Birational geometry of Fano direct products

Aleksandr V. Pukhlikov

Max-Planck-Institut für Mathematik Vivatsgasse 7 53111 Bonn GERMANY e-mail: pukh@mpim-bonn.mpg.de

Steklov Institute of Mathematics Gubkina 8 117966 Moscow RUSSIA e-mail: pukh@mi.ras.ru

Division of Pure Mathematics Department of Mathematical Sciences M&O Building, Peach Street The University of Liverpool Liverpool L69 7ZL ENGLAND e-mail: pukh@liv.ac.uk

April 23, 2004

Abstract

Combining the connectedness principle of Shokurov and Kollár with the technique of hypertangent divisors, we develop a new technique of studying birational maps of natural Fano fiber spaces. We prove that the only structures of a rationally connected fibration on direct products of typical Fano varieties are projections onto the factors. In particular, the groups of biregular and birational self-maps of Fano direct products coincide.

CONTENTS

Introduction

1. Effective divisors on rationally connected fibrations 1.1. Effective divisors and essentially non-effective divisors 1.2. Divisorially canonical varieties 1.3. Direct products

2. Fano direct products 2.1. The relative and absolute cases 2.2. Geometry of Fano direct products 2.3. Proof of Theorem 5

3. Divisorially canonical Fano varieties 3.1. 2n-inequality 3.2. The regularity conditions 3.3. Fano hypersurfaces are divisorially canonical 3.4. Fano double spaces 3.5. Further results

References

Introduction

The aim of this paper is to prove that Fano direct products

F_{1} × … × F_{K},

where F_{i} belong to a large class of primitive Fano varieties (which includes general Fano hypersurfaces V_{M} ⊂ ℙ^{M}, M ≥ 6, and Fano double spaces of index 1 and dimension ≥ 3) admit no other structures of a rationally connected fibration than those given by the projections onto direct factors F_{i_{1}} × … × F_{i_{k}}.

The proof is given in full detail. However, I do not explain the details of one of the basic facts underlying the proof, that is, the connectedness principle of Shokurov and Kollár and its immediate follow up, the inversion of adjunction ([K]), which are in their turn based on the Kawamata-Viehweg vanishing theorem [Kw,V]. Our method is based on the connectedness principle combined with the (modified) technique of hypertangent divisors (see, for instance, [P3]).

For the Fano hypersurfaces of low dimension, that is, the famous three-dimensional quartic [IM] and four-dimensional quintic [P1], the proof is not completed yet. There is, however, no doubt that general varieties of these two types are divisorially canonical (see Definition 4 below).

The main result of this paper remains true in slightly weaker assumptions: one should assume divisorial log-canonicity of the factors together with the fact that moving divisors are canonical. No modification of the argument is required.

The result was obtained in October 2003 during my stay at Max-Planck-Institut für Mathematik in Bonn. However, it was not until Easter holidays (April 2004) that I could start writing up. I am very grateful to the Institute for the excellent conditions of work.

1 Effective divisors on rationally connected fibrations

1.1 Effective divisors and essentially non-effective divisors

Let X be an (irreducible) algebraic variety, non-singular in codimension one, that is, codim Sing ≥ 2.

Definition 1. An algebraic family of irreducible curves on a variety X

C = (C_{t}, t ∈ T),

parameterized by an irreducible quasi-projective variety T, is said to be free, if for any closed subset Y ⊂ X of codimension 2 the set

T_{Y} = {t ∈ T | C_{t} ∩ Y ≠ ∅}

is a proper closed subset in T.

Definition 2. A divisor D on the variety X is said to be essentially non-effective, if on X there is a free algebraic family of irreducible curves C = (C_{t}, t ∈ T), the general curve of which satisfies the inequality

(D · C_{t}) < 0.

Remark 1. Since the variety X by assumption is non-singular in codimension 1, whereas the family of curves C is free, for a general t ∈ T the curve C_{t} lies entirely in the smooth part of the variety X, so that the intersection index (D · C_{t}) is well defined.

Definition 3. We say that a smooth rationally connected variety X (respectively, a rationally connected fiber space π: X → S with X, S smooth) satisfies the necessary condition of effectivity of divisors or briefly the condition (E) (respectively, the relative necessary condition of effectivity of divisors or briefly the condition (R)), if

• for any irreducible divisor D,

lD ∈ | − nK_{X} + Z|,

where Z is an effective divisor on X (respectively, Z = π^{∗}Z_{S} is the pull back of an effective divisor Z_{S} on the base S) and n ∈ ℤ_{+}, l ≥ 1 a positive integer (in the relative case l = 1),

• for any non-trivial sequence of blow ups

φ: X̃ → X

the divisor

Z^{♯} = lD̃ + nKX̃

is not essentially non-effective, where D̃ ⊂ X̃ is the strict transform of the irreducible divisor D on the variety X̃.

Remark 2. Let E be the set of exceptional divisors of the birational morphism φ. Let a_{E} ∈ ℤ_{+} be the discrepancy of the divisor E ∈ E. There is a set of non-negative integers d_{E} ∈ ℤ_{+}, E ∈ E, such that

D̃ ∼ φ^{∗}D − ∑_{E∈E} d_{E}E.

Therefore

Z^{♯} = φ^{∗}Z + ∑_{E∈E}(na_{E} − ld_{E})E.

If the condition (E) or (R) holds, then for any free family of irreducible curves C = (C_{t}, t ∈ T) on X̃ the inequality

(Z^{♯} · C_{t}) ≥ 0

holds. On the other hand, if the condition (E)(or (R)) does not hold, then for some family C we have (Z^{♯} · C_{t}) < 0. Since the divisor Z is effective, in this case there is an exceptional divisor E ∈ E satisfying the Noether-Fano inequality

ld_{E} > n · a_{E}.

1.2 Divisorially canonical varieties

Consider a smooth primitive Fano variety V of dimension dim V ≥ 3, Pic V = ℤK_{V}.

Definition 4. We say that the variety V is divisorially canonical or, briefly speaking, satisfies the condition (C), if for any effective divisor

D ∈ | − nK_{V} |,

n ∈ ℤ_{+}, the pair (V, \frac{1}{n}D) has canonical singularities.

In other words, V satisfies the condition (C), if for any effective divisor D ∈ | − nK_{V} |, any non-trivial sequence of blow ups

φ: Ṽ → V

and any exceptional divisor E ⊂ Ṽ the inequality

ν_{E}(D) ≤ na(E),

holds, where ν_{E}(D) = ord_{E} φ^{∗}D is the multiplicity of the divisor D with respect to the discrete valuation ν_{E}(·), a(E) is the discrepancy of the divisor E.

