On the degree and the birationality of the second adjunction mapping

Mauro C. Beltrametti, Andrew J. Sommese

Abstract Let L be a very ample line bundle on M, a projective manifold of dimension n ≥ 3. Under the assumption that K_{M} + (n − 2)L has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(K_{M} + (n − 2)L)|. In particular we show that under a number of conditions, e.g., n ≥ 5 or K_{M} + (n − 3)L having nonnegative Kodaira dimension, the degree ϕ is one, i.e., ϕ is birational. We also show that under a mild condition on the linear system |K_{M} + (n − 2)L| satisfied for all known examples, ϕ is birational unless (M, L) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ = 2. 1991 Mathematics Subject Classification. Primary 14C20, 14E05; Secondary 14J99, 14M99. Keywords and phrases. Smooth complex polarized n-fold, very ample line bundle, adjunction theory, log-general type, second adjunction mapping, birational map.

Introduction

Let L be a very ample line bundle on an n-dimensional projective manifold M. We assume throughout this introduction that n ≥ 3 and also that κ(K_{M}+(n−2)L) = n, i.e., that the Kodaira dimension of K_{M}+(n−2)L is n. This condition on the Kodaira dimension is satisfied except for a short list of special varieties (M, L) (see e.g., [2]).

The condition κ(K_{M} + (n − 2)L) = n implies that M is the blowing up of a projective manifold M, π : M → M, at a finite set such that K_{M} + (n − 2)L ≅ π^{∗}(K_{M} + (n − 2)L), where L := (π_{∗}L)^{∗∗} is an ample line bundle, K_{M} + (n − 1)L is very ample, and K_{M} := K_{M} + (n − 2)L is nef. Thus the a priori meromorphic map associated to |2(K_{M} + (n − 2)L)| factors as π composed with the mapping associated to |2K_{M}|. It is a theorem of the second author (see [2, (13.2.5)]) that in the situation of this paper 2K_{M} is spanned. Thus we are reduced to describing the structure of the morphism ϕ : M → ℙ_{ℂ} associated to |2K_{M}|. The morphism ϕ factors as r ∘ s, where r : M → Y is a birational morphism with connected fibers onto a normal projective variety Y and s is finite. The structure of r is completely understood; cf., [2, 12] for a description of this map.

In Theorem (2.1) we show that under the added assumption that there is a smooth divisor A ∈ |K_{M}| the map s (and hence ϕ) is birational unless n = 3, K_{M}^{3} = 1, K_{M}^{2} · L = 3, K_{M} · L^{2} = 9, and A is a Del Pezzo surface with K_{A}^{2} = 1. There is moreover a degree 27 threefold M ⊂ ℙ^{13} with these invariants and with deg ϕ = 2 (see (2.2)). Though we expect such a smooth A to exist always, this is not known. We show that, without any assumption on the existence of a smooth A ∈ |K_{M}|,

1) the same conclusion holds under the added hypothesis that n = dim M ≥ 5 (see Corollary (2.4));

2) the morphism ϕ is birational if κ(K_{M} + (n − 3)L) ≥ 0 (see Theorem (2.3)); and

3) deg ϕ ≤ 7 if n = 3 (see Theorem (3.1)).

The authors would like to thank the Max-Planck-Institut für Mathematik in Bonn, where a large part of the work was done. The second author would also like to thank the Alexander von Humboldt Stiftung for its support.

1 Background material

We work over the complex field 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.

≈, the linear 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;

Γ(L), the space of the global sections of a line bundle, L, on a variety V; we say that L is spanned if it is spanned at all points of V by Γ(L);

q(V) = h^{1}(O_{V}), the irregularity, for V smooth;

κ(D), the Kodaira dimension of the line bundle associated to a Cartier divisor D on V;

κ(V) := κ(K_{V}), the Kodaira dimension of V, for V smooth.

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

1.2 Genus formula. For a line bundle L on an irreducible normal 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 contained in the smooth points of V, then g(L) = g(C) = 1 − χ(O_{C}), the arithmetic genus of C.

1.3 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., [6, 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}) , (1)

where [x] means the greatest integer ≤ x.

