗}L)^{∗∗} of (X, L). Recall that X′ has terminal 2-Gorenstein singularities, and 2L′ and K′ := K_{X′} + L′ are line bundles. In this section we will show that 3(K_{X′} + L′) gives an embedding on X′ (see Corollary (5.4) below). We will derive this fact from the following more general result.

Theorem 5.1. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L), π : X̂ → X, (X′, L′), φ : X → X′ be the first and the second reduction as in (1.3). Let L be a line bundle on X′. Let Ŝ be a smooth surface in |L̂| and let S := π(Ŝ), S′ = φ(S). Let φ_{S} : S → S′ and L_{S′} be the restrictions of φ and L to S and S′ respectively. Assume that

i) the image of Γ(φ^{∗}L) → Γ(φ_{S}^{∗}L_{S′}) is surjective and spans φ_{S}^{∗}L_{S′}; ii) the morphism, σ, defined by |φ_{S}^{∗}L_{S′}| factors as σ = s ∘ φ_{S}, where s is an

embedding.

Then L is very ample.

Proof. First note that L, φ^{∗}L are spanned by global sections. To see this, take a point x ∈ X and let x̂ ∈ X̂ such that x = π(x̂). By Bertini’s type theorem (see e.g., [3, (1.7.9)]) there exists a smooth Â ∈ |L̂| passing through x̂. Then the image A := π(Â) is a smooth divisor in |L| passing through x. Let A′ := φ(A). Since by the assumption i) the image of Γ(φ^{∗}L) → Γ(φ_{A}^{∗}L_{A′}) spans φ_{A}^{∗}L_{A′}, we see that φ^{∗}L is spanned at x. Since this is true for any point x ∈ X, we conclude that φ^{∗}L is spanned, and hence L is also spanned.

Thus we have a commutative diagram

\begin{matrix}X \\ φ ↓ & ↘^{Φ} \\ X′ & ⟶^{Φ′} & ℙ_{ℂ}, \end{matrix} (19)

where Φ, Φ′ are the morphisms given by φ^{∗}L and L respectively. To show that L is very ample is equivalent to show that Φ′ is an embedding.

By condition ii) we have a commutative diagram

\begin{matrix}S \\ φ_{S} ↓ & ↘^{σ} \\ S′ & ⟶^{s} & ℙ_{ℂ}, \end{matrix} (20)

where s is an embedding. Since Γ(φ^{∗}L) ≅ Γ(L) and Γ(φ_{S}^{∗}L_{S′}) ≅ Γ(L_{S′}), the assumption ii) gives a surjection Γ(L) → Γ(L_{S′}), and hence s = Φ′_{S′} is the restriction of Φ′ to S′.

Take two distinct points x′_{1}, x′_{2} ∈ X′ and let x̂_{1}, x̂_{2} ∈ X̂ such that π(x̂_{i}) := x_{i} and φ(x_{i}) = x′_{i}, i = 1, 2. By Bertini’s type theorem (see e.g., [3, (1.7.9)]) we know that there exists a smooth divisor Â ∈ |L̂| passing through x̂_{1}, x̂_{2}. Then the image A := π(Â) is a smooth divisor in |L| passing through x_{1}, x_{2}. Furthermore A′ := φ(A) is a surface passing through x′_{1}, x′_{2}. Now, since s is an embedding, we have s(x′_{1}) ≠ s(x′_{2}). But s(x′_{i}) = σ(x_{i}), i = 1, 2, by the commutativity of diagram (20). Therefore σ(x_{1}) ≠ σ(x_{2}). Since σ(x_{i}) = Φ(x_{i}) = Φ′(x′_{i}), i = 1, 2, by the commutativity of diagram (19), we thus conclude that Φ′(x_{1}) ≠ Φ′(x_{2}), and hence Φ′ separates x_{1}, x_{2}. This shows that Φ′ is one-to-one.