Theorem 1. If the primitive Fano variety V satisfies the condition (C), then it satisfies the condition (E), too.

Proof. Assume the converse: the condition (E) does not hold. Let H = −K_{V} be the positive generator of the Picard group. In accordance with Definition 3, there is an effective divisor D,

lD ∈ |(n + m)H|,

where n, m ∈ ℤ_{+}, l ≥ 1 and a non-trivial sequence of blow ups φ: Ṽ → V such that the divisor

Z^{♯} = lD̃ + nKṼ

is essentially non-effective on Ṽ. Let C = (C_{t}, t ∈ T) be a free family of irreducible curves on Ṽ, on which Z^{♯} is negative. Let E be the set of exceptional divisors of the birational morphism φ. Then

lD̃ = (n + m)H − ∑_{E∈E} d_{E}E, KṼ = −H + ∑_{E∈E} a_{E}E,

where a_{E} is the discrepancy and we write H instead of φ^{∗}H. If the number γ = −(KṼ · C_{t}) ≤ 0, then the inequality (Z^{♯} · C_{t}) < 0 implies that

(D̃ · C_{t}) < 0,

which is impossible, since D̃ is effective and C is a dense family of irreducible curves. Therefore γ > 0. Besides, n ≥ 1.

Now compare the inequalities

(Z^{♯} · C_{t}) = m(H · C_{t}) − ∑_{E∈E}(ld_{E} − na_{E})(E · C_{t}) < 0  (1)

and

−(KṼ · C_{t}) = (H · C_{t}) − ∑_{E∈E} a_{E}(E · C_{t}) > 0  (2)

Obviously, (H · C_{t}) > 0 and (E · C_{t}) ≥ 0 for all E ∈ E, whereas for some E ∈ E we have (E · C_{t}) > 0, otherwise the inequality (1) would not be possible. Set

E^{+} = {E ∈ E|(E · C_{t}) > 0}.

Lemma 1. There exists an exceptional divisor E ∈ E^{+} such that

ld_{E} > (n + m)a_{E}  (3)

Proof. Rewrite the inequality (1) in the following way

(Z^{♯} · C_{t}) = m[(H · C_{t}) − ∑_{E∈E^{+}} a_{E}(E · C_{t})] + ∑_{E∈E^{+}}(−ld_{E} + (n + m)a_{E})(E · C_{t}) < 0

Taking into consideration the inequality (2), we obtain the claim of the lemma.

The inequality (3) is the Noether-Fano inequality for the divisor lD. Thus the pair (V, \frac{l}{n+m}D) is not canonical and the condition (C) is not satisfied. A contradiction.

Q.E.D. for the theorem.

1.3 Direct products

Theorem 2. Let S be a smooth rationally connected variety, F a smooth primitive Fano variety. Assume that:

(1) the variety S satisfies the condition (E),

(2) the variety F satisfies the condition (C).

Then the Fano fiber space

π: F × S → S

(π is the projection onto the second direct factor) satisfies the condition (R).

Proof. Assume the converse: the condition (R) is not satisfied by the Fano fiber space π: V = F × S → S. Therefore there exists a prime divisor D,

D ∈ | − nK_{V} + Z|

(recall that in the relative case l = 1), where Z = π^{∗}Z_{S} = F × Z_{S} is an effective divisor on the base S and a non-trivial sequence of blow ups

φ: Ṽ → V,

such that the class Z^{♯} = D̃ +nKṼ is negative on the free family of irreducible curves C = (C_{t}, t ∈ T).

First of all, let us see to what extent this situation depends on the choice of the sequence of blow ups φ. Let

ψ: V^{+} → V

be another model of the variety V, that is, ψ is a birational morphism. Assume that the birational map

γ = ψ^{−1} ∘ φ: Ṽ − − → V^{+}

is an isomorphism in codimension one (in other words, γ is biregular at the generic point of each exceptional divisor E ⊂ Ṽ of the morphism φ). Then there exist open sets

U ⊂ Ṽ, U^{+} ⊂ V^{+}

such that γ(U) = U^{+}, γ: U → U^{+} is an isomorphism and

codimṼ(Ṽ \ U) ≥ 2.

Since the family of curves C is free, for a general t ∈ T we get

C_{t} ⊂ U.

Set T^{+} = {t ∈ T | C_{t} ⊂ U}. Then the family of irreducible curves

C^{+} = (C_{t}^{+} = γ(C_{t}), t ∈ T^{+})

is a free family of irreducible curves on V^{+}. Set D^{+} to be the strict transform of the divisor D on V^{+}, K^{+} = K_{V^{+}}. Obviously,

((D^{+} + nK^{+}) · C_{t}^{+}) = ((D̃ + nKṼ) · C_{t})

for t ∈ T^{+}, so that the class Z^{+} = D^{+} + nK^{+} is negative on the curves C_{t}^{+} and thus essentially non-effective.

From what was said it follows that the sequence of blow ups φ can be replaced by another sequence of blow ups ψ: V^{+} → V provided that the map ψ^{−1} ∘φ is biregular at the generic point of each exceptional divisor E ⊂ Ṽ of the morphism φ. Note also that for the condition of being essential non-effective only those exceptional divisors E ⊂ Ṽ matter, for which (E · C_{t}) ≠ 0.

By what was said, we may assume from the start that the sequence of blow ups φ has the following structure:

φ: Ṽ →^{γ} V^{+} →^{μ} V,

φ = μ∘γ, where the properties of birational morphisms γ and μ are described below. The morphism μ is a part of the commutative diagram

V^{+} ⟶^{μ} V
π ↓  ↓ π
S^{+} ⟶^{μ_{S}} S,

where μ_{S}: S^{+} → S is a sequence of blow ups on the base S, μ = (id_{F} , μ_{S}), V^{+} = F ×S^{+}, the symbol π in both cases stands for the projection onto the second factor. Let E_{S} be the set of exceptional divisors of the birational morphism μ_{S}. Then the exceptional divisors of the morphism μ are of the form F × E = π^{∗}E for E ∈ E_{S}.

The morphism γ: Ṽ → V^{+} is a sequence of blow ups with the set of exceptional divisors E^{+}. We denote the composition π ∘ γ: Ṽ → S^{+} by the same symbol π. For each exceptional divisor E ∈ E^{+} we have π^{−1}(π(E)) ≠ E, that is, it is not a pull back of a divisor on the base. For any exceptional divisor E ∈ E^{+} satisfying the condition (E · C_{t}) ≠ 0, one of the following two cases realize:

1) either π(E) = S^{+},

2) or π(E) ⊂ S^{+} is a prime divisor on S^{+}.

