Projective manifolds with small pluridegrees^{∗}

M.C. Beltrametti, A.J. Sommese

Abstract

Let L be a very ample line bundle on a connected complex projective manifold M of dimension n ≥ 3. Except for a short list of degenerate pairs (M, L), it follows that κ(K_{M} +(n−2)L) = n and that there exists a morphism π : M → M expressing M as the blowup of a projective manifold M at a finite set B, with K_{M} := K_{M} + (n − 2)L nef and big for the ample line bundle L := (π_{∗}L)^{∗∗}. The projective geometry of (M, L) is largely controlled by the pluridegrees d_{j} := L^{n−j} · (K_{M} + (n − 2)L)^{j} for j = 0, …, n, of (M, L). For example d_{0} + d_{1} is equal to the the genus of a curve section of (M, L), and d_{2} is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of (M, L). In this article a detailed analysis is made of the pluridegrees of (M, L). The restrictions found are used to give a new lower bound for the dimension of the space of sections of K_{M}. The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to |2(K_{M} + (n − 2)L)|.

Introduction

Let L be a very ample line bundle on a connected complex projective manifold M of dimension n ≥ 3. Except for a short list of degenerate pairs (M, L), it follows that κ(K_{M} +(n−2)L) = n and that there exists a morphism π : M → M expressing M as the blowup of a projective manifold M at a finite set B, with K_{M} := K_{M} + (n − 2)L nef and big for the ample line bundle L := (π_{∗}L)^{∗∗}. The projective geometry of (M, L) is largely controlled by the pluridegrees d_{j} := L^{n−j} · (K_{M} + (n − 2)L)^{j} for j = 0, …, n, of (M, L). For example d_{0} + d_{1} is equal to the the genus of a curve section of (M, L), and d_{2} is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of (M, L). In this article a detailed analysis is made of the pluridegrees of (M, L).

Here is a detailed summary of the results of this article. Theorems (2.1) details the exceptions to inequalities of the form d_{j} ≥ d_{j−1} + 2 in dimensions ≥ 5, and Theorem (2.2) details the exceptions to inequalities of the form 2d_{j} ≥ d_{j−1} in dimension 4.

Except for a short list of exceptions, K_{M} + (n − 3)K is nef and big if dim M ≥ 6. Theorem (3.1) shows that under the assumption dim M ≥ 6 we have with only a few exceptions that d_{j} ≥ 2 (\frac{n−2}{n+1}) d_{j−1} for j = 1, …, n.

Results that hold in the more difficult dimensions 3, 4, and 5 are presented in Theorems (4.1), (5.1), (6.2), and (7.1). As a first application of our results we give in Proposition (8.1) a new lower bound for the dimension of the space of sections of K_{M} when n ≥ 5.

∗1991 Mathematics Subject Classification. Primary 14J40; Secondary 14M99, 14C20 .

Keywords and phrases. Smooth complex polarized n-fold, very ample line bundle, adjunction theory, log-general type, pluridegrees.

In a sequel, we will use the results of this article to give sharp bounds for the degree of the morphism associated to |2(K_{M} + (n − 2)L)|.

Both authors would like to thank the Duncan Chair of the University of Notre Dame for the research support it provided in March of 1997. The first author would like to thank the Max-Planck-Institute für Mathematik in Bonn, where the final parts of this work, including the preparation of the manuscript, were done in July 1997. The second author would like to thank the Alexander von Humboldt Stiftung for their support in July 1997.

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.

≈ (respectively ∼), the 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;

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);

p_{g}(V) = h^{0}(K_{V}), the geometric genus, for V smooth;

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

c_{n}(V), the n-th Chern class of the tangent bundle of V, for V smooth;

κ(V) := κ(K_{V}), the Kodaira dimension, 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 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 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}) ,  (1)

where [x] means the greatest integer ≤ x.

1.4 Reductions. (See e.g., [6, 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 p 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 [6, §§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 [10, §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 pair (X, K) is known as the second reduction of (M, L). The morphism φ is very well behaved (see e.g., [6, §§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 X which is φ-exceptional and dim φ(Δ) ≤ 1 (see [6, (7.5.7)]). Notice that from K_{M} + (n − 2)L ≈ φ^{∗}(K_{X} + (n − 2)D) and 2L ≈ φ^{∗}(2D) − Δ it follows that 2K_{M} ≈ φ^{∗}(2K_{X}) + 3Δ. For definition and properties of terminal singularities we also refer to [10].

If Y is an irreducible subvariety of X we will denote by D_{Y} the dual sheaf (D_{|Y})^{∗∗}, where D_{|Y} is the restriction of D to Y as a sheaf. Moreover, for any k-dimensional irreducible subvariety Y ⊂ X, we set

D^{k} · Y := \frac{(2D)^{k} · Y}{2^{k}} .

We say that (M, L) is of log-general type if κ(K_{M} + (n − 2)L) = n. Notice that this is equivalent to say that K_{M} + (n − 2)L is nef and big (see [6, (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 [6, (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 [6, (1.3)].

For further properties of log-general type polarized pairs see e.g., [6, §13.2] and [7, (0.10)].

We recall the following special useful fact from [3, (4.41)].

Proposition 1.5 Let M be a smooth n-fold and let L be a very ample line bundle on M. Assume that |L| embeds M in ℙ^{N} with 2n > N. Assume that the first reduction, (M, L), of (M, L) exists. If d̂:= L^{n} ≠ 2^{n} − 1, then (M, L) ≅ (M, L).

1.6 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 ≈ 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., [6, (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 [6] 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.  (5)

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 first assertion of the following proposition is a special case of the general formula of Kleiman [11, (I, 37), Section D, p. 313] (see also [6, (8.2.1), (13.1.5)]). The second assertion follows from the proof of Lemma (2.3) in [7] (see also [6, (13.10)]).

Proposition 1.7 (Double point formula for surfaces) Let M be a smooth n-fold polarized by a very ample line bundle L, n ≥ 2. Assume that |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|. Let (M, L), π : M → M, be the first reduction of (M, L) as in (1.4). Let S := π(Ŝ). Let d̂_{j}, d_{j}, j = 0, …, n, d̂_{0} = d̂, be the pluridegrees of (M, L), (M, L) respectively. Then we have the following inequalities (note that g(L) = g(L), χ(O_{S}) = χ(O_{S}).

1. d̂(d̂− 5) − 10(g(L) − 1) + 12χ(O_{S}) ≥ 2d̂_{2}, with equality if N = n + 2.

2. Assume d + d̂≥ 7, e.g., d̂≥ 4. Then

d(d − 5) − 10(g(L) − 1) + 12χ(O_{S}) ≥ 2d_{2},

or, by using the genus formula (1.2),

d(d − 10) + 12χ(O_{S}) ≥ 2d_{2} + 5d_{1}.

The following is a consequence of the double point formula above (see also [6, (13.1.6)]).

Proposition 1.8 (Double point formula) 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 d̂_{j}, d_{j}, 0 ≤ j ≤ 3, be the pluridegrees of (M, L), (M, L) respectively. Let γ be the number of points blown up under the first reduction map. Let S be a smooth element in |L|. Then

44h^{0}(K_{M} + L) + 58χ(O_{S}) + 2h^{0}(K_{M}) + 4 ≥ 12d_{2} + 17d_{1} + d_{3} + (20 − d̂)d̂+ 5γ.

The following is a consequence of the Tsuji inequality (see [14, (5.2)]), the log version of the usual Yau inequality (see also [6, (13.1.7), (13.1.8)]).

Proposition 1.9 (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. 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 1.10 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 [6, (8.1)], [13, (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.

We need the following useful fact.

Lemma 1.11 Let L be a line bundle on an irreducible projective variety V and let p : V → Y be a birational surjective morphism with Y normal. Then we have an inclusion

H^{0}(V, L) ⊂ H^{0}(Y, (p_{∗}L)^{∗∗}).

Proof. Let Z be the union of the singular locus of Y and the exceptional set of the morphism p. Then Z is a closed subset of codimension ≥ 2 such that Y \ Z is smooth, and p induces an isomorphism V \ p^{−1}(Z) ≅ Y \ Z. A section s ∈ H^{0}(V, L) gives a section s′ ∈ H^{0}(V \ p^{−1}(Z), L) ≅ H^{0}(Y, (p_{∗}L)_{Y \Z}^{∗∗}) ≅ H^{0}(Y, (p_{∗}L)^{∗∗}), where the last isomorphism follows from the facts that Z is of codimension ≥ 2 in Y and Y is normal. Q.E.D.

2 Exceptions to nefness and bigness

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 reduction of (M, L) as in (1.4). In this section we study the exceptions to nefness and bigness of K_{X} + (n − 3)K when n ≥ 5, and the exceptions to nefness and bigness of K_{X} + 3K for n = 4.