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

Except for an explicit list of well understood pairs (M, L) (see in particular [2, §§7.2, 7.3, 7.4]) we can assume:

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

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

Since by the above we can assume that K_{M} + (n − 2)L is nef and big, from the Kawamata-Shokurov base point free theorem (see [7, §3]) we know that |m(K_{M} + (n − 2)L)|, for m ≫ 0, gives rise to a morphism φ : M → X with connected fibers and normal image. Thus there is an ample line bundle K on X such that K_{M} + (n − 2)L ≈ φ^{∗}K. The morphism φ is very well behaved (see e.g., [2, §§7.5, 7.6, 7.7 and Chapter 12]). Furthermore X has terminal, 2-Gorenstein (i.e., 2K_{X} is a line bundle) isolated singularities and K ≈ K_{X} + (n − 2)D, where D := (φ_{∗}L)^{∗∗} is a 2-Cartier divisor such that 2L ≈ φ^{∗}(2D) − Δ for some effective divisor Δ on M which is φ-exceptional and dim φ(Δ) ≤ 1 (see [2, (7.5.7)]). The pair (X, D) is known as the second reduction of (M, L). For definition and properties of terminal singularities we also refer to [7].

We say that (M, L) is of log-general type if κ(K_{M} + (n − 2)L) = n. Notice that this is equivalent to saying that K_{M} + (n − 2)L is nef and big (see [2, (7.6.9)]). Let Ŝ be the smooth surface obtained as transversal intersection of n − 2 general members of |L| and let S := π(Ŝ) be the corresponding smooth surface in M. Since K_{M} + (n − 2)L is nef and big the canonical bundle K_{S} of S is nef and big, so that S is a minimal surface of general type (see also [2, (7.6.10)]). We have

K_{S} · K_{S} < 9χ(O_{S}).  (2)

Indeed, Miyaoka’s inequality yields K_{S} · K_{S} ≤ 9χ(O_{S}). Note that the equality cannot happen. Otherwise S is ball quotient and hence a K(π, 1), which contradicts [2, (1.3)].

For further properties of log-general type polarized pairs see, e.g., [2, §13.2] and [3, (0.10)].

1.5 Pluridegrees. Let (M, L), (M, L) be as in (1.4). Define the pluridegrees, for j = 0, …, n = dim M, by

d̂_{j} := (K_{M} + (n − 2)L)^{j} · L^{n−j} and d_{j} := (K_{M} + (n − 2)L)^{j} · L^{n−j}.

If γ denotes the number of points blown up under π : M → M, then because K_{M} + (n − 2)L ≈ π^{∗}(K_{M} + (n − 2)L) + ∑_{i} E_{i}, E_{i} the exceptional divisors, the invariants d̂_{j}, d_{j} are related by

d̂_{j} = d_{j} − (−1)^{j}γ.

We put d̂:= d̂_{0}, d := d_{0}. If K_{M} + (n − 2)L is nef, by the generalized Hodge index theorem (see e.g., [2, (2.5.1), (13.1)]) one has

d_{j}^{2} ≥ d_{j−1}d_{j+1}, j = 1, …, n − 1,  (3)

and the parity Lemma (13.1.1) of [2] says that

d_{j} ≡ d_{j+1} mod(2) for j = 0, …, n − 1 and j even.  (4)

Moreover if K_{M} + (n − 2)L is nef and big, i.e., if (M, L) is of log-general type, the d_{j}’s are positive.

If the second reduction, (X, D), φ : M → X, with D = (φ_{∗}L)^{∗∗}, K ≈ K_{X} + (n − 2)D, of (M, L) exists, then we define

d′_{j} := K^{j} · D^{n−j}, j = 0, …, n, d′ := d′_{0}.

Note that d_{j} = d′_{j} for j ≥ 2. To see this recall that 2L ≈ φ^{∗}(2D) − Δ for some effective Cartier divisor Δ which is φ-exceptional (see (1.4)) and compute, for j ≥ 2,

2^{n−j}d_{j} = (K_{M} + (n − 2)L)^{j} · (2L)^{n−j}
= (φ^{∗}K)^{j} · (φ^{∗}(2D) − Δ)^{n−j} = 2^{n−j}K^{n−j} · D^{j} = 2^{n−j}d′_{j},

where the last but one equality follows from the fact that dim φ(Δ) ≤ 1.

The following is a consequence of the Tsuji inequality (see [13, (5.2)]), the log version of the usual Yau inequality (see also [2, (13.1.7), (13.1.8)]).

Proposition 1.6 (Tsuji inequality) Let M be a smooth 3-fold polarized by a very ample line bundle L. Assume that the first reduction, (M, L), of (M, L) exists. Let d_{j}, 0 ≤ j ≤ 3, be the pluridegrees of (M, L). Assume that K_{M} + L is nef. Let S be a smooth element in |L|. Then we have

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

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

The following lower bound for the degree is not optimal, but it is sufficient for our purposes.

Lemma 1.7 Let M be a smooth n-fold polarized by a very ample line bundle L. Assume that (M, L) is of log-general type. Let (M, L) be the first reduction of (M, L). Then either |L| embeds M in ℙ^{n+1} or d ≥ d̂:= L^{n} ≥ 8.