Existence of this modification of the morphism φ follows from the remark above. Indeed, let φ_{1}: V_{1} → V be an arbitrary sequence of blow ups, for which the condition (E) is violated. Let E_{1} be the set of exceptional divisors. Let us break the set E_{1} into two subsets:

E_{1} = E_{1}^{♯} ∪ E_{1}^{∗},

where φ_{1}(E) covers the base S for E ∈ E_{1}^{♯} and π ∘ φ_{1}(E) ⊂ S is a proper closed subset for E ∈ E_{1}^{∗}. Now we construct an arbitrary sequence of blow ups

μ_{S}: S^{+} → S,

such that for any discrete valuation ν_{E}, E ∈ E_{1}^{∗}, its center

B_{E} = centre(ν_{E}, V^{+})

on V^{+} = F × S^{+} covers a divisor on the base:

π(B_{E}) ⊂ S^{+}

is a prime divisor. Now let γ: Ṽ → V be an arbitrary sequence of blow ups, the centers of which are not pulled back from the base S^{+}, such that all the discrete valuations

ν_{E}, E ∈ E_{1} = E_{1}^{♯} ∪ E_{1}^{∗},

are realized by prime divisors on Ṽ. It is easy to see that φ = μ ∘ γ, μ = (id_{F} , μ_{S}) satisfies the properties described above.

Let D^{+} be the strict transform of the prime divisor D on V^{+}, K^{+} = K_{V^{+}} the canonical class.

Lemma 2. The algebraic family of curves

C_{S} = π_{∗}γ_{∗}C = (πγ(C_{t}), t ∈ T)

on S^{+} is free unless all the curves C_{t}, t ∈ T, lie in the fibers of the morphism π.

Proof. Let Y ⊂ S_{+} be an irreducible closed subset of codimension ≥ 2. We have

π^{−1}(Y) = ⋃_{i=1}^{N} Y_{i} ⊂ Ṽ.

It is clear that the curve π(C_{t}) intersects Y if and only if the curve C_{t} intersects at least one of the sets Y_{i}, i = 1, ..., N. If codim Y_{i} ≥ 2, then C_{t} ∩ Y_{i} = ∅, since the family C is free. If Y_{i} ⊂ Ṽ is a prime divisor, then Y_{i} = E ∈ E^{+} is an exceptional divisor of the morphism γ. However, by the properties of the morphism γ

(C_{t} · E) = 0,

since codim π(E) ≥ codim Y ≥ 2. Once again we obtain that for a general t ∈ T the curve C_{t} does not intersect E = Y_{i}. Consequently, for a general t ∈ T

π(C_{t}) ∩ Y = ∅

Q.E.D. for the lemma.

Corollary 1. The class

Z^{+} = D^{+} + nK^{+}

is non-negative on the dense (although generally speaking, not free) family of curves

C^{+} = γ_{∗}C = (γ(C_{t}), t ∈ T).

Proof. Assume the converse:

(Z^{+} · C_{t}^{+}) < 0  (4)

C^{+} = γ(C_{t}). Clearly, in this case the curves C_{t} do not lie in the fibers of the projection π. Explicitly, we get

D^{+} = μ^{∗}D − ∑_{E∈E_{S}} d_{E}π^{∗}E,

K^{+} = μ^{∗}K_{V} + ∑_{E∈E_{S}} a_{E}π^{∗}E,

where a_{E} is the discrepancy of the exceptional divisor E ⊂ S^{+} with respect to S. Taking into account that

D + nK_{V} = π^{∗}Z_{S}

for some effective class Z_{S} on S, we obtain that the class Z^{+} = π^{∗}Z_{S}^{+}, where

Z_{S}^{+} = Z_{S} − ∑_{E∈E_{S}}(d_{E} − na_{E})E,

is negative on the dense family of curves C^{+} and thus the class Z_{S}^{+} is negative on the free family of curves C_{S}.

Since V = F × S, we get

Pic V = π_{F}^{∗} Pic F ⊕ π^{∗} Pic S = ℤK_{V} ⊕ π^{∗} Pic S,

where K_{V} = π_{F}^{∗} K_{F} + π^{∗}K_{S}. In particular, for the divisor D we obtain the presentation

D = −nπ_{F}^{∗} K_{F} + π^{∗}D_{S},

where D_{S} = −nK_{S} + Z_{S}. Let z ∈ F be a point of general position. Set

D_{z} = D ∩ {z} × S.

Obviously, D_{z} is an effective reduced divisor on S, and moreover

D_{z} ∈ |D_{S}|,

and for a sufficiently general point z ∈ F we have

D_{z}^{+} ∈ |μ_{S}^{∗}D_{S} − ∑_{E∈E_{S}} d_{E}E|,

where D_{z}^{+} is the strict transform of the divisor D_{z} on S^{+}. Setting

K_{S}^{+} = K_{S^{+}} = μ_{S}^{∗}K_{S} + ∑_{E∈E} a_{E}E

to be the canonical class of the variety S^{+}, we obtain the presentation

Z_{S}^{+} = D_{S}^{+} + nK_{S}^{+}.

Note that in the situation under consideration the sequence of blow ups μ_{S} cannot be trivial, since the class Z_{S} is effective.

Thus assuming that the inequality (4) is true, we obtain a non-trivial sequence of blow ups μ_{S}: S^{+} → S such that the class (D_{S}^{+} + nK_{S}^{+}) is essentially non-effective, whereas the divisor D_{z} ∼ D_{S} = −nK_{S} + Z_{S} is effective and Z_{S} is an effective class. The only problem which prevents us from applying the condition (E) is that the divisor D_{z} can be reducible. Let us consider this problem.

For a general point z ∈ F the divisor D_{z} can be reducible although the divisor D is irreducible.

Consider the set of divisors

D = {D_{z} | z ∈ F, {z} × S ⊄ D}.

These divisors have no common irreducible components: otherwise the divisor D would have been reducible (or pulled back from the base S, which is impossible). Therefore the linear system |D_{S}| has no fixed components. In a similar way, the linear system of divisors

|D_{S}^{+}|

is free from fixed components. Let

ε: S − − → ℙ^{N}

be the rational map defined by this linear system. Two cases are possible: dim ε(S) = 1 and dim ε(S) ≥ 2.

In the second case the general divisor Δ^{+} ∈ |D_{S}^{+}| is irreducible. Set

Δ = μ_{S}(Δ^{+}) ⊂ S.