First assume n ≥ 5. From [6, (7.7.2), (7.7.3), (7.7.5), (7.7.6)] we know that K_{X} + (n − 3)K is nef and big unless either

i) n = 6, (X, K) ≅ (ℙ^{6}, O_{ℙ^{6}}(1)); or

ii) n = 5, (X, K) ≅ (Q, O_{Q}), Q a quadric in ℙ^{6}; or

iii) n = 5, X is a ℙ^{4}-bundle over a smooth curve, K_{F} ≈ O_{ℙ^{4}}(1) for any fiber F; or

iv) n = 5, X is a singular 2-Gorenstein Fano 5-fold described in [6, (7.7.5)]; or

v) (X, K) is a scroll over a normal 4-fold Y, ψ : X → Y, K_{|F} ≈ O_{ℙ^{4}}(1) for a general fiber F; or

vi) (X, K) is a quadric fibration, ψ : X → Y, over a normal 3-fold Y; or

vii) (X, K) is a Del Pezzo fibration, ψ : X → Y, over a normal surface Y; or

viii) (X, K) is a Mukai fibration, ψ : X → Y, over a smooth curve Y; or ix) X is a Fano n-fold with K_{X} ≈ −(n − 3)K if n is even and 2K_{X} ≈ −2(n − 3)K if n

is odd.

In cases i), ii) one has d_{n} = d_{6} = 1, d_{n} = d_{5} = 2 respectively.

In cases v), vi), vii), viii) we have K_{X} + (n − 3)K ≈ ψ^{∗}H if n is even and 2(K_{X} + (n − 3)K) ≈ ψ^{∗}L if n is odd, for some ample line bundle H on Y. In both cases, since K ≈ K_{X} + (n − 2)D, we conclude that K − D is numerically effective. In this case, dotting with K^{n−1}, we get the relation

d′_{n} > d′_{n−1} on X,

or, recalling (5),

d_{n} > d_{n−1} on M.  (6)

By using the Hodge index relation (3), we find

d_{n} > d_{n−1} > d_{n−2} > ⋯ > d_{1} > d.  (7)

According to whether n is odd or even we have the following bounds:

d_{k} ≥ d_{k−1} + 2 for k = 1, …, n, and hence d_{n} ≥ d + 2n, if n is odd,

and

d_{n} ≥ d_{n−1} + 1, d_{k} ≥ d_{k−1} + 2 for k = 1, …, n − 1, and hence d_{n} ≥ d + 2n − 1, if n is even.

Assume n is odd. By parity (4) we have

d_{n} ≥ d_{n−1} + 2.

Let t be the biggest integer, t < n, such that d_{t} < d_{t−1} + 2. Since d_{t} > d_{t−1}, thus we get d_{t} = d_{t−1} + 1. Therefore d_{t−1}d_{t+1} ≤ d_{t}^{2} = d_{t}d_{t−1} + d_{t}, or

d_{t−1}(d_{t+1} − d_{t}) ≤ d_{t}.

By the assumption on t one has d_{t+1} − d_{t} ≥ 2. Then we find 2d_{t−1} ≤ d_{t}, whence 2d_{t−1} ≤ d_{t} = d_{t−1} + 1. This gives d_{t−1} = 1 and hence d = 1 by the Hodge index inequalities. Therefore d̂= 1 and (M, L) ≅ (ℙ^{n}, O_{ℙ^{n}}(1)), contradicting the log-general type assumption.

Assume n is even. By parity (4) we have

d_{n−1} ≥ d_{n−2} + 2.

Let t be the biggest integer, t ≤ n, such that d_{t} < d_{t−1} + 2. Since d_{t} > d_{t−1}, then we have d_{t} = d_{t−1} + 1. It may happen that d_{n} = d_{n−1} + 1. If this is not the case we may assume t < n and exactly the same argument as above leads to a contradiction.

The arguments above show the following result.

Theorem 2.1 Let M be a smooth connected n-fold polarized by a very ample line bundle L, with n ≥ 5. Assume that (M, L) is of log-general type. Let (M, L), (X, D), K ≈ (n − 2)D be the first and the second reduction of (M, L) as in (1.4). Let d_{j}, j = 0, …, n, d_{0} = d, be the pluridegrees of (M, L). Assume that K_{X} + (n − 3)K is not nef and big. Then either 1. d_{k} ≥ d_{k−1} + 2 for k = 1, …, n, and hence d_{n} ≥ d + 2n, if n is odd

and

d_{n} ≥ d_{n−1} + 1, d_{k} ≥ d_{k−1} + 2 for k = 1, …, n − 1, and hence d_{n} ≥ d + 2n − 1, if n is even; or

2. n = 6, (X, K) ≅ (ℙ^{6}, O_{ℙ^{6}}(1)); or

3. n = 5, (X, K) ≅ (Q, O_{Q}), Q a quadric in ℙ^{6}; or

4. n = 5, X is a ℙ^{4}-bundle over a smooth curve, K_{|F} ≈ O_{ℙ^{4}}(1) for any fiber F; or

5. n = 5, X is a singular 2-Gorenstein Fano 5-fold described in [6, (7.7.5)]; or

6. X is a Fano n-fold with K_{X} ≈ −(n − 3)K if n is even and 2K_{X} ≈ −2(n − 3)K if n is odd.

We consider now the case n = 4. From [4, (2.1)] we know that K_{X} + 3K is nef and big unless either

i) (X, K) ≅ (ℙ^{4}, O_{ℙ^{4}}(1)); or

ii) X is a Gorenstein Fano 4-fold with −K_{X} ≈ 3K and −4K ≈ 6D; or

iii) there exists a holomorphic map ψ : X → C, where C is a smooth curve, 4K_{X} +6D ≈ ψ^{∗}H for some ample line bundle H on C. In this case K_{X} + 3K ≈ ψ^{∗}H, i.e., (X, K) is a quadric fibration over C; or

iv) there exists a holomorphic map ψ : X → S, where S is a smooth surface, 4K_{X} +6D ≈ ψ^{∗}H for some ample line bundle H on S. In this case K_{X} + 3K ≈ ψ^{∗}H, i.e., (X, K) is a scroll over S.

In case i), d_{4} = 2. In cases iii) and iv), 4K_{X} + 6D is effective. From [6, (7.6.1)] we know that

H^{0}(X, 4K_{X} + 6D) ≅ H^{0}(M, 4K_{M} + 6L),

so that 4K_{M} + 6L = 4(K_{M} + 2L) − 2L is effective on M. Therefore, dotting with (K_{M} + 2L)^{j} · L^{3−j}, j = 0, 1, 2, 3, we obtain

2d_{j} > d_{j−1}, j = 1, 2, 3, 4.

The arguments above show the following result.

Theorem 2.2 Let M be a smooth connected 4-fold polarized by a very ample line bundle L. Assume that (M, L) is of log-general type. Let (M, L), (X, D), K ≈ 2D be the first and the second reduction of (M, L) as in (1.4). Let d_{j}, j = 0, 1, 2, 3, 4, d_{0} = d, be the pluridegrees of (M, L). Assume that K_{X} + 3K is not nef and big. Then either

1. 2d_{j} > d_{j−1}, j = 1, 2, 3, 4; or

2. (X, K) ≅ (ℙ^{4}, O_{ℙ^{4}}(1)), d_{4} = 1; or

3. X is a Gorenstein Fano 4-fold with −K_{X} ≈ 3K and −4K_{X} ≈ 6D.

3 The case n ≥ 6

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 reduction of (M, L) as in (1.4). In this section we consider the general case when n ≥ 6. From the results of §2 we can assume that

• K_{X} + (n − 3)K is nef and big.

It should be noted that K_{X} +(n−3)K is always nef under the log-general type assumption if n ≥ 7 (see [2, (3.1)]). We can prove the following general result.

Theorem 3.1 Let M be a smooth connected n-fold polarized by a very ample line bundle L, with n ≥ 6. 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 reduction of (M, L). Assume that K_{X} + (n − 3)K is nef and big. Let d_{j} be the pluridegrees of (M, L), j = 0, …, n, d_{0} = d. Then either

1. d_{j} ≥ 2 (\frac{n − 1}{n + 1}) d_{j−1}, j = 1, …, n, and hence d_{n} ≥ (\frac{2n − 2}{n + 1})^{n} d; or

2. n = 8 and there is a birational morphism ψ : X → ℙ^{8} such that K_{X} + (n − 3)D ≈ ψ^{∗}O_{ℙ^{8}}(1); or

3. n = 6, d̂= L^{6} = 364 and d_{j} ≥ \frac{13}{16}d_{j−1}, j = 1, …, 6.

Proof. First assume n ≥ 7. Thus Theorem (3.1) of [2] applies to say that (n + 1)K_{M} + n(n − 3)L is effective unless n = 8 and we are as in case 2). From [6, (7.6.1)] we know that

H^{0}(M, (n + 1)K_{M} + n(n − 3)L) ≅ H^{0}(M, (n + 1)K_{M} + n(n − 3)L).

Therefore, up to the special case with n = 8,

(n + 1)K_{M} + n(n − 3)L = (n + 1)(K_{M} + (n − 2)L) − 2(n − 1)L

is effective. Thus, dotting with (K_{M} + (n − 2)L)^{j−1} · L^{n−j}, we get

d_{j} ≥ 2 (\frac{n − 1}{n + 1}) d_{j−1}, j = 1, …, n.

Hence in particular

d_{n} ≥ (\frac{2n − 2}{n + 1})^{n} d.

Assume now n = 6. Thus from [2, (4.1)] we know that 13K_{M} + 36L is effective. If moreover d̂= L^{6} ≠ 364, then 7K_{M} + 18L is effective.