Proof. We can assume that Γ(L) embeds M in ℙ^{N} with N ≥ n + 2. Let Ŝ be the smooth surface obtained as transversal intersection of n − 2 general members of |L|. Since Ŝ is of general type we have by a result of Castelnuovo (see [2, (8.1)], [9, (0.6)]) that d̂ > 2(N − n) + 2 ≥ 6. Thus d ≥ d̂ ≥ 7. Assume d̂ = 7. Then N = n + 2 since otherwise d̂ > 2(N − n) + 2 ≥ 8. Thus by Castelnuovo’s bound we conclude that g(C) ≤ 6 for any smooth curve section C of M ⊂ ℙ^{n+2}. Therefore, by the genus formula, 7 + d_{1} ≤ 7 + d̂_{1} = 2g(C) − 2 ≤ 10, or d_{1} ≤ 3. But then d_{2} ≤ 1 by the Hodge index relation d_{1}^{2} ≥ dd_{2}. Since (M, L) is of log-general type we know that d_{j} ≥ 1 for each j = 1, …, n. Therefore we conclude that d_{2} = 1 and d_{2}^{2} ≥ d_{1}d_{3} implies d_{1} = 1. This gives the absurdity that d = 1. Q.E.D.

2 Two birationality results for the second adjunction mapping

Let M be a smooth connected n-fold polarized by a very ample line bundle L, n ≥ 3. Assume that (M, L) is of log-general type. Let π : (M, L) → (M, L), φ : (M, L) → (X, D), K ≈ K_{X} + (n − 2)D, be the first and the second reductions of (M, L) as in (1.4).

Set K_{M} := K_{M} + (n − 2)L. In this section we study the birationality of the map given by the complete linear system |2K_{M}| under the assumption that |K_{M}+(n−2)L| contains a smooth element (see (2.1)). We consider this result to be one of the most important results in this paper since it is likely optimal. We also prove the birationality of |2K_{M}| in the case when κ(K_{M} + (n − 3)L) ≥ 0 (see (2.3)).

Note that, since K_{M} ≈ φ^{∗}K, the map associated to |2K| is a birational morphism if and only if the map associated to |2K_{M}| is a birational morphism.

Let us recall the following basic result of the second author (see e.g., [2, (13.2.5)]).

• With the notations as above, assume that (M, L) is of log-general type. Then 2K_{M} is spanned by its global sections.

The first result we have is the following.

Theorem 2.1 Let M be a smooth connected n-fold polarized by a very ample line bundle L, n ≥ 3. Assume that (M, L) is of log-general type. Let π : (M, L) → (M, L) be the first reduction of (M, L). Let K_{M} := K_{M} + (n − 2)L. Assume that there exists a smooth element A in the complete linear system |K_{M}|. Then the map associated to |2K_{M}| is a birational morphism unless n = 3, d_{3} = 1, d_{2} = 3, d_{1} = 9 and A is a Del Pezzo surface with K_{A}^{2} = 1.

Proof. Let L_{A} be the restriction of L to A. Consider on M the exact sequence

0 → K_{M} + (n − 2)L → 2K_{M} → K_{A} + (dim A − 1)L_{A} → 0.

Since 2K_{M} is spanned by •) and h^{1}(K_{M} + (n − 2)L) = 0 we see that K_{A} + (dim A − 1)L_{A} is spanned and defines a morphism ψ_{A} which coincides with the restriction, ϕ_{A}, of the morphism ϕ associated to |2K_{M}|. Note that the connected part of the morphism ψ_{A} is the first reduction map, π_{A} : A → A′, of the pair (A, L_{A}) and that the positive dimensional fibers of π_{A} are contained in the exceptional set of ϕ. We have ψ_{A} = s_{A} ∘ π_{A}, where s_{A} : A′ → ℙ_{ℂ} is a finite-to-one morphism.

Assume first n ≥ 4. Then by a well known result due to the second author and Van de Ven (see e.g., [2, (11.3.1)]), s_{A} is an embedding. Let 𝔱 be the sheet number of ϕ, i.e., ϕ is a generically 𝔱-to-one morphism. Choose a general point x ∈ M. Then ϕ^{−1}(ϕ(x)) consists of 𝔱 distinct points. Note that, since 2A belongs to the linear system defining ϕ, a smooth element A ∈ |K_{M}| containing x contains all the 𝔱 points. Therefore ψ_{A}^{−1}(ψ_{A}(x)) consists of 𝔱 distinct points and hence, since s_{A} is an embedding, π_{A}^{−1}(π_{A}(x)) consists of 𝔱 distinct points. This contradicts the fact that π_{A} has connected fibers. Thus we conclude that 𝔱 = 1.