Obviously, Δ^{+} is the strict transform of the divisor Δ on S^{+}, Δ ∼ D_{z}, Δ^{+} ∼ D_{z}^{+}. Now we come to a contradiction with the condition (E) and conclude that the inequality (4) cannot be true.

In the first case note that the image ε(S) is a rational curve (recall that the variety S is rationally connected). Therefore for a general point z ∈ F

D_{z} = D_{1}(z) + D_{2}(z) + … + D_{k}(z),
D_{z}^{+} = D_{1}^{+}(z) + D_{z}^{+}(z) + … + D_{k}^{+}(z).

The number of components k ≥ 2 does not depend on z (for a general z) and

D_{1}(z) ∼ D_{2}(z) ∼ … ∼ D_{k}(z),

and similarly

D_{1}^{+}(z) ∼ D_{2}^{+}(z) ∼ … ∼ D_{k}^{+}(z),

that is, D_{i}(z) ∈ Π, D_{i}^{+}(z) ∈ Π^{+}, where Π is a pencil of divisors on S, and Π^{+} its strict transform on S^{+}. Thus

D_{z} ∼ kD_{1}(z), D_{z}^{+} ∼ kD_{1}^{+}(z)

and we obtain an irreducible divisor D_{1}(z) such that

kD_{1}(z) ∼ −nK_{S} + Z_{S}

and the class

kD_{1}^{+}(z) + nK_{S}^{+}

is essentially non-effective. Once again we come to a contradiction with the condition (E).

Q.E.D. for Corollary 1.

Remark 3. It is clear from the last part of the proof of Corollary 1, why do we need to introduce this additional parameter l ≥ 1. It can be avoided if one works with ℚ-divisors, that is, in the presentation

D ∼ −nK_{V} + Z

one allows that n ∈ ℚ_{+} and Z is a ℚ-divisor (however, the divisor D remains to be integral).

Remark 4. We require that the divisor D is irreducible for the only reason: one cannot allow that D had components pulled back from the base.

Let us continue the proof of Theorem 2.

For the divisor Z^{♯} on Ṽ we have the presentation

Z^{♯} = γ^{∗}Z^{+} + ∑_{E∈E^{+}}(−d_{E}^{+} + na_{E}^{+})E,

where a_{E}^{+} is the discrepancy of the exceptional divisor E ⊂ Ṽ with respect to V^{+}, d_{E}^{+} is the multiplicity of the divisor D^{+} with respect to the discrete valuation ord_{E}(·). By assumption, (Z^{♯} · C_{t}) < 0. On the other hand,

(γ^{∗}Z^{+} · C_{t}) = (Z^{+} · γ(C_{t})) ≥ 0.

Therefore there exists an exceptional divisor E ∈ E^{+} such that

d_{E}^{+} > na_{E}^{+}  (5)

and

(E · C_{t}) > 0.

By the properties of the morphism γ we get that the closed set πγ(E) ⊂ S^{+} is either the whole variety S^{+} or an exceptional divisor on S^{+}. In any case for a point of general position s ∈ πγ(E) we get

F_{s} = F × {s} = π^{−1}(s) ⊄ D^{+},

since the divisor D^{+} is irreducible. On the other hand, γ(E) intersects F_{s}, because s ∈ πγ(E). The Noether-Fano inequality (5) means that the pair

(V^{+}, \frac{1}{n}D^{+})

is not canonical along γ(E) ∩ F_{s}. By the inversion of adjunction this implies that the pair

(F_{s}, \frac{1}{n}D_{s}^{+}),

where D_{s}^{+} = D^{+}|_{F_{s}}, is not log-canonical and the more so not canonical. However, F_{s} ≅ F and D_{s}^{+} ∈ |nK_{F} | by construction, since K_{V}^{+}|_{F_{s}} = K_{F_{s}} and the divisor Z is pulled back from the base.

This gives a contradiction with the condition (C).

Q.E.D. for the theorem.

2 Fano direct products

2.1 The relative and absolute cases

Let F_{1}, …, F_{K}, K ≥ 2, be primitive Fano varieties, Pic F_{i} = ℤK_{F_{i}}, i = 1, …, K. Let us denote by the symbol H_{i} the positive generator of the Picard group, H_{i} = −K_{F_{i}}. Set

V = F_{1} × … × F_{K} = ∏_{i=1}^{K} F_{i},

S_{j} = ∏_{\begin{matrix}i=1 \\ i≠j \end{matrix}}^{K} F_{i}, j = 1, …, K,

so that V = F_{j} × S_{j}, j = 1, …, K. The projections of the variety V onto F_{j} and S_{j} we denote by λ_{j} and π_{j}, respectively.

Theorem 3. Assume that each of the Fano fiber spaces π_{j}: V → S_{j}, j = 1, …, K satisfies the condition (R). Then the variety V satisfies the condition (E).

Proof. Obviously,

Pic V = λ_{j}^{∗} Pic F_{j} ⊕ π_{j}^{∗} Pic Sj = ⨁_{i=1}^{K} λ_{i}^{∗} Pic F_{i} = ⨁_{i=1}^{K} ℤ(λ_{i}^{∗}H_{i}).

Besides,

K_{V} = λ_{j}^{∗}K_{F_{j}} + π_{j}^{∗}K_{S_{j}} = − ∑_{i=1}^{K} λ_{i}^{∗}H_{i}.

Let D,

lD ∼ −nK_{V} + Z,

be an irreducible divisor, where Z is an effective divisor on V. To simplify the notations, we write H_{i} instead of λ_{i}^{∗}H_{i}. We get

Z = ∑_{i=1}^{K} m_{i}H_{i}

with the coefficients m_{i} ∈ ℤ_{+}. Setting

m = m_{e} = min_{i∈{1,...,K}} m_{i},

we write Z in the form

Z = −mK_{V} + π_{e}^{∗}Y,  (6)

where

Y = ∑_{\begin{matrix}i=1 \\ i≠e \end{matrix}}^{K} (m_{i} − m)H_{i}

is an effective divisor on S_{e}.

The remaining part of the arguments is similar to the proof of Theorem 1. Fix a non-trivial sequence of blow ups

φ: Ṽ → V

with the set of exceptional divisors E. Let D̃ ⊂ Ṽ be the strict transform of the divisor D. We get

lD̃ ∼ −nKṼ + Z^{♯},

where

Z^{♯} = φ^{∗}Z − ∑_{E∈E}(ld_{E} − na_{E})E.

Taking into account the presentation (6), rewrite the latter formula in the form

Z^{♯} = −mKṼ + φ^{∗}π_{e}^{∗}Y − ∑_{E∈E}(ld_{E} − (n + m)a_{E})E.