Let us consider first the general case d̂≠ 364. From [6, (7.6.1)] we know that

H^{0}(M, 7K_{M} + 18L) ≅ H^{0}(M, 7K_{M} + 18L).

Therefore 7K_{M} + 18L = 7(K_{M} + 4L) − 10L is effective. Thus dotting with 7(K_{M} + (n − 2)L)^{j−1} · L^{6−j}, we get

d_{j} ≥ \frac{10}{7}d_{j−1} = (\frac{2n − 2}{n + 1}) d_{j−1}, j = 1, …, 6,  (8)

as in 1). In the special case d̂= 364, the same argument gives

13d_{j} ≥ 16d_{j−1}, j = 1, …, 6,  (9)

as in case 3). Q.E.D.

We have stronger bounds in special cases.

Remark 3.2 Notation and assumptions as in (3.1). Assume that either X is not Gorenstein or D is not Cartier. Then we know from [2, (3.1)] that K_{M} + (n − 4)L is effective. By using again [6, (7.6.1)] we thus conclude that K_{M} + (n − 3)L = K_{M} + (n − 2)L − 2L is effective. Therefore, dotting with (K_{M} + (n − 2)L)^{j−1} · L^{n−j}, we get the stronger bound d_{j} > 2d_{j−1}, j = 1, …, n, and hence d_{n} > 2^{n}d.

We have the following numerical conditions, expressed in terms of pluridegrees, for a given variety V to be a k-fold section of the polarized pair (M, L).

Remark 3.3 Let M be a smooth connected n-fold polarized by a very ample line bundle L with either n ≥ 7 or n = 6 and L^{6} ≠ 364, as in (3.1). Let V be the k-fold section of M obtained as transversal intersection of n − k general members of |L|, k = 2, …, n − 1. Let L_{V} be the restriction of L to V. Let π : (M, L) → (M, L), φ : (M, L) → (X, D) be the first and the second reduction of (M, L). Let V := π(V), Y := φ(V). Let L_{V} and D_{Y} be the restrictions of L and D to V and Y respectively. Thus intersections are compatible with reductions, i.e., (V, L_{V}), (Y, D_{Y}) are the first and the second reduction of (V, L_{V}).

Let d′_{j}s be the pluridegrees of (M, L) and let

d_{j}(V) := (K_{V} + (k − 2)L_{V})^{j} · L_{V}^{k−j}(= (K_{M} + (n − 2)L^{j}) · L^{n−j})

be the pluridegrees of (V, L_{V}). We have

d_{k} = d_{k}(V), k ≥ 1, and d = L^{n} = L_{V}^{k} .

Assuming n ≠ 8 and L^{n} ≠ 364, we have from (3.1)

d_{k}(V) = d_{k} ≥ (\frac{2n − 2}{n + 1})^{k} d,  (10)

i.e., V cannot be a k-fold section of (M, L) as soon as d_{k}(V) = d_{k} < (\frac{2n−2}{n+1})^{k} d.

In particular, for k = 2, inequality (10) gives a lower bound for the Euler characteristic of the surface section V = S. Indeed Miyaoka’s inequality d_{2} < 9χ(O_{S}) combined with (10) gives

χ(O_{S}) > \frac{4}{9} (\frac{n − 1}{n + 1})^{2} d.  (11)

4 The case n = 5

Let M be a smooth connected 5-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} + 3D, be the first and the second reduction of (M, L) as in (1.4). From the results of §2 we can assume that

• K_{X} + 2K is nef and big.

We can prove the following result.

Theorem 4.1 Let M be a smooth connected 5-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} + 3D, be the first and the second reduction of (M, L). Assume that K_{X} +2K is nef and big. Let d_{j}’s, d_{0} = d, j = 0, …, 5, be the pluridegrees of (M, L). Then either

1. d ≥ 14, d_{1} ≥ 19, d_{2} ≥ 24, d_{3} ≥ 30, d_{4} ≥ 37, d_{5} ≥ 44; or

2. M is the complete intersection of a quadric and a quintic in ℙ^{7}, L ≅ O_{M}(1), (M, L) ≅ (M, L), d = 10, d_{j} = 2^{j}d, j = 1, 2, 3, 4, 5; or

3. M is the complete intersection of a quadric and a sextic in ℙ^{7}, L ≅ O_{M}(1), (M, L) ≅ (M, L), d = 12, d_{j} = 3^{j}d, j = 1, 2, 3, 4, 5; or

4. M is the complete intersection of a cubic and a quartic in ℙ^{7}, L ≅ O_{M}(1), (M, L) ≅ (M, L), d = 12, d_{j} = 2^{j}d, j = 1, 2, 3, 4, 5.

Proof. Consider the line bundle F := 2(K_{X} + 2D) on X. Since K_{X} + 2K ≈ 3(K_{X} + 2D) is nef and big we conclude that

F is nef and big.  (12)

Moreover

2F − K_{X} = 3K_{X} + 8D = K_{X} + 2D + 2(K_{X} + 3D)

is nef and big since by the above K_{X} + 2D is nef and big and K ≈ K_{X} + 3D is ample. Thus the Kawamata-Shokurov basepoint free theorem (see e.g., [6, (1.5.1)]) applies to say that

tF is spanned by global sections for sufficiently large integers t.  (13)

From [6, (7.2.9)] (see also [2, (0.2.8)]) we know that h^{i}(t(K_{X} + 2D) = 0 for all positive integers t and for i > 0. Then in particular

h^{i}(tF) = 0 for i > 0, t > 0.  (14)

Thus in view of conditions (12), (13), (14), Proposition (0.5) of [2] applies to say that either

h^{0}(K_{X} + 5F) = h^{0}(11K_{X} + 20D) > 0,

or there exists a birational morphism ψ : X → ℙ^{5} such that F = 2(K_{X} + 2D) ≈ ψ^{∗}O_{ℙ^{5}}(1). The latter case is easily ruled out. Indeed we have (ψ_{∗}K_{X})^{∗∗} ≈ O_{ℙ^{5}}(−6) and (ψ_{∗}D)^{∗∗} ≈ O_{ℙ^{5}}(a) for some integer a. Hence we get the numerical contradiction −12 + 4a = 1.

Thus we conclude that 11K_{X} + 20D is effective. We claim that 2(11K_{M} + 20L) is effective. To see this recall that 2K_{M} ≈ φ^{∗}(2K_{X}) + 3Δ and 2L ≈ φ^{∗}(2D) − Δ, for some effective Cartier divisor Δ (see (1.4)). Thus

22K_{M} + 40L ≈ φ^{∗}(22K_{X} + 40D) + (33 − 20)Δ.

Hence 2(11K_{M} + 20L) ≈ 22K_{M} − 26L is effective, where K_{M} := K_{M} + 3L. Thus dotting with K_{M}^{j−1} · L^{5−j}, we get

11d_{j} ≥ 13d_{j−1}, j = 1, 2, 3, 4, 5.  (15)

From Lemma (1.10) we know that d ≥ d̂≥ 8. Then inequalities (15) give d_{1} ≥ 10.

Assume that |L| embeds M in ℙ^{N}, N ≥ 9. Suppose d̂:= L^{5} = 14. Then d ≥ 14 and d_{1} ≥ 17 by (15). Castelnuovo’s bound (1) gives

g := g(L) ≤ Castel(14, 5) ≤ 15.

Then we find the numerical contradiction 31 ≤ d_{1} + d = d̂+ d̂_{1} = 2g − 2 ≤ 28.

Exactly the same argument, by using (15) and (1), rules out the low degree cases 8 ≤ d̂≤ 13.

Thus d ≥ d̂≥ 15. From (15) we obtain

d ≥ 15, d_{1} ≥ 18, d_{2} ≥ 22, d_{3} ≥ 26, d_{4} ≥ 31, d_{5} ≥ 37.

If d_{1} = 18, then d ≥ 16 by parity, contradicting d_{1}^{2} = 324 ≥ dd_{2} ≥ 330. Therefore d_{1} ≥ 19. Then d_{2}^{2} ≥ d_{1}d_{3} ≥ 494 yields d_{2} ≥ 23. If d_{3} = 26, then d_{2} ≥ 24 by parity, contradicting d_{3}^{2} = 676 ≥ d_{2}d_{4} ≥ 744. Therefore d_{3} ≥ 27. If d_{3} = 27, then d_{2} ≥ 25 by parity, which contradicts d_{3}^{2} = 729 ≥ d_{2}d_{4} ≥ 775. Thus d_{3} ≥ 28, so that (15) gives d_{4} ≥ 34 and d_{5} ≥ 41. Summarizing, we have

d ≥ 15, d_{1} ≥ 19, d_{2} ≥ 23, d_{3} ≥ 28, d_{4} ≥ 34, d_{5} ≥ 41.

From d_{2}^{2} ≥ d_{1}d_{3} ≥ 532 we get d_{2} ≥ 24, so that (15) gives d_{3} ≥ 29, d_{4} ≥ 35, d_{5} ≥ 42. If d_{3} = 29, then d_{2} ≥ 25 by parity, contradicting d_{3}^{2} = 841 ≥ d_{2}d_{4} ≥ 875. Thus d_{3} ≥ 30, so that (15) yields d_{4} ≥ 36, d_{5} ≥ 43. Then we have

d ≥ 15, d_{1} ≥ 19, d_{2} ≥ 24, d_{3} ≥ 30, d_{4} ≥ 36, d_{5} ≥ 43.