It remains to consider the case when n = 3. In this case exactly the same argument works unless the polarized surface (A, L_{A}) is one of the four exceptional pairs described e.g., in [2, (10.3.1)].

The first possibility is that A is a degree 9 Del Pezzo surface, K_{A}^{2} = 1, L_{A} ≈ −3K_{A}. Then L_{A} ≈ −3(2K_{M} + L)_{A}, or (6K_{M} + 4L)_{A} ≈ O_{A}. It thus follows that (3K_{M} − L)_{A} ≈ O_{A} and hence, since A ∈ |K_{M}|, we have (3K_{M} − L) · K_{M} · D = 0 for any effective divisor D on M. Therefore, by taking D = A and D = L, we find 3d_{3} = d_{2} and 3d_{2} = d_{1} respectively. Thus d_{1} = 9d_{3}. If d_{3} = 1, we have d_{1} = 9, d_{2} = 3 and we find the 3-dimensional exceptional case as in the statement. Note that the case d_{3} ≥ 2 cannot occur, since in this case d_{1} ≥ 18 and we get the numerical contradiction 9 = L_{A} · L_{A} = L · L · K_{M} = d_{1} ≥ 18.

The second possibility is that A is a degree 8 Del Pezzo surface, K_{A}^{2} = 2, L_{A} ≈ −2K_{A}. The same argument as in the case considered above gives now (4K_{M} −L)_{A} ≈ O_{A} and hence d_{1} = 4d_{2}, d_{2} = 4d_{3}. Thus we find the numerical contradiction 8 = L_{A} · L_{A} = L · L · K_{M} = d_{1} = 16d_{3}.

The third possibility is when A is the blowing up at one point, r : A → A′, of a degree 8 Del Pezzo surface (A′, L′) with L′ ≈ −2K_{A′}, K_{A′}^{2} = 2. In this case 2K_{A} + L_{A} ≈ r^{∗}(2K_{A′} + L′) + ℓ ≈ ℓ, where ℓ is the exceptional line of r. Therefore 2K_{A} + L_{A} has a section. Since A ∈ |K_{M}| it follows from K_{A} ≈ (2K_{M} − L)_{A} that (4K_{M} + 3L)_{A} ≈ (4K_{M} − L)_{A} has a section. Then from

(4K_{M} − L) · K_{M}^{2} = (4K_{M} − L)_{A} · K_{M|A} = (2K_{A} + L_{A}) · K_{M|A} = ℓ · K_{M|A} = 0

and

(4K_{M} − L) · K_{M} · L = (2K_{A} + L_{A}) · L_{A} = ℓ · L_{A} = 1

we get 4d_{3} = d_{2} and 4d_{2} = d_{1} + 1 respectively. But d_{1} = K_{M} · L · L = L_{A} · L_{A} = 7, so that we find the numerical contradiction 7 = d_{1} = 16d_{3} − 1 ≥ 15.

The fourth possibility is when A is a ℙ^{1}-bundle over a smooth elliptic curve of invariant e = −1, L_{A} ≈ 3E, E a section of minimal self-intersection. Let f ≅ ℙ^{1} be a fiber of the ℙ^{1}-bundle. Then K_{A} ≈ −2E +f and (K_{A} +L_{A})·L_{A} = (E +f)·3E = 6. Thus from 2L_{A} · K_{M|A} = (K_{A} + L_{A}) · L_{A} = 6 we get L_{A} · K_{M|A} = L · K_{M} · K_{M} = d_{2} = 3 . Moreover d_{1} = L_{A} · L_{A} = 9, so that the Hodge inequality d_{1} · d_{3} ≤ d_{2}^{2} yields d_{3} = 1. This gives the contradiction q(M) = 0 ([4, (3.2)]). Q.E.D.

Example 2.2 Notation as in (2.1). Note that the exceptional case of a polarized 3-fold (M, L) with the morphism associated to |2(K_{M} + L)| not birational and with invariants d_{3} = 1, d_{2} = 3, d_{1} = 9 as in (2.1) really occurs. The example in [8] has L = 3H very ample with K_{M} ≅ −2H. Thus all reduction maps are isomorphisms with K_{M} = K_{M} +L ≈ H having h^{0}(K_{M}) = 3, d_{3} = K_{M}^{3} = 1, d_{2} = K_{M}^{2} ·L = 3H^{3} = 3, d_{1} = K_{M}^{2} · L = 3H^{3} = 9. The morphism ϕ associated to |2K_{M}| has degree 2 (see also [11, (5.3)]). Indeed, from the exact sequence

0 → K_{M} ≅ H → 2H → 2H_{H} → 0,

we obtain h^{0}(2K_{M}) = 7 and ϕ : M → ℙ^{6} maps M surjectively onto the cone ℙ(O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(2)) over (ℙ^{2}, O_{ℙ^{2}}(2)). Since (2K_{M})^{3} = 8 and the tautological bundle of ℙ(O_{ℙ^{2}} ⊕ O_{ℙ^{2}}(2)) satisfies ξ^{3} = 4 we conclude that ϕ has degree 2.