Assume now that the class Z^{♯} is negative on the free family of irreducible curves C = (C_{t}, t ∈ T) on Ṽ. First of all, we get

−(KṼ · C_{t}) > 0,

otherwise (D̃ · C_{t}) < 0, which is impossible. It follows that the class

Y^{♯} = φ^{∗}π_{e}^{∗}Y − ∑_{E∈E}(ld_{E} − (n + m)a_{E})E

is negative on the curves of the family C. Since

lD̃ ∼ −(n + m)KṼ + Y^{♯},
lD ∼ −(n + m)K_{V} + π_{e}^{∗}Y,

we obtain a contradiction with the condition (R) for the Fano fiber space π_{e}: V → S_{e}. Therefore the class Z^{♯} cannot be negative on the curves of the family C. This means that the variety V satisfies the condition (E). Q.E.D. for the theo-

rem.

2.2 Geometry of Fano direct products

Theorem 4. Assume that each of the varieties F_{j}, i = 1, …, K, satisfies the condition (C) of divisorial canonicity. Then their direct product V = F_{1} × … × F_{K} satisfies the condition (E).

Proof. Let us argue by induction on K ≥ 1. For K = 1 the condition (E) is proved in Theorem 1. Assume that the claim of the theorem is proved for the smaller values of K. Then all the varieties S_{j}, j = 1, …, K satisfy the condition (E). Since the varieties F_{j}, j = 1, …, K are divisorially canonical, Theorem 2 implies that each of the Fano fiber spaces

π_{j}: V → S_{j}

satisfies the relative condition (R). Now we apply Theorem 3 and conclude that V satisfies the absolute condition (E). Q.E.D. for the theorem.

Theorem 5. In the assumptions of Theorem 4 let

χ: V − − → V^{♯}

be a birational map, where β: V^{♯} → S^{♯} is a rationally connected fibration. Then there exists an index e ∈ {1, …, K} such that the following diagram of maps is commutative:

V − − →^{x} V^{♯}
π_{e} ↓  ↓ β
S_{e} − − →^{α} S^{♯}

for some rational dominant map α: S_{e} − − → S^{♯}, that is, the map χ is fiber-wise with respect to the structures of rationally connected fibrations V/S_{e} and V^{♯}/S^{♯}.

Corollary 2. (i) Let ρ: V − − → Y be a rational dominant map, the general fiber of which is a rationally connected variety. Then there is a set of indices

I = {i_{1}, …, i_{k}} ⊂ {1, …, K},

such that Y is birational to

∏_{i∈I} F_{i}

and the map ρ is equivalent to the projection with respect to the direct decomposition

V ≅ ∏_{i∈I} F_{i} × ∏_{i∉I} F_{i}.

(ii) The groups of birational and biregular self-maps of the variety V coincide:

Bir V = Aut V.

(iii) The variety V is non-rational.

Proof of the corollary. Let us prove (i). The other two claims are obvious. Let ρ: V − − → Y be a rational dominant map, the general fiber of which is rationally connected. Theorem 5 implies that ρ factors through the map π_{e}: V → S_{e} for some e ∈ {1, …, K}, that is, ρ = ρ_{1} ∘π_{e} for some rational dominant map ρ_{1}: S_{e} − − → Y. Obviously, for a general fiber of the map ρ_{1} we get

ρ^{−1}(y) = F_{e} × ρ_{1}^{−1}(y),

so that the general fiber of the map ρ_{1} is rationally connected. Iterating this procedure, we obtain the claim (i). Q.E.D. for the corollary.

Remark 5. Let us break the set of varieties F_{1}, …, F_{K} into subsets of pair-wise isomorphic varieties:

I = {1, …, K} = ⋁_{j=1}^{l} I_{j},

where F_{i} ≅ F_{j} if and only if {i, j} ⊂ I_{k} for some k ∈ {1, …, l}. It is easy to see that

Aut V = ∏_{j=1}^{l} Aut(∏_{i∈I_{j}} F_{i}).

In particular, if the varieties F_{1}, …, F_{K} are pair-wise non-isomorphic, then

Aut V = ∏_{i=1}^{K} Aut F_{i}.

Now consider the opposite case when

F_{1} ≅ F_{2} ≅ … ≅ F_{K} ≅ F.

It is easy to see that in this case we get the exact sequence

1 → (Aut F)^{×K} → Aut V → S_{K} → 1,

where S_{K} is the symmetric group of permutations of K elements.

2.3 Proof of Theorem 5

Let χ: V − − → V^{♯} be a birational map, where β: V^{♯} → S^{♯} is a rationally connected fibration. Choose a very ample class H^{♯} on S^{♯} and a general divisor

Y ∈ |H^{♯}|

in the linear system |H^{♯}|. The divisor Y and its pull back β^{∗}Y on V^{♯} are irreducible. Let D be the strict transform of the divisor β^{∗}Y with respect to χ.

If there is an index e ∈ {1, …, K} such that

D = π_{e}^{∗}Δ

for some (irreducible) divisor Δ on S_{e}, then the claim of the theorem is true: χ factors through the projection π_{e}.

Assume the converse: D is not pulled back from S_{e} for some e ∈ {1, …, K}. It follows that

D ∼ ∑_{i=1}^{K} n_{i}H_{i},

where all the integers n_{i} ≥ 1. Therefore for

n = min_{i∈{1,...,K}} n_{i} ≥ 1

we get

D ∼ −nK + Z,

where Z is an effective divisor on V. (In fact, Z = π_{e}^{∗}X is pulled back from the base S_{e} for e such that n_{e} = n, but we do not need that.) Let

φ: Ṽ → V

be a resolution of singularities of the map χ. Consider a free system of irreducible rational curves

C^{♯} = (C_{t}^{♯}, t ∈ T^{♯})

on the variety V^{♯}, lying in the fibers of the morphism β. Obviously,

(C_{t}^{♯} · K_{V^{♯}}) < 0.

Let C = (C_{t}, t ∈ T) be the strict transform of the family C^{♯} on the variety Ṽ. We get

(C_{t} · KṼ) = (C_{t}^{♯} · K_{V^{♯}}) < 0.

Finally, let D̃ ⊂ Ṽ be the strict transform of the divisor D. We get

(D̃ · C_{t}) = (β^{∗}Y · C_{T}^{♯}) = 0.

Therefore we get

(Z^{♯} · C_{t}) = ((D̃ + nKṼ) · C_{t}) < 0,

where

D̃ ∼ −nKṼ + Z^{♯}.

The family of curves C is free. We obtain a contradiction with the condition (E). Q.E.D. for the theorem.