Finally, if d_{4} = 36, then d_{5} ≥ 44 by parity, which contradicts d_{4}^{2} = 1296 ≥ d_{3}d_{5} ≥ 1320. Therefore d_{4} ≥ 37 and (15) gives d_{5} ≥ 44, as in case 1).

Thus we can assume that |L| embeds M in ℙ^{N}, N ≤ 8. Then by Barth-Lefschetz theorem we know that Pic(M) ≅ ℤ[L]. Note that this condition implies M ≅ M ≅ X. Moreover L ≈ O_{M}(1), K_{M} ≈ O_{M}(r) for some r ∈ ℤ and hence

d_{j} = (r + 3)d_{j−1} = (r + 3)^{j}d, j = 1, 2, 3, 4, 5.  (16)

From (15) one has r + 3 ≥ 2. Note that for the sectional genus g we have the upper bound (see e.g., [1, p. 118])

2g − 2 ≤ \frac{d(d − 4)}{2}.

Therefore, recalling that r + 3 ≥ 2, we have 2g − 2 ≥ 3d and hence the inequality

3d ≤ \frac{d(d − 4)}{2},

which gives d ≥ 10. Hence d_{1} ≥ 12 by (15). Let V be the smooth 3-fold section of M obtained as transversal intersection of two general members of |L|. Let L_{V} be the restriction of L to V. Then V is embedded by |L_{V}| in ℙ^{N−2} as 3-fold of degree d = d̂= L_{V}^{3}. Moreover (V, L_{V}) is of log-general type since (M, L) is of log-general type and (V, L_{V}) coincides with its own first reduction since (M, L) ≅ (M, L). In what follows we use the lists from [5, Chapter 6] of 3-folds of degree ≤ 12 in ℙ^{5}.

Let d = 10. Then M lies in ℙ^{7} since otherwise g ≤ Castel(10, 4) ≤ 9, which leads to the contradiction 22 ≤ d+d_{1} = 2g −2 ≤ 16. Since by (16) all the pluridegrees are multiple of d = 10, we see that the only possible case is when V is the complete intersection of a quadric and a quintic in ℙ^{5}. Then M is the complete intersection of a quadric and a quintic in ℙ^{7}, and d_{j} = 2^{j}d, j = 1, 2, 3, 4, 5 as in case 2).

Thus we can assume d ≥ 11. Note that the case d = 11 doesn’t happen. Indeed in this case d_{1} must be odd by parity and hence by (16) we have d_{1} = (2k + 1)d for some positive integer k. Then in particular d_{1} ≥ 3d. Since d is odd we have the upper bound (see e.g., [1, p. 118])

2g − 2 ≤ \frac{(d − 1)(d − 3)}{2} − 2.

This gives the numerical contradiction 38 = 2g − 2 = d_{1} + d ≥ 4d ≥ 44.

Thus d ≥ 12. Assume that d = 12 and that |L| embeds M in ℙ^{7}. Then |L_{V}| embeds V in ℙ^{5} and, recalling (16), we see that either V is the complete intersection of a quadric and a sextic in ℙ^{5} or V is the complete intersection of a cubic and a quartic in ℙ^{5}. Accordingly, either M is the complete intersection of a quadric and a sextic in ℙ^{7}, with L ≅ O_{M}(1), d_{j} = 3^{j}d, j = 1, 2, 3, 4, 5, as in case 3), or M is the complete intersection of a cubic and a quartic in ℙ^{7}, with L ≅ O_{M}(1), d_{j} = 2^{j}d, j = 1, 2, 3, 4, 5, as in case 4).

Thus we can assume that d = 12 and that |L| embeds M in ℙ^{8}. Then Castelnuovo’s bound gives g ≤ Castel(12, 4) ≤ 15. Since, by (16), d_{1} = (r + 3)d and r + 3 ≥ 2 we find the contradiction 36 ≤ 3d ≤ d + d_{1} = 2g − 2 ≤ 30.

Thus d ≥ 13. Assume that d = 13 and that |L| embeds M in ℙ^{7}. Since d_{1} must be odd by parity, we have d_{1} ≥ (2k +1)d for some positive integer k from (16). Castelnuovo’s bound gives g ≤ Castel(13, 3) ≤ 30. If k ≥ 2, i.e., d_{1} ≥ 5d, we find the contradiction 78 ≤ 6d = d + d_{1} = 2g − 2 ≤ 58. Therefore k = 1, i.e., d_{1} = 3d, so that g = 2d + 1 = 27 doesn’t reach the maximum according to Castelnuovo’s bound. Thus, since the hyperplane curve section of M lies in ℙ^{3}, we can use Gruson-Peskine bound which gives (see e.g., [6, (1.4.9)])

g ≤ \frac{d(d − 3)}{6} + 1 = \frac{68}{3},

or g ≤ 22. This contradicts the above equality g = 27.

Therefore we have to consider the case when d = 13 and |L| embeds M in ℙ^{8}. Again, as above, d_{1} ≥ 3d. Castelnuovo’s bound yields g ≤ Castel(13, 4) ≤ 18 and then we get the contradiction 52 ≤ 4d ≤ d_{1} + d = 2g − 2 ≤ 34.

Thus we can assume d ≥ 14, so that (16) gives, since r + 3 ≥ 2,

d ≥ 14, d_{1} ≥ 28, d_{2} ≥ 56, d_{3} ≥ 112, d_{4} ≥ 224, d_{5} ≥ 448

and we fall again in case 1). Q.E.D.

.

5 The case n = 4

Let M be a smooth connected 4-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} + 2D, be the first and the second reduction of (M, L) as in (1.4). From the results of §2 we can assume that

• K_{X} + 3K is nef and big.

We can prove the following result.

Theorem 5.1 Let M be a smooth connected 4-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} + 2D, be the first and the second reduction of (M, L). Assume that K_{X} + 3K is nef and big. Let d_{j}’s, d_{0} = d, j = 0, 1, 2, 3, 4, be the pluridegrees of (M, L). Then either

1. d ≥ 11, d_{1} ≥ 9, d_{2} ≥ 7, d_{3} ≥ 5, d_{4} ≥ 3; or

2. M is the complete intersection of a quadric and a quartic in ℙ^{6}, L ≅ O_{M}(1), (M, L) ≅ (M, L), and d_{j} = 8, j = 0, 1, 2, 3, 4; or

3. M is the complete intersection of two cubics in ℙ^{6}, L ≅ O_{M}(1), (M, L) ≅ (M, L), and d_{j} = 9, j = 0, 1, 2, 3, 4; or

4. M is the complete intersection of of a quadric and a quintic in ℙ^{6}, L ≅ O_{M}(1), (M, L) ≅ (M, L), and d = 10, d_{j} = 2d_{j−1}, j = 1, 2, 3, 4.

Proof. Since K_{X} + 3K ≈ 4K − 2D is nef and big, dotting with K^{j} · D^{3−j}, j = 2, 3, we obtain d′_{4} > d′_{3}/2, d′_{3} > d′_{2}/2 on X. Therefore, by using (5), we have d_{4} > d_{3}/2, d_{3} > d_{2}/2 on M. Thus Hodge index inequalities (3) give d_{2} > d_{1}/2, d_{1} > d/2. Then we get

d_{4} > \frac{d_{3}}{2} > \frac{d_{2}}{4} > \frac{d_{1}}{8} > \frac{d}{16}.  (17)

In particular d_{4} ≥ 2 since otherwise d = d̂ = 1 and hence (M, L) ≅ (ℙ^{n}, O_{ℙ^{n}}(1)), contradicting the log-general type assumption. From Lemma (1.10) we know that d ≥ d̂:= L^{4} ≥ 8. Then inequalities (17) give

d ≥ 8, d_{1} ≥ 5, d_{2} ≥ 3, d_{3} ≥ 2, d_{4} ≥ 2.

By using repeatedly the Hodge index inequalities we obtain d_{1} ≥ 7, d_{2} ≥ 6, d_{3} ≥ 4. Then d_{4} > d_{3}/2 from (17) gives d_{4} ≥ 3, and hence d_{3}^{2} ≥ d_{2}d_{4} = 18 yields d_{3} ≥ 5. If d_{1} = 7, then d ≥ 9 by parity. This contradicts d ≤ d_{1}^{2}/d_{2} ≤ 49/6 < 9. Therefore d_{1} ≥ 8, so that d_{2}^{2} ≥ d_{1}d_{3} ≥ 40 gives d_{2} ≥ 7. Thus we have

d ≥ 8, d_{1} ≥ 8, d_{2} ≥ 7, d_{3} ≥ 5, d_{4} ≥ 3.  (18)

Assume now that |L| embeds M in ℙ^{N}, N ≥ 7. If d̂ = L^{4} = 8, 9, 10, Castelnuovo’s bound g := g(L) ≤ Castel(d̂, 4) gives respectively g ≤ 5, g ≤ 7, g ≤ 9. In each case 18 ≤ d_{1} + d = d̂_{1} + d̂ = 2g − 2 ≤ 16 leads to a numerical contradiction. Thus d ≥ d̂ ≥ 11 and d_{1}^{2} ≥ dd_{2} ≥ 77 gives d_{1} ≥ 9. Then, recalling (18), we are in case 1).