A second result we have is the following.

Theorem 2.3 Let M be a smooth connected n-fold polarized by a very ample line bundle L, n ≥ 3. Assume that κ(K_{M} + (n − 3)L) ≥ 0. Let π : (M, L) → (M, L) be the first reduction of (M, L). Let K_{M} := K_{M} + (n − 2)L. Then the map associated to |2K_{M}| is a birational morphism.

Proof. We claim that d_{2} := K_{S}^{2} ≥ 10, where S = π(Ŝ) and Ŝ is the transversal intersection of n − 2 general members of |L|. To see this consider the 3-fold section V = V_{3} of M obtained as transversal intersection of n − 3 general members of |L|. Let L_{V} be the restriction of L to V. Let V := π(V) and let L_{V} be the restriction of L to V. Then the reduction is compatible with the restriction, i.e., (V, L_{V}) is the first reduction of (V, L_{V}). Let d = d_{0}, d_{1}, d_{2}, d_{3} be the pluridegrees of (V, L_{V}) as in (1.5), so that d_{2} = K_{S}^{2}. Note that the assumption κ(K_{M} + (n − 3)L) ≥ 0 implies κ(V) ≥ 0. Then from [12, (1.5), (3.1)] we know that

d_{3} ≥ d_{2} ≥ d_{1} ≥ d.  (5)

From Lemma (1.7) we have d ≥ 8. First consider the case when |L_{V}| embeds V in ℙ^{N}, N ≥ 6. If d = 8, 9, Castelnuovo’s bound (1.3) gives g := g(L_{V}) ≤ 5, 7 respectively. Since d_{1} ≥ d by (5) the genus formula d+d_{1} = 2g −2 gives a numerical contradiction.

Thus we can assume that |L_{V}| embeds V in ℙ^{5}. By looking over the list of small degree 3-folds in ℙ^{5} (see e.g., [5, Chapter 6]) we see that the only possible cases with d = 8, 9 are when V is the complete intersection of either a quadric and a quartic or two cubics in ℙ^{5} and L_{V} = O_{V}(1). Accordingly, M is the complete intersection either of type (2, 4) or of type (3, 3) in ℙ^{n+2} with L ≈ O_{M}(1). In both cases K_{M} +(n−2)L is very ample. Thus from now on we can assume that d_{2} ≥ 10.

First assume n = 3, so that κ(M) ≥ 0. Recalling •), consider the morphism ϕ : M → Y associated to |2K_{M}|. Assume that ϕ is not birational, i.e., is not generically one-to-one. Thus there exists a dense open set U ⊂ Y such that for any point y ∈ Y, the fiber ϕ^{−1}(y) contains at least two points x_{1} := x_{1,y}, x_{2} := x_{2,y} (depending on y) such that ϕ(x_{1}) = ϕ(x_{2}) = y. From a Bertini’s type theorem (see e.g., [2, (1.7.9)]) we know that there is a smooth surface Ŝ ∈ |L| passing through π^{−1}(x_{1}), π^{−1}(x_{2}). Then the image S = π(Ŝ) is a smooth surface in |L| passing through x_{1}, x_{2}.

The exact sequence

0 → K_{M} + K_{M} → 2K_{M} → 2K_{S} → 0

gives a surjection Γ(2K_{M}) → Γ(2K_{S}) → 0. Thus we see that ϕ restricts to the bicanonical map ϕ_{S} associated to |2K_{S}|. Hence in particular ϕ_{S} is not an embedding. Recall that S is a minimal surface of general type under our assumptions (see (1.4)). Then we know by Reider’s theorem (see e.g., [2, (8.5.1)]) that there exists an effective curve C_{y} ⊂ S, C_{y} depending on y and containing x_{1}, x_{2}, such that

K_{S} · C_{y} − 2 ≤ C_{y} · C_{y} < \frac{K_{S} · C_{y}}{2} < 2.