We need the structure of the second reduction map described in Theorem (1.4). Let Z be the algebraic subset of X′, of dimension ≤ 1, such that φ is an isomorphism outside of Z. Let Z_{0} ⊂ Z be the subset of Z consisting of all 0-dimensional components of Z and let Z_{1} ⊂ Z be the set of points x′ belonging to the 1dimensional components of Z and such that φ^{−1}(x′) is a divisor (see (1.4), a)). Then from (1.4) we know that

• Z_{0} ∪ Z_{1} is a finite set of points, X′ is smooth outside of Z_{0} ∪ Z_{1} and φ^{−1}(x′) is a reduced (possibly reducible) divisor for x′ ∈ Z_{0} ∪ Z_{1}.

First consider the case when x′ ∈ X′ \ Z. Let v′ be a tangent direction at x′, i.e., v′ is a non-zero element of T_{X′,x′}, where T_{X′} denotes the tangent bundle of X′. Choose points x̂ ∈ X̂, x ∈ X such that π(x̂) = x, φ(x) = x′. Let v̂, v be nonzero elements of T_{X̂,x̂}, T_{X,x} respectively such that dπx̂(v̂) = v, dφ_{x}(v) = v′, where dπx̂, dφ_{x} denote the differential maps. Then d(φ ∘ π)x̂(v̂) = v′. By [3, (1.7.9)] there exists a smooth surface Â ∈ |L̂| with x̂ ∈ Â and v̂ ∈ T_{Â,x̂}. Therefore the image A := π(Â) is a smooth divisor in |L| passing through x and with v ∈ T_{A,x}. Furthermore A′ := φ(A) is a surface passing through x′, x′ is a smooth point of A′ and v′ ∈ T_{A′,x′}.

We want to show that dΦ′_{x′}(v′) ≠ 0, where dΦ′_{x′} is the differential map. Since s is an embedding, we have ds_{x′}(v′) ≠ 0. By the commutativity of diagram (20), one has ds_{x′}(v′) = dσ_{x}(v) = dΦ_{x}(v) and dΦ_{x}(v) = dΦ′_{x′}(v′) by the commutativity of diagram (19). Thus dΦ′_{x′}(v′) ≠ 0 and we are done.

Assume now that x′ ∈ Z \ (Z_{0} ∪ Z_{1}) and that v′ is a tangent direction at x′. Then by using the structure of the second reduction map given in (1.4), i.e., that the restriction φ_{X\φ^{−1}(Z\(Z_{0}∪Z_{1}))} is the blowup of the smooth curve Z \ (Z_{0} ∪ Z_{1}), we can find points x̂ ∈ X̂, x ∈ X and nonzero elements v̂, v of T_{X̂,x̂}, T_{X,x} respectively such that dπx̂(v̂) = v, dφ_{x}(v) = v′with all the same properties above. Then the same argument as above applies to give the result.

It remains to consider the case when x′ ∈ Z_{0} ∪ Z_{1}. By •) we know that in both cases D := φ^{−1}(x′) is a reduced divisor.

Look at diagram (19). Note that Φ′ being an embedding at x′ is equivalent to

Φ′^{∗}𝔪_{Φ′(x′)} = 𝔪_{x′}. (21)

Claim 5.2. Let D = φ^{−1}(x′) and let J_{D} be the ideal sheaf of D in X. Then (Φ^{∗}𝔪_{Φ(x)})_{x} = J_{D,x} for every x ∈ D implies relation (21).

Proof. Note that φ^{∗}𝔪_{x′} = J_{D}. Therefore, by the commutativity of diagram (19) and by the assumption, we have, for each x ∈ D,

((Φ′ ∘ φ)^{∗}𝔪_{Φ(x)})_{x} = (Φ^{∗}𝔪_{Φ(x)})_{x} = J_{D,x} = (φ^{∗}𝔪_{x′})_{x}. (22)

Now,

(Φ′ ∘ φ)^{∗}𝔪_{Φ(x)} = φ^{∗}(Φ′^{∗}𝔪_{Φ(x)}) = φ^{∗}Φ′^{∗}𝔪_{Φ′(x′)}. (23)

By combining (22) and (23) we get

(φ^{∗}Φ′^{∗}𝔪_{Φ′(x′)})_{x} = (φ^{∗}𝔪_{x′})_{x}, x ∈ D.