Thus we can assume that |L| embeds M in ℙ^{6}. Therefore by Barth-Lefschetz theorem we know that Pic(M) ≅ ℤ[L]. Note that this condition implies M ≅ M ≅ X. Moreover L ≅ O_{M}(1), K_{M} ≅ O_{M}(r) for some r ∈ ℤ and hence

d_{j} = (r + 2)d_{j−1} = (r + 2)^{j}d, j = 1, 2, 3, 4.  (19)

Let V be a general smooth member of |L|. Let L_{V} be the restriction of L to V. Then V is a smooth 3-fold of degree d = d̂ = L_{V}^{3} embedded by |L_{V}| in ℙ^{5}. Moreover (V, L_{V}) is of log-general type since (M, L) is of log-general type and (V, L_{V}) coincides with its own first reduction since (M, L) ≅ (M, L). In what follows we use the lists from [5, Chapter 6] of 3-folds of degree ≤ 12 in ℙ^{5}.

Assume d(= d̂) = 8. There are only two possible cases. In the first case K_{V} + L_{V} is not nef and big, contradicting the log-general type assumption. In the second case V is the complete intersection of a quadric and a quartic in ℙ^{5}, so that M is a complete intersection of type (2, 4) in ℙ^{6} and d_{j} = 8, j = 0, 1, 2, 3, 4, as in 2).

Let d(= d̂) = 9. Then from the lists we see that d̂_{1} = d_{1} ≤ 13 and hence (19) gives r + 2 = 1, i.e., d_{j} = d_{j−1}, j = 1, 2, 3, 4. This is not possible unless V is the complete intersection of two cubic hypersurfaces in ℙ^{5}. In this case M is the complete intersection of two cubics in ℙ^{6} and d_{j} = 9, j = 0, 1, 2, 3, 4, as in 3).

Assume d(= d̂) = 10. Then either V is the complete intersection of a quadric and a quintic in ℙ^{5}, so that M is a complete intersection of type (2, 5) in ℙ^{6}, K_{M} ≈ O_{M}, r = 0, d_{j} = 2d_{j−1}, j = 1, 2, 3, 4, and we are as in case 4), or d_{3} = 1, which contradicts (18), or d_{1} = d_{2} = d_{3} = d_{4} = 12. From (19) we have d_{j} = (r + 2)^{j}d = (r + 2)^{j}10, j = 1, 2, 3, 4, so we find again a contradiction.

Thus we conclude that d(= d̂) ≥ 11 and d_{1}^{2} ≥ dd_{2} ≥ 77 gives d_{1} ≥ 9. Then, recalling (18), we fall again in case 1). Q.E.D.

6 The case n = 3, I

Let M be a smooth connected 3-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} + D, be the first and the second reduction of (M, L) as in (1.4). Let d_{j}, j = 0, 1, 2, 3, d_{0} = d, be the pluridegrees of (M, L).

First, let us state some general relations. Let S be a smooth general member of |L|. Set K_{M} := K_{M} + L and h := h^{0}(K_{M} + L). Recall that h ≥ 2 from [7, (1.2)]. Since K_{M} is nef and big, the Riemann-Roch formula yields

h^{0}(K_{M} + tK_{M}) = −χ(tK_{M})
= \frac{t^{3}d_{3}}{6} + \frac{t^{2}(d_{3} − d_{2})}{4} + \frac{tK_{M} · ((K_{M} − L)^{2} + c_{2}(M))}{12} − χ(O_{M}).

Since K_{M} · (K_{M} − L)^{2} = d_{3} − 2d_{2} + d_{1},

K_{M} · c_{2}(M) = K_{M} · c_{2}(M) + L · c_{2}(M) = −24χ(O_{M}) + 12χ(O_{S}) − d_{2} − d_{1},

and χ(O_{M}) = χ(O_{S}) − h, we find

h^{0}(K_{M} + tK_{M}) = \frac{t^{3}d_{3}}{6} + \frac{t^{2}d_{3}}{4} − \frac{t^{2}d_{2}}{4} + \frac{td_{3}}{12} − \frac{td_{2}}{4} − (t + 1)χ(O_{S}) + h(2t + 1).

For t = 1 we have

2h^{0}(K_{M} + K_{M}) + 4χ(O_{S}) + d_{2} = d_{3} + 6h.  (20)

For t = 2 we have

2h^{0}(K_{M} + 2K_{M}) + 6χ(O_{S}) + 3d_{2} = 5d_{3} + 10h.  (21)

For t = 3 we have

h^{0}(K_{M} + 3K_{M}) + 4χ(O_{S}) + 3d_{2} = 7d_{3} + 7h.  (22)

We have the following four exceptions to K_{X} + 2K being nef from [2, (2.2)].

i) d_{3} = 1, (X, D) ≅ (ℙ^{3}, O_{ℙ^{3}}(5)).

ii) d_{3} = 2, (X, D) ≅ (Q, O_{Q}(1)), where Q is a quadric in ℙ^{4} and O_{Q}(1) is the restriction of O_{ℙ^{4}}(1) to Q, under the usual embedding Q ⊂ ℙ^{4}.

iii) d_{3} = 4, and X is the cone over (ℙ^{2}, O_{ℙ^{2}}(2)) with 2D ≈ 7ξ, where ξ is the tautological line bundle of the cone.

iv) X is a ℙ^{2}-bundle, p : X → C, over a smooth curve C with K_{X} + 3K ≈ p^{∗}H for some ample line bundle H on C.

Let’s now analyze the special case iv) of a variety M with second reduction a ℙ^{2}-bundle X over a curve C of genus q = h^{1}(O_{X}) and K_{|F} ≈ O_{ℙ^{2}}(1) for any fiber F ≅ ℙ^{2}.

Note that, since X has rational singularities, h^{i}(O_{M}) = h^{i}(O_{X}) = 0 for i = 2, 3. Hence

χ(O_{S}) − h = χ(O_{M}) = 1 − q,  (23)

i.e.,

h = p_{g}(S).  (24)

Note also that, since K_{X} + tK ≈ (φ_{∗}(K_{M} + tK_{M}))^{∗∗} for t ≥ 1, from Lemma (1.11) one has an inclusion

H^{0}(M, K_{M} + tK_{M}) ⊂ H^{0}(X, K_{X} + tK), t ≥ 1.  (25)

Then, since K_{X} + tK is not effective for t = 1, 2 in case iv), we conclude that h^{0}(K_{M} + K_{M}) = h^{0}(K_{M} + 2K_{M}) = 0. Thus (20) and (21) give respectively

4χ(O_{S}) + d_{2} = d_{3} + 6h

and

6χ(O_{S}) + 3d_{2} = 5d_{3} + 10h.

By subtracting we obtain

χ(O_{S}) + d_{2} = 2d_{3} + 2h.

The last two relations give

3χ(O_{S}) + d_{3} = 4h  (26)

as well as

10h = d_{2} + 7χ.  (27)

Lemma 6.1 Notation as at the beginning of this section. Assume that X is a ℙ^{2}-bundle, p : X → C, over a smooth curve C of genus q with K_{X} + 3K ≈ p^{∗}H for some ample line bundle H on C as in case iv) above. Let h := h^{0}(K_{M} + L) and let S be a smooth member of |L|. Then we have d_{3} ≥ 2 with equality implying q = 1, χ(O_{S}) = 2, h = 2, d_{2} = 6, d ≥ 24, d_{1} ≥ 13.

Proof. Assume d_{3} = 1. Then from (24) and (26) we get h = 4 − 3q. Thus

p_{g}(S) = 4 − 3q.  (28)

This implies that either q = 0 or q = 1. If q = 0 then h = 4. Since K is nef (indeed ample) on a projective bundle, then it is spanned. Hence |K| defines a morphism σ : X → ℙ^{3}. Since d_{3} = d′_{3} = (K′)^{3} = 1 and K is ample we conclude that σ is generically one-to-one and finite, and therefore σ is an isomorphism. This contradicts the assumption that X is a ℙ^{2}-bundle.

If q = 1 then from (28) we get p_{g}(S) = χ(O_{S}) = 1. Thus (26) gives the absurdity that h = 1 (see [7, (1.2)]). If d_{3} = 2, then we have from (24) and (26),

p_{g}(S) = 5 − 3q.  (29)

This implies that either q = 0 or q = 1. If q = 0, then h = p_{g}(S) = 5. As above, since K is nef on a projective bundle, then it is spanned. Hence |K| defines a morphism σ : X → ℙ^{4}. Since d_{3} = d′_{3} = (K′)^{3} = 2, the image of σ is a quadric in ℙ^{4} and σ is generically one-to-one. Since K is ample we conclude that σ is finite and hence X is isomorphic to a quadric in ℙ^{4}. Again this contradicts the fact that X is a ℙ^{2}-bundle.

If q = 1, then by (29), p_{g}(S) = χ(O_{S}) = 2. Thus (24) and (27) lead to h = 2 and d_{2} = 6. Since (M, L) is of log-general type we have d ≥ 8 by Lemma (1.10). Then d_{1}^{2} ≥ dd_{2} ≥ 48 gives d_{1} ≥ 7. If d_{1} = 7 then d = 8 by the above inequality, contradicting the parity condition. By combining the double point formula (1.7) with Hodge index inequalities (3) we obtain d ≥ 24, d_{1} ≥ 13. Q.E.D.