3 Divisorially canonical Fano varieties

3.1 2n-inequality

Let x ∈ S be a germ of a smooth surface, C ⊂ S a germ of an effective (possibly reducible) curve, x ∈ C. Consider a sequence of blow ups

φ_{i,i−1}: S_{i} → S_{i−1},

S_{0} = S, i = 1, …, N, φ_{i,i−1} blows up a point x_{i−1} ∈ S_{i−1}, E_{i} = φ_{i,i−1}^{−1}(x_{i−1}) ⊂ S_{i} is the exceptional line. For i > j set

φ_{i,j} = φ_{i,i+1,j} ∘ … ∘ φ_{i,i−1}: S_{i} → S_{j},

φ = φ_{N,0}, S̃ = S_{N}. We assume that the points x_{i} lie one over another, that is, x_{i} ∈ E_{i}, and that x_{0} = x, so that all the points x_{i}, i ≥ 1, lie over x:

x_{i,0}(x_{i}) = x ∈ S.

Let Γ be the graph with the vertices 1, …, N and oriented edges (arrows)

i → j, that connect i and j if and only if i > j and

x_{i−1} ∈ E_{j}^{i−1},

where for a curve Y ⊂ S_{j} its strict transform on S_{a}, a ≥ j, is denoted by the symbol Y^{a}. Assume that the point x is an isolated non-log-canonical singularity of the pair

(S, \frac{1}{n}C)

for some n ≥ 1. This means that for some exceptional divisor E ⊂ S̃ the inequality

ν_{E}(C) = ord_{E} φ^{∗}C > n(a_{E} + 1),  (7)

holds, where a_{E} is the discrepancy of E. Without loss of generality we may assume that E = E_{N} is the last exceptional divisor. The inequality (7) is called the log Noether-Fano inequality.

For i > j let the symbol p_{ij} denote the number of paths in the graph Γ from the vertex i to the vertex j. For i < j set p_{ij} = 0. Set also p_{ii} = 1. In terms of the numbers p_{ij} the log Noether-Fano inequality (7) takes the traditional form

∑_{i=1}^{N} p_{Ni}μ_{i} > n(∑_{i=1}^{N} p_{Ni} + 1),  (8)

where

μ_{i} = mult_{x_{i−1}} C^{i−1}.

Proposition 1. Either μ_{1} > 2n (that is, the first exceptional divisor E_{1} ⊂ S_{1} already gives a non-log-canonical singularity of the pair (S, (1/n)C)), or N ≥ 2 and the following inequality holds:

μ_{1} + μ_{2} > 2n.

Proof. If N = 1, then μ_{1} > 2n by means of log Noether-Fano inequality. Assume that μ_{1} ≤ 2n, then N ≥ 2. Obviously, μ_{1} > n. If μ_{2} ≥ n, then μ_{1} + μ_{2} > 2n, as we claim. So assume that μ_{2} < n. Then for each i ∈ {1, …, N}, i ≥ 2, we have μ_{i} ≤ μ_{2} < n (since the point x_{i−1} lies over x_{1}). Therefore from the inequality (8) we get

p_{N1}(μ_{1} − n) + ∑_{i=2}^{N} p_{Ni}(μ_{2} − n) > n.

However,

p_{N1} = ∑_{j→1} p_{Nj} ≤ ∑_{i=2}^{N} p_{Ni},

so that the more so

∑_{i=2}^{N} p_{Ni}(μ_{1} + μ_{2} − 2n) > n.

Therefore μ_{1} + μ_{2} > 2n. Q.E.D. for the proposition.

3.2 The regularity conditions

Let V = V_{M} ⊂ ℙ = ℙ^{M} be a smooth Fano hypersurface. For a point x ∈ V fix a system of affine coordinates z_{1}, …, z_{M} with the origin at x and let

f = q_{1} + q_{2} + … + q_{M}

be the equation of the hypersurface V, q_{i} = q_{i}(z_{∗}) are homogeneous polynomials of degree deg q_{i} = i. In order to prove that this variety is divisorially canonical, we need stronger regularity conditions (that is, the conditions of general position) for f than those that were used for proving birational rigidity in [P3].

(G1) The sequence

q_{1}, q_{2}, …, q_{M−1}

is regular in O_{x,ℙ}, that is, the system of equations

q_{1} = q_{2} = … = q_{M−1} = 0

defines a one-dimensional subset, a finite set of lines in ℙ, passing through the point x. This is the standard regularity condition, which was used in the previous papers (see [P3]).

(G2) The linear span of any irreducible component of the closed algebraic set

q_{1} = q_{2} = q_{3} = 0

in ℂ^{M} is the hyperplane q_{1} = 0 (that is, the tangent hyperplane T_{x}V).

(G3) The closed algebraic set

\overline{\{f_{1} = f_{2} = 0\} ∩ V} = \overline{\{q_{1} = q_{2} = 0\} ∩ V} ⊂ ℙ

(the bar ¯ means the closure in ℙ) is irreducible and any section of this set by a hyperplane P ∋ x is

• either also irreducible and reduced,

• or breaks into two irreducible components B_{1} + B_{2}, where B_{i} = V ∩ S_{i} is the section of V by a plane S_{i} ⊂ ℙ of codimension 3, and moreover mult_{x} B_{i} = 3,

• or is non-reduced and is of the form 2B, where B = V ∩ S is the section of V by a plane S of codimension 3, and moreover mult_{x} B = 3.

Proposition 2. For M ≥ 6 a general (in the sense of Zariski topology) hypersurface V satisfies the conditions (G1), (G2) and (G3) at every point x ∈ V.

Proof. For the condition (G1) see [P3]. A general hypersurface satisfies the condition (G1) at each point; in particular, through each point x ∈ V there are at most finitely many lines lying on V.

The conditions (G2) and (G3) are checked by straightforward computations. We omit these easy considerations.

Note that for M ≥ 8 a general hypersurface V satisfies the condition (G3) with the first option only.

3.3 Fano hypersurfaces are divisorially canonical

Theorem 6. Assume that the Fano hypersurface V = V_{M} ⊂ ℙ = ℙ^{M} satisfies the conditions (G1),(G2) and (G3) at every point x ∈ V. Then the primitive Fano variety V is divisorially canonical.

Proof. Let Δ ∈ |nH| be a prime divisor, n ≥ 1, where H ∈ Pic V is the class of a hyperplane section, K_{V} = −H. We must prove that the pair (V, \frac{1}{n}Δ) has canonical singularities.