Therefore φ^{∗}Φ′^{∗}𝔪_{Φ′(x′)} = φ^{∗}𝔪_{x′} and hence since X′ is normal and φ has connected fibers Φ′^{∗}𝔪_{Φ′(x′)} = 𝔪_{x′}. This proves the claim.

Thus we are reduced to show that, for each x ∈ X,

(Φ^{∗}𝔪_{Φ(x)})_{x} = J_{D,x}. (24)

Recall that Φ is the morphism defined by φ^{∗}L. Write M := φ^{∗}L. Since sections of O_{ℙ_{ℂ}}(1) vanishing at Φ(x) pull back to sections of M ⊗ J_{D,x}, it is enough to show that Γ(M ⊗ J_{D}) spans M ⊗ J_{D} in a neighborhood U of D (i.e., that Γ(M ⊗ J_{D}) spans M ⊗ J_{D,x} at x for each x ∈ D).

Claim 5.3. Notation as above. Let N := N_{D/X} be the normal bundle of D in X. If Γ(M⊗J_{D}) spans M⊗N^{∗} then Γ(M⊗J_{D}) spans M⊗J_{D} in a neighborhood U of D.

Proof. This is an immediate consequence of Nakayama’s lemma.

Thus by Claim (5.3) we are reduced to show that Γ(M⊗J_{D}) spans M⊗N_{D/X}^{∗} or, equivalently, that

• Γ(M ⊗ J_{D}) spans M ⊗ N_{D/X}^{∗} at x for each x ∈ D.

Let ℓ := D ∩ S be the transversal intersection of D with a general S ∈ |L| that contains x. We know that ℓ is either a smooth rational curve with L · ℓ = 1 or a reducible conic, i.e., the union of two distinct lines meeting in x. Let M_{S} be the restriction of M = φ^{∗}L to S, let J_{ℓ} be the ideal sheaf of ℓ in S, and let N_{ℓ} be the normal bundle of ℓ in S, N = N_{D/X}. We claim that the result follows from the following three facts.

a) Γ(M_{S} ⊗ J_{ℓ}) spans M_{S} ⊗ N_{ℓ}^{∗} at x, for each x ∈ D;

b) the restriction map Γ(M ⊗ J_{D}) → Γ(M_{S} ⊗ J_{ℓ}) is onto;

c) (M ⊗ N^{∗})_{ℓ} ≅ M_{S} ⊗ N_{ℓ}^{∗}. Indeed, we have the diagram

Γ(M_{S} ⊗ J_{ℓ}) ⟶^{α} (M_{S} ⊗ N_{ℓ}^{∗})/𝔪_{x} ≅ ℂ
β ↑  ↑ γ
Γ(M ⊗ J_{D}) ⟶^{ρ} (M ⊗ N^{∗})/𝔪_{x} ≅ ℂ

where 𝔪_{x} is the ideal sheaf of x in S, the morphisms α, β are onto by conditions a),

b) respectively. Furthermore by c) there exists a map γ which makes the diagram commute. Hence in particular γ is an isomorphism and therefore ρ is onto. This gives •).

Thus it remains to prove conditions a), b), c). Proof of a). By the assumption ii), we know that |M_{S}| = |φ_{S}^{∗}L_{S′}| defines a morphism, σ, which factors as σ = s ∘ φ_{S}, where s is an embedding. In particular s is an embedding at the points x′ ∈ S′ such that D := φ^{−1}(x′), ℓ = D ∩ S = φ_{S}^{−1}(x′) (see diagram (20)). This means that the map

Γ(L_{S′} ⊗ 𝔪_{x′}) → Γ(L_{S′} ⊗ (𝔪_{x′}/𝔪_{x′}^{2})) → 0

is onto, where 𝔪_{x′} is the ideal sheaf of x′ in S′. Since M_{S} ⊗ J_{ℓ} ≅ φ_{S}^{∗}(L_{S′} ⊗ 𝔪_{x′}), this implies that Γ(M_{S} ⊗ J_{ℓ}) spans M ⊗ N_{ℓ}^{∗} at x for each x ∈ D. Proof of b). Let M_{ℓ} be the restriction of M to ℓ. We have a commutative diagram