As a consequence of the results above, we can describe the case d_{3} = 1.

Theorem 6.2 Let M be a smooth connected 3-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} +D, be the first and the second reduction of (M, L). Let h := h^{0}(K_{M} +L) and let S be a smooth member of |L|. If d_{3} = 1 then either (X, D) ≅ (ℙ^{3}, O_{ℙ^{3}}(5)) or X is a Gorenstein Del Pezzo threefold with K_{X} ≈ −2K, h = 3, χ(O_{S}) = 4, d ≥ 9, d_{1} ≥ 6, d_{2} = 3.

Proof. If K_{X} + 2K is not nef we are in one of the special cases i)—iv). In case i) we are done, cases ii), iii) are not possible since we are assuming d_{3} = 1, and case iv) as well is not possible by Lemma (6.1).

Thus we can assume that K_{X} + 2K is nef, so that K_{X} + 3K = 4K − D is ample. Therefore (4K − D) · K · K > 0, i.e., 4d′_{3} − d′_{2} > 0 and hence d′_{2} = d_{2} ≤ 3.

If d_{2} = 1 we have, by the Hodge index relations, d_{1} = d = 1 and hence (M, L) ≅ (ℙ^{3}, O_{ℙ^{3}}(1)), contradicting the log-general type assumption. Since d_{2} ≠ 2 by parity, we conclude that d_{2} = 3. Also, the log-general type assumption gives d ≥ 8 (see Lemma (1.10)) and therefore d_{1}^{2} ≥ dd_{2} ≥ 24 yields d_{1} ≥ 5. But d_{1} = 5 implies d = 8, which contradicts again parity. Then d_{1} ≥ 6.

Assume K_{X} + K ≈ 2K − D effective. Then (2K − D) · K · K ≥ 0 gives 2d′_{3} > d′_{2}, or 2d_{3} > d_{2}. This contradicts d_{3} = 1, d_{2} = 3. Thus we can assume h^{0}(K_{X} + K) = 0 and therefore (20) and (25) yield 4χ(O_{S}) + d_{2} = d_{3} + 6h, or 2χ(O_{S}) + 1 = 3h. Since h = p_{g}(S) by (24), this gives 3 = h + 2q. Since h ≥ 2 we conclude that

q = 0, h = 3, and χ(O_{S}) = 4.

Then using the double point formula (1.7) we have

d(d − 10) ≥ 2d_{2} + 5d_{1} − 12χ(O_{S}) ≥ −12,

and hence d ≥ 9. From d_{1}^{2} ≥ dd_{2} ≥ 27 we obtain d_{1} ≥ 6.

Assume now h^{0}(K_{X} + 2K) ≥ 1. Note that K_{X} + 2K is a rank 1 reflexive sheaf (since K_{X} is a reflexive sheaf and K is a line bundle). Then a section s ∈ H^{0}(X, K_{X} + 2K) vanishes on a Cartier divisor. But the ampleness of K and the equality

(K_{X} + 2K) · K · K = (3K − D) · K · K = 3d_{3} − d_{2} = 0

imply that s vanishes on a subvariety Y ⊂ X of codimension ≥ 2. This gives a contradiction, unless K_{X} + 2K ≈ O_{X} (and h^{0}(K_{X} + 2K) = 1). Thus X is a Gorenstein Del Pezzo threefold with K_{X} ≈ −2K.

To conclude the proof it remains to show that h^{0}(K_{X} + 2K) ≥ 1. In view of (25) it suffices to show that h^{0}(K_{M} + 2K_{M}) ≥ 1. If χ(−2K_{M}) ≠ 0 we have, since K_{M} is nef and big,

χ(−2K_{M}) = −χ(K_{M} + 2K_{M}) = −h^{0}(K_{M} + 2K_{M}) ≠ 0,

so we are done. Thus we can assume χ(−2K_{M}) = 0, i.e., h^{0}(K_{M} + 2K_{M}) = 0 and hence also

h^{0}(K_{M} + K_{M}) = χ(K_{M} + K_{M}) = −χ(−K_{M}) = 0,

i.e., χ(−K_{M}) = 0. Moreover χ(K_{M}) = χ(K_{M} + L) = h^{0}(K_{M} + L) = h = 3. Therefore from the exact sequence

0 → −L → O_{M} → O_{S} → 0

we obtain

χ(O_{M}) = χ(O_{S}) + χ(−L) = 4 − χ(K_{M} + L) = 1.

Now a standard polynomial argument, by using χ(K_{M}) = 3, χ(O_{M}) = 1, χ(−K_{M}) = 0 and χ(−2K_{M}) = 0 and the bigness of K_{M} leads to a numerical contradiction. Q.E.D.

Remark 6.3 Notation as in (6.2). Note that the case of X being a Gorenstein Del Pezzo threefold with K_{X} ≈ −2K, of invariants h = 3, χ(O_{S}) = 4, d ≥ 9, d_{1} ≥ 6, d_{2} = 3 is possible. The example in [12] has L = 3H very ample with K_{M} ≈ −2H and degree L^{3} = 27. Thus all reductions are isomorphic with K = H having h^{0}(K) = 3, d_{3} = K^{3} = 1.

7 The case n = 3, II

Let M be a smooth connected 3-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} + D, be the first and the second reduction of (M, L) as in (1.4). Let d_{j}, j = 0, 1, 2, 3, d_{0} = d, be the pluridegrees of (M, L). Let h := h^{0}(K_{M} + L) and let S be a general member of |L|. Set also K_{M} := K_{M} + L.

In this section we study the case d_{3} = 2. We will work under the assumption that K_{X} + 2K is nef. If this is not the case, in view of the exceptions listed in §6, we know that either (X, D) ≅ (Q, O_{Q}(1)), Q smooth quadric in ℙ^{4}, or X is as in Lemma (6.1) (see also Remark (6.3)).

Since K_{X} + 2K is nef we have (K_{X} + 2K) · K · K = (3K − D) · K · K ≥ 0 and hence

3d_{3} ≥ d_{2}.

Therefore, by parity, d_{2} = 2, 4, 6. If d_{2} = 2, then d_{2}^{2} ≥ d_{1}d_{3} = 2d_{1} gives d_{1} = 2 and hence d = 2. Then we would be in the special case of the quadric in ℙ^{4}, contradicting the nefness assumption.

Thus either d_{2} = 6 or d_{2} = 4. We have the following result, whose proof is given in the two paragraph below, where the cases d_{2} = 6 and d_{2} = 4 are treated separately. Recall that d ≥ 8 by Lemma (1.10).

Theorem 7.1 Let M be a smooth connected 3-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} +D, be the first and the second reduction of (M, L). Let d_{j}, j = 0, 1, 2, 3, d_{0} = d, be the pluridegrees of (M, L). Let h := h^{0}(K_{M} + L) and let S be a general member of |L|. Assume that K_{X} + 2K is nef and big. Further assume that d_{3} = 2. Then either

1. χ(O_{S}) = 5, h = 4, d_{2} = 6, d_{1} ≥ 9, d ≥ 11, or

2. χ(O_{S}) = 4, h = 3, d_{2} = 4 and (d, d_{1}) = (11, 7), (12, 8), (14, 8), (16, 8).

7.2 The case d_{2} = 6. The nefness of K_{X} +2K and the ampleness of K imply nefness and bigness of 2(K_{X} + 2K) − K_{X}. Thus, by the Kawamata-Shokurov basepoint free theorem, we conclude that N(2(K_{X} + 2K)) has a section for N ≫ 0, and hence N(K_{X} + 2K) has a section for N ≫ 0, i.e., K_{X} + 2K is ℚ-effective. From

(K_{X} + 2K) · K^{2} = (3K − D) · K^{2} = 3d′_{3} − d′_{2} = 3d_{3} − d_{2} = 0

it thus follows that N(K_{X} + 2K) ≈ O_{X} and therefore −K_{X} is ample. This implies

q := q(S)(= q(X)) = 0.  (30)

From d_{1}^{2} ≥ dd_{2} ≥ 48 we get d_{1} ≥ 7. If d_{1} = 7, then by parity d ≥ 9, so that d_{1}^{2} ≥ dd_{2} yields to a contradiction. Therefore

d_{1} ≥ 8.  (31)

From

(K_{X} + K) · K^{2} = (2K − D) · K^{2} = 2d′_{3} − d′_{2} = 2d_{3} − d_{2} < 0

we conclude that h^{0}(K_{X} + K) = 0. Recalling (25) we have h^{0}(K_{M} + K_{M}) = 0. Then relation (20) gives 4χ(O_{S}) + d_{2} = d_{3} + 6h, or

2χ(O_{S}) + 2 = 3h.  (32)

Noether inequality gives d_{2} ≥ χ(O_{S}) − 6, so that χ(O_{S}) ≤ 6 and hence (32) yields either χ(O_{S}) = h = 2, or χ(O_{S}) = 5, h = 4.