Assume the converse. Taking into account that for any curve C ⊂ V the following inequality holds,

mult_{C} Δ ≤ n,

we obtain, by Proposition 1, that there is a point x ∈ V and a hyperplane B ⊂ E, where E ⊂ Ṽ is the exceptional divisor of the blow up

φ: Ṽ → V

of the point x, E ≅ ℙ^{M−2}, such that the inequality

mult_{x} Δ + mult_{B} Δ̃ > 2n,  (9)

holds, where Δ̃ ⊂ Ṽ is the strict transform of the divisor Δ.

Let 𝕋 = \overline{T_{x}V} ⊂ ℙ be the tangent hyperplane at the point x. The divisor E can be naturally identified with the projectivization

ℙ(T_{x}𝕋) = ℙ(T_{x}V).

There is a unique hyperplane 𝔹 ⊂ 𝕋, x ∈ 𝔹, such that

B = ℙ(T_{x}𝔹)

with respect to the above-mentioned identification. Let Λ_{𝔹} be the pencil of hyperplanes in ℙ, containing 𝔹, and Λ_{B} = Λ_{𝔹}|_{V} ⊂ |H| its restriction onto V. Consider a general divisor R ∈ Λ_{B}. It is a hypersurface of degree M in ℙ^{M−1}, smooth at the point x. Let R̃ ⊂ Ṽ be the strict transform of the divisor R. Obviously,

R̃ ∩ E = B.

Set Δ_{R} = Δ|_{R} = Δ ∩ R. It is an effective divisor on the hypersurface R. Lemma 3. The following estimate holds:

mult_{x} Δ_{R} > 2n.  (10)

Proof. We have

(Δ̃ ∘ R̃) = Δ̃_{R} + Z,

where Z is an effective divisor on E. According to the elementary rules of the intersection theory [F],

mult_{x} Δ_{R} = mult_{x} Δ + deg Z,

since mult_{x} R = 1. However, Z contains B with multiplicity at least mult_{B} Δ̃. Therefore, the inequality (9) implies the estimate (10). Q.E.D. for the lemma.

Lemma 4. The divisor

T_{R} = T_{x}R ∩ R

on the hypersurface R is irreducible and has multiplicity exactly 2 at the point x.

Proof. The irreducibility is obvious (for instance, for M ≥ 6 one can apply the Lefschetz theorem). By the condition (G2) the quadric

{q_{2}|_{E} = 0}

does not contain a hyperplane in E as a component, in particular, it does not contain the hyperplane B ⊂ E. Thus the quadratic component of the equation of the divisor T_{R}, that is, the polynomial

q_{2}|_{B},

is non-zero. Q.E.D. for the lemma.

Let us continue our proof of Theorem 6. By Lemmas 3 and 4 we can write

Δ_{R} = aT_{R} + Δ_{R}^{♯},

where a ∈ ℤ_{+} and the effective divisor Δ_{R}^{♯} ∈ |n^{♯}H_{R}| on the hypersurface R satisfies the estimate

mult_{x} Δ_{R}^{♯} > 2n^{♯}.

Moreover, Δ_{R}^{♯} does not contain the divisor T_{R} as a component. Without loss of generality we can assume the divisor Δ_{R}^{♯} to be irreducible and reduced.

Now consider the second hypertangent system

Λ_{2}^{R} = |s_{0}f_{2} + s_{1}f_{1}|_{R},

where s_{i} are homogeneous polynomials of degree i in the linear coordinates z_{∗}. Its base set

S_{R} = {q_{1}|_{R} = q_{2}|_{R} = 0}

is by condition (G3) of codimension 2 in R and either irreducible and of multiplicity 6 at the point x, or breaks into two plane sections of R, each of multiplicity 3 at the point x. In any case, for a general divisor D ∈ Λ_{2}^{R} we get Δ_{R}^{♯} ⊄ Supp D, so that the following effective cycle of codimension two on R,

Δ_{D} = (D ∘ Δ_{R}^{♯}),

is well defined. Since mult_{x} D = 3 and Λ_{2}^{R} ⊂ |2H_{R}|, the cycle Δ_{D} satisfies the estimate

\frac{mult_{x}}{deg} Δ_{D} > \frac{3}{M} .  (11)

We can replace the cycle Δ_{D} by its suitable irreducible component and thus assume it to be an irreducible subvariety of codimension 2 in R. Comparing the estimate (11) with the description of the set S_{R} given above, we see that

Δ_{D} ⊄ S_{R}.

This implies that Δ_{D} ⊄ T_{R}. Indeed, if this were not true, we would have got

f_{1}|_{Δ_{D}} ≡ q_{1}|_{Δ_{D}} ≡ 0.  (12)

However, Δ_{D} ⊂ D, so that for some s_{0} ≠ 0, s_{1} ≠ 0 (the divisor D is chosen to be general) we have

(s_{0}f_{2} + s_{1}f_{1})|_{Δ_{D}} ≡ 0.

By (12) this implies that

f_{2}|_{Δ_{D}} ≡ (q_{1} + q_{2})|_{Δ_{D}} ≡ 0

(since s_{0} ≠ 0 is just a constant), so that

Δ_{D} ⊂ S_{R}.

A contradiction.

Thus Δ_{D} ⊄ T_{R}. Therefore the effective cycle

Δ^{+} = (Δ_{D} ∘ T_{R})

is well defined. It satisfies the estimate

\frac{mult_{x}}{deg} Δ^{+} > \frac{6}{M} .  (13)

The effective cycle Δ^{+} as a cycle on V is of codimension 4. In [P3] it was shown that the inequality (13) for an effective cycle of codimension 4 is impossible provided that the regularity condition (G1) holds.

Q.E.D. for the theorem.

3.4 Fano double spaces

Theorem 6 together with Proposition 2 give examples of divisorially canonical Fano varieties of any dimension starting with 5. Let us give an example embracing all dimensions starting with 3. Let σ: V → ℙ = ℙ^{M} be a double space of index 1, that is, σ is branched over a smooth hypersurface W = W_{2M} ⊂ ℙ of degree 2M, M ≥ 3, see [P2]. The variety V is obviously a primitive Fano variety of index 1.

Theorem 7. A general variety V (corresponding to a general hypersurface W ⊂ ℙ) is divisorially canonical.

Proof. Let D ∈ |nH| be an irreducible divisor, where H = −K_{V} = σ^{∗}H_{ℙ}, H_{ℙ} is the class of a hyperplane in ℙ. In [P4] it was proved that

mult_{C} D ≤ n

for any curve C ⊂ V. Assume now that the singularities of the pair (V, \frac{1}{n}D) are not canonical. Then there is a point x ∈ V and a hyperplane B ⊂ E in the exceptional divisor E ⊂ Ṽ of the blow up φ: Ṽ → V of the point x, such that

mult_{x} D + mult_{B} D̃ > 2n,

where D̃ ⊂ Ṽ is the strict transform of D.