Therefore, since K_{S} is nef, we have 0 ≤ K_{S}·C_{y} ≤ 3. The case K_{S}·C_{y} = 3 is excluded. Indeed in this case C_{y}^{2} = 1 and the Hodge index relation d_{2} ≤ K_{S}^{2}C_{y}^{2} ≤ (K_{S} · C_{y})^{2} contradicts the assumption d_{2} ≥ 10.

Note that when y varies in U, the curves {C_{y}}_{y∈U} cover a dense open set ϕ^{−1}(U) of M. This is clear since the points x_{1,y}, x_{2,y} cover ϕ^{−1}(U).

Assume K_{S} · C_{y} = 0. Then, by the genus formula, C_{y}^{2} = −2 and C_{y} is the union of rational curves. Thus by the above, there is a dense open set U ⊂ M covered by rational curves. This contradicts the assumption κ(M) ≥ 0 (see e.g., [10, Part I, (5.8)]). Since κ(M) ≥ 0, a multiple tK_{M} of the canonical bundle K_{M} is effective for some t ≫ 0. Since the curve C_{y} moves it thus follows that K_{M} · C_{y} ≥ 0.

If K_{S} ·C_{y} = 1, 2, from K_{S} ·C_{y} = (K_{M} +L)·C_{y} ≥ L·C_{y} we obtain L·C_{y} ≤ 2. Since L is very ample outside of a finite set of points, this implies that C_{y} is a rational curve. Thus we conclude as in the previous case.

It remains to consider the general case when n ≥ 4. Let V_{t} be the t-fold section of M obtained as transversal intersection of n−t general members of |L|, t = 2, …, n, V_{n} := M, V_{2} := Ŝ. Let L_{t} be the restriction of L to V_{t}. let V_{t} := π(V_{t}) and let L_{t} be the restriction of L to V_{t}. Reductions are compatible with restrictions, i.e., (V_{t}, L_{t}) is the first reduction of (V_{t}, L_{t}). Set K_{t} := K_{V_{t}} + (t − 2)L_{t}. Consider the exact sequence, for t = 3, …, n − 1,

0 → K_{V_{t}} + K_{t} + (n − 3)L_{t} → 2K_{t} → 2K_{t−1} → 0,

which gives a surjective map Γ(2K_{t}) → Γ(2K_{t−1}). Assuming ϕ is not birational and using induction on t, we conclude as above that the bicanonical map of the surface section V_{2} := S of M with respect to L is not an embedding. Recalling that the assumption κ(K_{M} + (n − 3)L) ≥ 0 implies κ(V_{3}) = κ(V_{3}) ≥ 0, the above argument shows that we can reduce to the 3-fold case. Q.E.D.

By using results from [1] we obtain from Theorem (2.3) the following rather strong consequence.

Corollary 2.4 Let M be a smooth connected n-fold polarized by a very ample line bundle L. Assume that (M, L) is of log-general type. Let π : (M, L) → (M, L), φ : (M, L) → (X, D), K ≈ K_{X} + (n − 2)D, be the first and the second reductions of (M, L). Let K_{M} := K_{M} + (n − 2)L. If n ≥ 5, then the map associated to |2K_{M}| is a birational morphism.

Proof. First assume n ≥ 7. Then by [1, (3.1)] we know that K_{X} + (n − 3)K is nef. Thus t(K_{X} + (n − 3)K) is effective for some integer t ∈ ℤ, i.e., K_{X} + (n − 3)K is ℚ-effective. Therefore κ(K_{X} + (n − 3)K) ≥ 0. Hence from [2, (7.6.1), (7.6.2)] we have

κ(K_{X} + (n − 3)D) = κ(K_{M} + (n − 3)L) = κ(K_{M} + (n − 3)L) ≥ 0.

Thus Theorem (2.3) applies to give the result.

Assume n = 6. Then from [1, (4.1)] we know that K_{X} + 3K = K_{X} + (n − 3)K is nef unless (X, K) ≅^{σ} (ℙ^{6}, O_{ℙ^{6}}(1)). In this case K is very ample, so |2K_{M}| = |σ^{∗}O_{ℙ^{6}}(2)| defines a birational morphism. Thus the same argument as above lets us conclude that κ(K_{M} + 3L) ≥ 0 and hence that |2K_{M}| defines a birational morphism by (2.3).

Assume n = 5. Then from [1, (4.1)] we know that K_{X} + 3K is nef unless either (X, K) ≅^{σ} (Q, O_{Q}(1)), Q hyperquadric in ℙ^{6}, or L^{5} = 121(≠ \frac{3^{5}−1}{2}) and X is a singular 2-Gorenstein Fano 5-fold with 2K_{X} ≈ −7K.