Note that the case χ(O_{S}) = h = 2 is not possible. Indeed from χ(O_{S}) = 2 and q = 0 we have p_{g}(S) = 1. Then we see that h = h^{0}(K_{M} + L) = 2 is not possible by considering the cohomology sequence associated to the exact sequence

0 → K_{M} → K_{M} + L → K_{S} → 0.

Therefore χ(O_{S}) = 5, h = 4. Since q = 0 one has p_{g}(S) = 4. From the double point formula (1.7), 2) we have d(d − 10) + 48 ≥ 5d_{1}. Recalling (31) we obtain d ≥ 10.

Assume that |L| embeds M in ℙ^{N} with N ≥ 6. Then d ≥ 11. Indeed, if d = 10, Castelnuovo’s bound gives g(L) ≤ Castel(10, 4) = 9, and hence we find the contradiction

18 ≤ d + d_{1} = d̂+ d̂_{1} = 2g(L) − 2 ≤ 16.

Assume now that |L| embeds M in ℙ^{5}. Since d ≥ 10, [3, (4.4.1)] applies to say that M ≅ M. Thus the double point formula (1.7), 2) for d = 10 gives 12χ(O_{S}) = 2d_{2} + 5d_{1}, whence the numerical contradiction 60 = 12 + 5d_{1}. Therefore we conclude that d ≥ 11.

From d_{1}^{2} ≥ dd_{2} ≥ 66 we also get d_{1} ≥ 9. Summarizing, we have the following numerical bounds:

d_{3} = 2, d_{2} = 6, d_{1} ≥ 9, d ≥ 11.

Example 7.3 Let us construct an explicit example satisfying the numerical bounds above. Let p : M → ℙ^{3} be a finite double cover of ℙ^{3} branched along a smooth quartic surface B. Then K_{M} ≈ p^{∗}O_{ℙ^{3}}(−2). Let L := p^{∗}O_{ℙ^{3}}(3), so that K_{M} + L ≈ p^{∗}O_{ℙ^{3}}(1). Note that L and K_{M} + 2L ≈ p^{∗}O_{ℙ^{3}}(4) are very ample (see e.g., [12]). Hence in particular (M, L) is isomorphic to its own first reduction (M, L). The invariants are

d = 2O_{ℙ^{3}}(3)^{3} = 54, d_{1} = 2O_{ℙ^{3}}(1)·O_{ℙ^{3}}(3)^{2} = 18, d_{2} = 2O_{ℙ^{3}}(1)^{2}·O_{ℙ^{3}}(3) = 6, d_{3} = 2O_{ℙ^{3}}(1)^{3} = 2.

Moreover h = h^{0}(K_{M} + L) = h^{0}(p^{∗}O_{ℙ^{3}}(1)) = 4. To see this, consider the exact sequence

0 → O_{ℙ^{3}}(−3) → O_{ℙ^{3}}(1) → O_{ℙ^{3}}(1)_{|B} → 0.

Then h^{0}(O_{ℙ^{3}}(1)_{|B}) = h^{0}(O_{ℙ^{3}}(1)) = 4. On M we have the exact sequence (identifying B with p^{−1}(B))

0 → p^{∗}O_{ℙ^{3}}(−2) → O_{M} → O_{B} → 0.

Tensoring with K_{M} + L and computing the cohomology we obtain

h = h^{0}(K_{M} + L) = h^{0}((K_{M} + L)_{|B}) = h^{0}(O_{ℙ^{3}}(1)_{|B}) = 4.

Note also that K_{M} + L is ample and spanned and gives a two-to-one map (see [12]).

7.4 The case d_{2} = 4. Let us first prove the following

Claim 7.5 h^{0}(K_{X} + K) = 0.

Proof. Assume otherwise. Then from

(K_{X} + K) · K^{2} = (2K − D) · K^{2} = 2d′_{3} − d′_{2} = 2d_{3} − d_{2} = 0

we see that 2(K_{X} + K) ≈ O_{X}. Hence we have a section of 2(K_{X} + K) and hence a section of K_{X} + K which doesn’t vanish at regular points of X. This implies that K_{X} + K ≈ O_{X}.

Thus X is Gorenstein, q(X) = 0, h^{0}(K_{X} + K) = 1, h^{2}(O_{X}) = h^{3}(O_{X}) = 0 by Kodaira vanishing and χ(O_{X}) = 1. Since X is Gorenstein we have

K_{M} + K_{M} ≈ φ^{∗}(K_{X} + K) + J,

where J is an integral divisor. Therefore J is a line bundle since M is smooth. Recalling that h^{0}(K_{M} + K_{M}) ≤ h^{0}(K_{X} + K) = h^{0}(φ_{∗}(K_{M} + K_{M})^{∗∗}) by Lemma (1.11), we conclude from h^{0}(K_{X} + K) = 1 that h^{0}(K_{M} + K_{M}) = 1. Thus relation (20) yields

2 = 2h^{0}(K_{M} + K_{M}) = d_{3} + 6h − d_{2} − 4χ(O_{S}).

Since χ(O_{S}) = h + 1 from (23), and d_{3} = 2, d_{2} = 4, we find

3h = 2(χ(O_{S}) + 1).

Noether formula gives 4 = d_{2} ≥ 2χ(O_{S}) − 6, i.e., χ(O_{S}) ≤ 5. Recalling that h ≥ 2 we infer that either h = 4, χ(O_{S}) = 5, or h = χ(O_{S}) = 2. Exactly the same argument as in paragraph (7.2) rules out the case h = χ(O_{S}) = 2. Then d_{2} = 2χ(O_{S}) − 6, so that S is a Horikawa surface. Then from [9, Lemma (1.1)] we know that K_{S} is spanned since d_{2} = K_{S}^{2} is even. Thus from the exact sequence

0 → K_{M} → K_{M} + L → K_{S} → 0,

by using h^{2}(O_{M}) = h^{1}(K_{M}) = 0, we infer that K_{M} = K_{M} + L is spanned. Let ψ : M → ℙ^{h−1} = ℙ^{3} be the surjective map associated to |K_{M}|. Since (K_{M} + L)^{3} = d_{3} = 2, the morphism ψ is generically one-to-one. Since K_{M} ≈ φ^{∗}K, ψ factors through the second reduction map φ, ψ = f ∘ φ, f : X → ℙ^{3}, and f is finite since K is ample. We have K ≈ f^{∗}O_{ℙ^{3}}(1) and h = h^{0}(K_{M}) = h^{0}(K) = 4. Let B be the branch locus of ψ. Then O_{ℙ^{3}}(B) ≈ O_{ℙ^{3}}(2b), b := deg B. One has L ≈ ψ^{∗}O_{ℙ^{3}}(a) for some integer a, K_{M} ≈ ψ^{∗}O_{ℙ^{3}}(b − 4) and hence

ψ^{∗}O_{ℙ^{3}}(a + b − 4) ≈ K_{M} ≈ ψ^{∗}O_{ℙ^{3}}(1).

Therefore a + b = 5. If b = 1, then a = 4, which gives the contradiction d_{2} = K_{M}^{2} · L = 2O_{ℙ^{3}}(1)^{2} · O_{ℙ^{3}}(3) = 6.

To conclude the proof of the claim it remains to show that the case b ≥ 3 does not occur. To see this consider the exact sequences (identifying B with p^{−1}(B))

0 → O_{ℙ^{3}}(5 − 2b) → O_{ℙ^{3}}(5 − b) → O_{ℙ^{3}}(5 − 2b)_{|B} → 0

and

0 → L − B ≈ ψ^{∗}O_{ℙ^{3}}(5 − 2b) → L → L_{B} → 0.

If b ≥ 3 then 5 − 2b < 0, so that

h^{0}(O_{ℙ^{3}}(5 − b)) = h^{0}(O_{ℙ^{3}}(5 − 2b)_{|B}) = h^{0}(L_{B}) ≥ h^{0}(O_{ℙ^{3}}(5 − b)).

Thus h^{0}(L) = h^{0}(O_{ℙ^{3}}(5−b)), i.e., all sections of L pullback from ℙ^{3}. Therefore |L| doesn’t give a birational map. This contradicts the fact that L is very ample outside of a finite set of points. □

Recalling also Lemma (1.11), from now on we can thus assume that h^{0}(K_{X} + K) = h^{0}(K_{M} + K_{M}) = 0. Then since d_{3} = 2, d_{2} = 4, from relation (20) we find

2χ(O_{S}) + 1 = 3h.  (33)

Noether formula d_{2} ≥ 2χ(O_{S}) − 6 gives again χ(O_{S}) ≤ 5. Thus, recalling that h ≥ 2, (33) gives

χ(O_{S}) = 4, h = 3.

From d_{3}d_{1} = 2d_{1} ≤ d_{2}^{2} = 16 we have d_{1} ≤ 8. Then dd_{2} = 4d ≤ d_{1}^{2} ≤ 64 yields d ≤ 16. From (1.7), 2) one has d(d−10)+40 ≥ 5d_{1}. Since d_{1} ≥ 6, this implies d ≥ 9. If d = 9, then d_{1} ≥ 7 by parity and hence the inequality above gives again a contradiction. Therefore d ≥ 10. Then d_{1}^{2} ≥ dd_{2} ≥ 40 yields d_{1} ≥ 7. Thus we have

d_{3} = 2, d_{2} = 4, 8 ≥ d_{1} ≥ 7, 16 ≥ d ≥ 10.  (34)

Assume d = 10. Then d̂ ≤ 10 and d_{1} = 8 by parity. Suppose that |L| embeds M in ℙ^{N} with N ≥ 6. Then g(L) ≤ Castel(10, 4) = 9. This gives the numerical contradiction

16 ≥ 2g(L) − 2 = d̂+ d̂_{1} = d + d_{1} ≥ 18.