0 → Γ(M ⊗ J_{D}) →^{α} Γ(M) →^{α′} Γ(M_{D}) ≅ Γ(O_{D}) ≅ ℂ → 0
β ↓  γ ↓  δ ↓
0 → Γ(M_{S} ⊗ J_{ℓ}) →^{ρ} Γ(M_{S}) →^{ρ′} Γ(M_{ℓ}) ≅ Γ(O_{ℓ}) ≅ ℂ → 0

where α′, ρ′ are onto since M_{S′}, M_{ℓ} are spanned because M = φ^{∗}L is spanned. Therefore δ is an isomorphism. Take a ∈ Γ(M_{S} ⊗J_{ℓ}). Then since γ is surjective by assumption ρ(a) = γ(b) for some b ∈ Γ(M). But α′(b) = 0 since α′(b) = ρ′(γ(b)) = ρ′ρ(a) = 0. Thus b ∈ Γ(M ⊗ J_{D}) and therefore a = β(b), i.e., β is onto.

Proof of c). Since D, S are both Cartier divisors on X, and ℓ is the transversal intersection ℓ = D ∩ S, we have

(N_{D/X}^{∗})_{|ℓ} ≅ (N_{D/X}^{∗})_{|S} ≅ O_{D}(−D)_{S} ≅ O_{ℓ}(−ℓ) ≅ N_{ℓ}^{∗}.

Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L), π : X̂ → X, be the first reduction of (X̂, L̂). For a general smooth surface Ŝ ∈ |L̂|, let S = π(Ŝ). Then

h^{1}(2K_{S}) = 0. (25)

To see this note that since K_{X} + L is nef and big, the restriction K_{S} is nef and big on S. Therefore h^{1}(2K_{S}) = 0 by the Kawamata-Viewheg vanishing theorem.

Corollary 5.4. Let (X̂, L̂) be a smooth threefold polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L), π : X̂ → X, (X′, L′), φ : X → X′ be the first and the second reduction as in (1.3). Then 3(K_{X′} + L′) is very ample.

Proof. Let K′ := K_{X′} + L′. Then φ^{∗}K′ ≈ K. Since the canonical model S′ := φ(S) of Ŝ has rational Gorenstein singularities we have K′_{S′} ≈ K_{S′}. From [18, (2.2.2)] we know that 2(K_{X} + L) is spanned. Then 3(K_{X} + L) ≈ φ^{∗}(3K′) is spanned. Since h^{1}(K_{X} + 2K) = 0, from the exact sequence

0 → K_{X} + 2K → 3K → 3K_{S} → 0

we infer that

Γ(φ^{∗}(3K′)) ≅ Γ(3K) → Γ(φ_{S}^{∗}(3K′_{S′})) ≅ Γ(3K_{S})

is onto.

By using Lemma (1.12), Corollary (1.17) and (25) we know from the canonical embedding theorem of surfaces (see [7, (1.2)]) that 3K′_{S′} ≈ 3K_{S′} gives an embedding, s, for any surface S′ = φ(S), S smooth member of |L|. This means that the morphism, σ, defined by |φ_{S}^{∗}(3K′_{S′})| factors as σ = s ∘ φ_{S}. Then Theorem (5.1) applies to say that 3K′ is very ample.

Remark 5.5. Notation as in (5.4). Let Ŝ be a smooth surface in |L̂| and let S := π(Ŝ), S′ := φ(S). The same arguments as above show that if |tK_{S′}| gives an embedding for each smooth S ∈ |L| and if the restriction map ρ_{t} : Γ(t(K_{X} + L)) → Γ(tK_{S}) is onto for t ≥ 1, then t(K_{X′} + L′) is very ample.

Note that for t ≥ 2 the surjectivity of the map ρ_{t} follows from the exact sequence

0 → tK_{X} + (t − 1)L → t(K_{X} + L) → tK_{S} → 0.