Case 1. The point z = σ(x) ∉ W. In this case let P ⊂ ℙ be the hyperplane such that P ∋ z and

Ṽ_{P} ∩ E = B,

where V_{P} = V ∩ P. Such hyperplane is unique. The divisor V_{P} is obviously irreducible and satisfies the equality

mult_{x} V_{P} + mult_{B} Ṽ_{P} = 2.

Since V_{P} ∈ |H|, this implies that V_{P} ≠ D. Therefore,

Z = (V_{P} ∘ D)

is a well defined effective cycle of codimension two on V. We get

deg Z = 2n, mult_{x} Z ≥ mult_{x} D + mult_{B} D̃ > 2n,

which is impossible. Thus the case z ∉ W does not realize.

Case 2. The point z = σ(x) ∈ W. In this case let

P = T_{z}W ⊂ ℙ

be the tangent hyperplane. Set again V_{P} = V ∩ P. We get V_{P} ∈ |H|, mult_{x} V_{P} ≥ 2. The divisor V_{P} is irreducible, so that if V_{P} ≠ D, then for the cycle Z = (V_{P} ∘ D) we get again

deg Z = 2n, mult_{x} Z ≥ 2 mult_{x} D > 2n,  (14)

since mult_{x} D ≥ mult_{B} D̃. Since there is no cycle Z, satisfying the condition(14), we conclude that

V_{P} = D.

Up to this moment we have never used the arguments of general position. Lemma 5. For a general W ⊂ ℙ the pair

(V, V_{P})

is canonical at every point z ∈ W.

Proof is straightforward and we omit it.

Now the proof of Theorem 7 is complete: we have come to a contradiction. Remark 5. It is easy to see that the claim of Theorem 7 remains true if one

admits simple singularities, for instance, non-degenerate factorial double points in dimension three.

The methods developed above make it possible to obtain the following result. Theorem 8. A general Fano double hypersurface of index 1 and a general Fano

complete intersection

V_{d_{1}·...·d_{k}} ⊂ ℙ^{M+k}

of index 1, where M ≥ 3k + 3, k ≥ 2, is divisorially canonical.

Conjecture. A general Fano complete intersection V ⊂ ℙ in a weighted projective space is divisorially canonical in any dimension ≥ 3.

3.5 Further results

The technique developed above makes it possible to prove the following fact.

Theorem 9. Let F_{1}, …, F_{K} be general Fano hypersurfaces of index 1 and dimension ≥ 5. Let

χ: V = F_{1} × … × F_{K} − − → V^{♯}

be a birational map, where π^{♯}: V^{♯} → S^{♯} is a K-trivial fibration. Then there exist: (i) a partition {1, …, K} = I ∨ I^{+} into two disjoint subsets; (ii) for each i ∈ I a pencil of hyperplane sections Λ_{i} on F_{i}, defining a rational

map

λ_{i}: F_{i} − − → ℙ^{1},

the fiber of which is K-trivial; (iii) a birational map

α: ∏_{i∈I} ℙ^{1} × ∏_{i∈I^{+}} F_{i} − − → S^{♯}

such that

π^{♯} ∘ χ = α ∘ (∏_{i∈I} λ_{i} × ∏_{i∈I^{+}} id_{F_{i}}) ,

that is, all structures of a K-trivial fibration on V are given by rational maps

∏_{i∈I} λ_{i} × ∏_{i∈I^{+}} id_{F_{i}}: V − − → ∏_{i∈I} ℙ^{1} × ∏_{i∈I^{+}} F_{i}.

A modification of the proof of Theorem combined with the arguments that prove Sarkisov’s theorem [S] make it possible to prove the following fact.

Theorem 10. Let F be a divisorially canonical primitive Fano variety of dimension ≥ 3. Then all structures of a rationally connected fibration on the variety

V = F × ℙ^{1}

are given by the projections

π_{l}: V → F and π_{r}: V → ℙ^{1}.

A more sophisticated argument makes it possible to prove the following fact. Theorem 11. Let F be a divisorially canonical primitive Fano variety of di-

mension ≥ 3. Then all structures of a rationally connected fibration on the variety

V = F × ℙ^{2}

are either given by the projections

π_{l}: V → F and π_{r}: V → ℙ^{2},

or are of the form

λ: F × ℙ^{2} − − → F × ℙ^{1}
↓  ↓
F  ⟷  F,

where for a general x ∈ F the fibers of the corresponding map

λ_{x}: ℙ^{2} − − → ℙ^{1}

are rational curves, or, finally, are obtained by taking the composition of a map λ just described with the projection of F × ℙ^{1} onto ℙ^{1}.

In particular, the variety V is non-rational. Theorems 10 and 11 can be considered as the first step in solving the problem of stable rationality for Fano varieties.

As we mentioned above, for a smooth point x ∈ F = F_{M} ⊂ ℙ on a Fano hypersurface of index 1 failure to satisfy the regularity conditions (G1)-(G3) gives a high codimension ≥ M + 1 for the hypersurface F as a point in the space

ℙ(H^{0}(ℙ, O_{ℙ}(M))).

Therefore for a general Fano fiber space π: V → ℙ^{1} with the fiber F = F_{M} ⊂ ℙ, M ≥ 6, each fiber F satisfies these conditions at every smooth point x ∈ F. Therefore the methods developed above give the following fact.

Theorem 12. Let π: V → ℙ^{1} be a general Fano fiber space with the fiber F = F_{M} ⊂ ℙ = ℙ^{M}, M ≥ 6, and moreover

Pic = ℤK_{V} ⊕ π^{∗} Pic ℙ^{1} = ℤK_{V} ⊕ ℤF.

Let D ∈ | − nK_{V} + lF | be an irreducible divisor, where l ∈ ℤ_{+} and n ≥ 1. Then the pair

(V, \frac{1}{n}D)

has canonical singularities. In particular, if

χ: V − − → V^{♯}

is a birational map, where π^{♯}: V^{♯} → S^{♯} is a rationally connected fibration, Σ^{♯} is a movable linear system on V^{♯}, pulled back from the base S^{♯}, and Σ its strict transform on V, then either Σ = |F | is a pencil of fibers or Σ ⊂ | − nK_{V} + lF | with l ≤ −1.

Similar results are true for fiber spaces V/ℙ^{1}, the fiber of which is a Fano double hypersurface and Fano complete intersection from the class described above.

The detailed proofs will be given in another paper.

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