In the first case, K is very ample, so that |2K_{M}| = |σ^{∗}O_{Q}(2)| gives a birational morphism again by (2.3).

In the second case, note that X is not Gorenstein since otherwise K_{X} ≈ −7A for some ample line bundle A on X. This contradicts the well known fact that X has index ≤ dim X + 1 = 6 (see also [2, (3.3.2)]). Thus, since X is not Gorenstein, [2, (0.3.3)] applies to say that h^{0}(K_{M} + 2L) = h^{0}(K_{M} + 2L) > 0.

Write 2K_{M} = K_{M} + (K_{M} + 2L) + L. Since K_{M} ≈ φ^{∗}(K), |L| gives a birational map (L is very ample off a finite set of points) and K_{M} + 2L is effective, we see that 2K_{M} is the sum of a nef and big and an effective line bundle. Thus we conclude that |2K_{M}| gives a birational morphism. Q.E.D.

The following two lemmas are not essentially used in the paper, but have some interest in themselves.

Lemma 2.5 Let M be a smooth connected n-fold polarized by a very ample line bundle L, n ≥ 3. Assume that (M, L) is of log-general type. Let π : (M, L) → (M, L) be the first reduction of (M, L). Let K_{M} := K_{M} + (n − 2)L. If h^{0}(K_{M} + K_{M} + tL) ≠ 0 for t ≤ n − 3, then the morphism given by |2K_{M}| is birational.

Proof. Write 2K_{M} as 2K_{M} = K_{M} + K_{M} + tL + (n − 2 − t)L. Since K_{M} + K_{M} + tL is effective, |L| gives a birational map and n − 2 − t > 0 we are done. Q.E.D.

Lemma 2.6 Let M be a smooth connected n-fold polarized by a very ample line bundle L, n ≥ 3. Assume that (M, L) is of log-general type. Let π : (M, L) → (M, L) be the first reduction of (M, L). Let K_{M} := K_{M} + (n − 2)L. Let V be the t-fold section of (M, L) obtained as transversal intersection of n−t general members of |L|. Let L_{V} be the restriction of L to V and let K_{V} := K_{V} + (t − 2)L_{V}. If t ≥ 3 and the map given by |2K_{M}| is not birational, then

h^{0}(K_{V} + K_{V} + jL_{V}) = 0 for 0 ≤ j ≤ t − 3.

Proof. Consider on M the Koszul complex

0 → −(n − t)L → ⋯ → ⊕^{n−t−1}(−2L) → ⊕^{n−t}(−L) → O_{M} → O_{V} → 0.

By tensoring with K_{M} + K_{M} + (n − t + j)L we get the exact sequence

0 → K_{M} + K_{M} + jL → ⋯ → K_{M} + K_{M} + (n − t + j)L → K_{V} + K_{V} + jL_{V} → 0.

Note that h^{1}(K_{M} + K_{M} + jL) = 0 by Kodaira vanishing and h^{0}(K_{M} + K_{M} + (n − t + j)L) = 0 by Lemma (2.5) since n−t+j ≤ n−3. Thus h^{0}(K_{V} +K_{V} +jL_{V}) = 0.Q.E.D.

As a consequence of (2.5) and (2.6) we have the following general fact.

Remark 2.7 Let M be a smooth connected n-fold polarized by a very ample line bundle L, n ≥ 3. Assume that (M, L) is of log-general type. Let π : (M, L) → (M, L) be the first reduction of (M, L). Let K_{M} := K_{M} +(n−2)L. If the morphism associated to |2K_{M}| is not birational, then h^{i}(O_{M}) = 0, i ≥ 3.

To see this, assume first n = 3. Then h^{3}(O_{M}) = h^{0}(K_{M}). If h^{0}(K_{M}) > 0, we would have an inclusion 0 → L → K_{M} + L, so that the morphism associated to |K_{M}| = |K_{M} + L| would be birational.

To show the statement in the general case, assume e.g., n = 4. Let V be the general smooth element of |L| and consider the exact sequence

0 → −L → O_{M} → O_{V} → 0.

Look at the cohomology associated to this sequence. Note that h^{i}(−L) = 0, i = 0, 1, 2, 3. Moreover h^{4}(O_{M}) = h^{0}(K_{M}) = 0 since otherwise we would have an inclusion

0 → (n − 2)L = 2L → K_{M} + 2L = K_{M},

and hence the same contradiction as above. Let L_{V} be the restriction of L to V. Set K_{V} := K_{V} + 2L_{V}. By the assumption, it follows that also the morphism associated to |2K_{M}| is not birational. Therefore Lemma (2.6) yields h^{0}(K_{V} +K_{V}) = 0 and thus h^{0}(K_{V}) = h^{3}(O_{V}) = 0. Then h^{3}(O_{M}) = 0.