Indeed, h^{1}(tK_{X} +(t−1)L) = h^{1}(K_{X} +(t−1)(K_{X} +L)) = 0 by Kawamata-Viewheg vanishing theorem since K_{X} + L is nef and big.

We conclude this section by pointing out an extension to higher dimension of the above results.

A straightforward modification of the proof of Theorem (5.1), which is in fact easier since the structure of the second reduction is simpler in dimension n ≥ 4 (see [3, §7.5]), proves the following.

Theorem 5.6. Let (X̂, L̂) be a smooth n-fold, n ≥ 4, polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L), π : X̂ → X, (X′, L′), φ : X → X′ be the first and the second reduction as in (1.3). Let L be a line bundle on X′. Let Ŝ be a smooth surface obtained as transversal intersection of n − 2 general members of |L̂| and let S := π(Ŝ), S′ = φ(S). Let φ_{S} : S → S′ and L_{S′} be the restrictions of φ and L to S and S′ respectively. Assume that

i) the image of Γ(φ^{∗}L) → Γ(φ_{S}^{∗}L_{S′}) is surjective and spans φ_{S}^{∗}L_{S′}; ii) the morphism, σ, defined by |φ_{S}^{∗}L_{S′}| factors as σ = s ∘ φ_{S}, where s is an

embedding. Then L is very ample.

The theorem above gives rise to the following n-dimensional version of Corollary (5.4).

Corollary 5.7. Let (X̂, L̂) be a smooth n-fold, n ≥ 4, polarized by a very ample line bundle L̂. Assume that (X̂, L̂) is of log-general type. Let (X, L), π : X̂ → X, (X′, L′), φ : X → X′ be the first and the second reduction as in (1.3). Then 3(K_{X′} + (n − 2)L′) is very ample.

Proof. Denote by X̂_{n−1} a general smooth element of |L̂| and let L̂_{n−1} := L̂X̂_{n−1}. Similarly X̂_{n−2} denotes a general smooth element of |L̂_{n−1}| and in general

X̂_{i} denotes a smooth general element of |L̂_{i+1}|, where L̂_{i+1} := L̂X̂_{i+1}, i = 2, …, n−

1, and L̂_{n} = L̂, X̂_{n} = X̂. Then X̂_{2} coincides with the surface Ŝ obtained as transversal intersection of n−2 general members of |L̂|. Let π be the first reduction map and let X_{i} = π(X̂_{i}), i = 2, …, n − 2. Then X_{2} coincides with the surface S := π(Ŝ). From the exact sequences

0 → K_{X_{i+1}} + (m − 1)(K_{X_{i+1}} + μL_{i+1}) → m(K_{X_{i+1}} + (i − 1)L_{i+1}) → mK_{X_{i}} → 0,

by noting that μ := \frac{m(i−1)−1}{m−1} ≥ i − 1, so that K_{X_{i+1}} + \frac{m(i−1)−1}{m−1}L_{i+1} is nef and big since K_{X_{i+1}} + (i − 1)L_{i+1} ≈ (K_{X} + (n − 2)L)_{X_{i+1}} is nef and big, we conclude that the restriction map Γ(m(K_{X} +(n−2)L)) → Γ(mK_{S}) is surjective and therefore the restriction, φ_{S}, of φ to S coincides with the m-canonical map associated to |mK_{S}|, m ≥ 2.

Let K′ := K_{X′} + (n − 2)L′. Then φ^{∗}K′ ≈ K. Since the canonical model S′ := φ(S) of Ŝ has rational Gorenstein singularities we have K′_{S′} ≈ K_{S′}. Now the same argument as in the proof of (5.4), by using Theorem (5.6), gives the result.

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Dipartimento di Matematica, Università degli Studi di Genova, I-16146 Genova, Italy

E-mail address: beltrame@dima.unige.it

Department of Mathematics, University of Notre Dame, Notre Dame, Indiana, 46556, U.S.A

E-mail address: sommese.1@nd.edu
URL: http://www.nd.edu/∼sommese/index.html