Thus if d = 10 we see that |L| embeds M in ℙ^{5}. By combining Lemma (1.10) and Proposition (1.5) we conclude that (M, L) ≅ (M, L), d = d̂ = 10, d_{1} = d̂_{1} = 8. Since there are no 3-folds in ℙ^{5} of degree 10 with d_{1} = 8 (see [5, Chapter 6]), we conclude that d ≥ 11.

Recalling (34) and by using parity, we have the following possibilities for d, d_{1}. Precisely, (d, d_{1}) = (11, 7), (12, 8), (14, 8), (16, 8). Note that if d = 13, then d_{1} = 7 by parity and d_{1}^{2} ≥ dd_{2} ≥ 52 gives a contradiction. Similarly, if d = 15, then d_{1} = 7, and we contradict again d_{1}^{2} ≥ dd_{2}.

8 A lower bound for the number of sections of the second adjoint bundle

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 reduction of (M, L) as in (1.4). Let h := h^{0}(K_{M} + (n − 2)L). In [7, (1.2), (3.3)] we showed that h ≥ 2 under the log-general type assumption and moreover, if κ(M) ≥ 0, that h ≥ 5 with equality only if n = 3 and (M, L) a degree 5 hypersurface in ℙ^{4}. In this section we prove that for n ≥ 5, under the further assumption that K_{X} +(n−3)K is nef and big (see §2) and by using the pluridegrees bounds stated in the previous sections, one has the general lower bound

h ≥ 5.  (35)

Let V be the smooth 5-fold section obtained as transversal intersection of n − 5 general members of |L| and let L_{V} be the restriction of L to V. The Kodaira vanishing theorem yields

h^{0}(K_{M} + (n − 2)L) ≥ h^{0}(K_{V} + 3L_{V}).

Thus, since we have better bounds as soon as n increases, we can assume n = 5 and (35) will be a consequence of the following result.

Proposition 8.1 Let M be a smooth connected 5-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} + 3D, be the first and the second reduction of (M, L). Let d_{j}’s, j = 0, 1, 2, 3, 4, 5, d_{0} = d, be the pluridegrees of (M, L). Assume that K_{X} + 2K is nef and big. Then h := h^{0}(K_{M} + 3L) ≥ 5.

Proof. Let S be the smooth surface obtained as transversal intersection of n − 2 = 3 general members of |L|. Then S is a minimal surface of general type. Since d_{2} ≥ 24 by (4.1), from Miyaoka’s inequality (2) we infer that

χ(O_{S}) ≥ 3.

Assume χ(O_{S}) = 3. Then since d_{1} ≥ 19, d_{2} ≥ 24 from (4.1), the double point formula (1.7), 2) gives d(d − 10) ≥ 107, and hence d ≥ 17. Therefore (15) gives

d ≥ 17, d_{1} ≥ 21, d_{2} ≥ 25, d_{3} ≥ 30, d_{4} ≥ 36, d_{5} ≥ 43.

If d_{2} = 25, then d_{3} ≥ 31 by parity, so that we find the contradiction d_{2}^{2} = 625 ≥ d_{1}d_{3} ≥ 651. Therefore d_{2} ≥ 26 and hence d_{1}^{2} ≥ dd_{2} ≥ 442 yields d_{1} ≥ 22. Thus (1.7), 2) gives d(d − 10) ≥ 126, or d ≥ 18. If d_{2} = 26, then d_{3} ≥ 31 by (15) and hence d_{3} ≥ 32 by parity, so that d_{2}^{2} ≥ d_{1}d_{3} ≥ 704 gives d_{2} ≥ 27. Then d_{1}^{2} ≥ dd_{2} = 486 yields d_{1} ≥ 23, as well as d_{3} ≥ 32, d_{4} ≥ 37, d_{5} ≥ 44 by (15). Summarizing, we have

d ≥ 18, d_{1} ≥ 23, d_{2} ≥ 27, d_{3} ≥ 32, d_{4} ≥ 37, d_{5} ≥ 44.

From d_{2}^{2} ≥ d_{1}d_{3} ≥ 736 we get d_{2} ≥ 28. Thus by (15) we obtain d_{3} ≥ 34, d_{4} ≥ 41, d_{5} ≥ 49 and therefore d_{3}^{2} ≥ d_{2}d_{4} ≥ 1184 yields d_{3} ≥ 35. Again, (15) gives d_{4} ≥ 42, d_{5} ≥ 50. From d_{2}^{2} ≥ d_{1}d_{3} ≥ 805 we find d_{2} ≥ 29. If d_{1} = 23 then d ≥ 19 by parity, so that d_{2} ≤ d_{1}^{2}/d leads to a contradiction. Thus we have

d ≥ 18, d_{1} ≥ 24, d_{2} ≥ 29, d_{3} ≥ 35, d_{4} ≥ 42, d_{5} ≥ 50.  (36)

Note that Miyaoka inequality χ(O_{S}) > d_{2}/9 yields χ(O_{S}) ≥ 4. Let V be the smooth 3-fold section of M obtained as transversal intersection of two general members of |L| and let L_{V} be the restriction of L to V. Let h′ := h^{0}(K_{V} + L_{V}). Then by Kodaira vanishing one has h ≥ h′. Thus it suffices to show that h′ ≥ 5.

By using (36), and recalling that d_{j} equals the pluridegrees d_{j}(V) of (V, L_{V}), j = 0, 1, 2, 3, Tsuji’s inequality (1.9) reads h′ ≥ 4. Clearly we can assume χ(O_{S}) = 4 since otherwise (1.9) gives

h′ ≥ \frac{5}{2} + 1 + \frac{35}{64}, i.e., h′ ≥ 5,

and we would be done.

Note also that we can assume h^{0}(K_{V}) = 0, since otherwise a section of Γ(K_{V}) gives an embedding Γ(L_{V}) ↪ Γ(K_{V} + L_{V}) and hence h′ := h^{0}(K_{V} + L_{V}) ≥ 5. Then double point formula (1.8) yields

44h′ + 58χ(O_{S}) + 4 ≥ 12d_{2} + 17d_{1} + d_{3} + (20 − d)d,

or, with h′ = χ(O_{S}) = 4, (d − 20)d ≥ 269. This implies d ≥ 30 and therefore d_{1} ≥ 36, d_{2} ≥ 43, d_{3} ≥ 51 by (15). Thus Tsuji’s inequality (1.9) gives

h ≥ 2 + \frac{36}{24} + \frac{51}{64}, or h ≥ 5.

Q.E.D.

References

[1] E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Volume I, Grundlehren Math, Wiss. 267, Springer-Verlag, (1995).

[2] M.C. Beltrametti, M.L. Fania, and A.J. Sommese, “On the adjunction theoretic classification of projective varieties,” Math. Ann. 290 (1991), 31–62.

[3] M.C. Beltrametti, M. Schneider, and A.J. Sommese, “Threefolds of degree 11 in ℙ^{5},” in Complex Projective Geometry, ed. by G. Ellingsrud, C. Peskine, G. Sacchiero, and S.A. Stromme, London Math. Soc. Lecture Note Ser. 179 (1992), 59–80.

[4] M.C. Beltrametti and A.J. Sommese, “Special results in adjunction theory in dimension four and five,” Ark. Mat. 31 (1993), 197–208.

[5] M.C. Beltrametti, M. Schneider, and A.J. Sommese, Some special properties of the adjunction theory for 3-folds in ℙ^{5}, Memoirs of A.M.S., v. 116, n. 554 (1995).

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

[7] M.C. Beltrametti and A.J. Sommese, “On the dimension of the adjoint linear system for threefolds,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. (4), XXII (1995), 1–24.

[8] J. Harris, Curves in projective space, with the collaboration of D. Eisenbud, Université de Montreal, Montreal (Québec), Canada, (1982).

[9] E. Horikawa, “Algebraic surfaces of general type with small c_{1}^{2}, I,” Ann. Math. 104 (1976), 357–387.

[10] Y. Kawamata, K. Matsuda, and K. Matsuki, “Introduction to the Minimal Model Problem,” in Algebraic Geometry, Sendai 1985, Adv. Stud. Pure Math. 10 (1987), 283–360.

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

[12] A. Lanteri, M. Palleschi, and A.J. Sommese, “On triple covers of ℙ^{n} as very ample divisors,” in Classification of Algebraic Varieties, Proceedings L’Aquila, 1992, ed. by C. Ciliberto, E.L. Livorni, and A.J. Sommese, Contemp. Math. 162 (1994), 277–292.

[13] E.L. Livorni and A.J. Sommese, “Threefolds of non-negative Kodaira dimension with sectional genus less than or equal to 15,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. (4) 13 (1986), 537–558.

[14] H. Tsuji, “Stability of tangent bundles of minimal algebraic varieties,” Topology 27 (1988), 429–442.

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

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