In the general case the same argument as above gives the result, starting from the Koszul complex

0 → −(n − 3)L ⋯ → ⊕^{n−4}(−2L) → ⊕^{n−3}(−L) → O_{M} → O_{V} → 0.

3 A bound for the degree of the second adjunction mapping in the case n = 3

We keep the notation and the assumptions as at the beginning of §2. This section is devoted to the proof of the following result.

Theorem 3.1 Let M be a smooth connected threefold polarized by a very ample line bundle L. Assume that (M, L) is of log-general type. Let π : (M, L) → (M, L) be the first reduction of (M, L). Let K_{M} := K_{M} + L. Let 𝔱 be the degree of the morphism, ϕ, associated to |2K_{M}|. Then 𝔱 ≤ 7.

Proof. Let d_{j}, j = 0, 1, 2, 3, d_{0} = d, be the pluridegrees of (M, L). Let S be a smooth general member of |L|.

Note that we can assume h^{0}(K_{M} + K_{M}) = h^{0}(2K_{M} − L) = 0. Indeed otherwise Γ(2K_{M} − L) gives an embedding Γ(L) ↪ Γ(2K_{M}) and hence Γ(2K_{M}) gives a birational morphism, given on a Zariski open set by sections of Γ(L). Therefore a Riemann-Roch computation (see [4, §6]) gives the general relation

4χ(O_{S}) + d_{2} = d_{3} + h,  (6)

where h := h^{0}(K_{M} +L). Note also that K_{M} is nef and big. Then the exact sequence

0 → K_{M} + K_{M} → 2K_{M} → 2K_{S} → 0

yields h^{0}(2K_{M}) = d_{2} + χ(O_{S}). Thus we obtain

(2K_{M})^{3} = 8d_{3} = 𝔱 deg ϕ(M) ≥ 𝔱(d_{2} + χ(O_{S}) − 3).  (7)

Reasoning by contradiction, assume 𝔱 ≥ 8. Then (7) yields d_{3} ≥ d_{2} + χ(O_{S}) − 3, which, combined with (6), reads χ(O_{S}) ≥ 2h − 1. From this and Tsuji’s inequality

(1.6) we infer that \frac{d_{1}}{12} + \frac{d_{3}}{32} ≤ 1, i.e.,

\frac{d_{1}}{12} + \frac{d_{3}}{32} = 1.  (8)

Hence in particular d_{3} ≥ 2 since d_{1} is integer.

Recall that h ≥ 2 from [3, (1.2)]. Thus (6) and the above inequality χ(O_{S}) ≥ 2h − 1 give

\frac{d_{3} − d_{2}}{2} = 2χ(O_{S}) − 3h ≥ h − 2 ≥ 0,

so that d_{3} ≥ d − 2.

Recall that d ≥ 8 by Lemma (1.7) and d_{3} ≥ 2 by the above. Then simply starting from d ≥ 8, d_{1} ≥ 1, d_{2} ≥ 1 and d_{3} ≥ 2 and by using repeatedly the Hodge index inequalities (3) we obtain d ≥ 8, d_{1} ≥ 8, d_{2} ≥ 8, d_{3} ≥ 8.

Assume d̂:= L^{3} = 8. Then |L| embeds M in ℙ^{5}. Indeed otherwise Castelnuovo’s bound (1) gives g(L) ≤ Castel(8, 4) ≤ 5 and hence we would have the numerical contradiction 16 ≤ d̂ + d̂_{1} = d + d_{1} = 2g(L) − 2 ≤ 8. By looking over the list of 3-folds in ℙ^{5} of degree ≤ 12 (see [5, Chapter 6]) we see that the only degree 8 log-general type case is the complete intersection of a quadric and a quartic. In this case K_{M} ≈ O_{M}(1) and therefore 𝔱 = 1, contradicting the present assumption 𝔱 ≥ 8.

Thus we can assume d̂ ≥ 9, so that d ≥ 9. Then starting from d ≥ 9, d_{1} ≥ 1, d_{2} ≥ 1, d_{3} ≥ 2 and using repeatedly the Hodge index inequalities (3) we obtain d ≥ 9, d_{1} ≥ 9, d_{2} ≥ 9, d_{3} ≥ 9. Now use relation (8) to get the numerical contradiction

1 = \frac{d_{1}}{12} + \frac{d_{3}}{32} ≥ \frac{9}{12} + \frac{9}{32} > \frac{9}{12} + \frac{8}{32} = 1.

Thus we conclude that 𝔱 ≤ 7. Q.E.D.

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

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