Certain examples of birationally rigid varieties with a pencil of double quadrics

A.V.Pukhlikov

Institute for Systems Analysis Prospekt 60-letya Oktyabrya 9 117312 Moscow RUSSIA and Max-Planck-Insitut für Mathematik Gottfried-Claren-Strasse 26 53225 Bonn GERMANY

February 3, 1998

Abstract We prove that double covers of P^{1} × Q, where Q ⊂ P^{k+1} is a non-degenerate quadric and the branch divisor is cut out by a hypersurface of the type (2λ, 2k − 2), are birationally superrigid. In particular, they have only one structure of a Fano fibration. Therefore they are non-rational.

Introduction

The aim of the present paper is to prove that certain higher-dimensional varieties with a pencil of Fano double quadrics are birationally superrigid. The double quadrics themselves were studied in [P1]. Informally speaking, we prove a very particular case of the following “general principle”:

if a Fano fibration is “sufficiently twisted” over the base, then “birational geometry of the entire variety reduces to the birational geometry of the generic fiber”, that is, a Fano variety over a nonclosed field.

All the known examples of birationally rigid Fano fibrations [P2,S] can be considered as realizations of this principle. Here we give one more example.

The paper consists of three sections. In the first section the necessary definitions are collected. We also prove there some elementary general facts. In Section 2 we give an explicit construction of a certain family of algebraic varieties. After that, we formulate the principal result — that is, birational superrigidity of any variety from this family, and start to prove it. Section 3 contains the key part of the proof — that is, exclusion of infinitely near maximal singularities. We do it by means of the technique, which is a higher-dimensional generalization [P2] of the method of V.A.Iskovskikh and Yu.I.Manin [IM].

The result was obtained during my stay at Max-Planck-Institut für Mathematik in Bonn. I would like to thank the staff of the Institute for hospitality.

1 Fano fibrations and their birational correspondences

Here we introduce a large class of higher-dimensional algebraic varieties which are (at least, in principle) to be studied by means of the method of maximal singularities.

Definition 1. A projective algebraic variety V, equipped with a morphism

π: V → S,

is said to be a Fano fibration, if the following conditions are satisfied:

(i) V is non-singular in codimension 1:

codim Sing V ≥ 2,

(ii) the dimension of the base S is strictly smaller than dim V,

(iii) the fiber F_{η} over the generic (non-closed) point of the base S is irreducible and canonical adjunction terminates on S, i.e. for any Weil divisor D on F_{η} there is α ∈ R_{+} such that the linear system

|aD + bK_{F_{η}}|

is empty for any nonnegative integers a, b ∈ Z_{+} such that b > αa.

Obviously, the (principal) condition (iii) is equivalent to the following condition: for a general point t ∈ S canonical adjunction terminates on the corresponding fiber F_{t} = π^{−1}(t).

Definition 2. Let D ⊂ V be a Weil divisor on the Fano fibration (V, π). The least α ∈ R_{+} for which

|aD + bK_{V} | = ∅

for any a, b ∈ Z_{+} such that b > αa, is said to be the threshold of canonical adjunction (or, briefly, just threshold) and denoted by c(D) = c(V, D).

Examples of Fano fibration. (i) Let S = ∗ be a point, V be a smooth (or with mild singularities) Fano variety, i.e. the anticanonical class (−K_{V}) is ample. For instance, V_{M} ⊂ P^{M} is a smooth hypersurface of degree M. Here Pic V = ZK_{V}.

(ii) Standard conic bundles

π: V → S,

V, S are smooth, E = π_{∗}O_{V} (−K_{V}) is a locally free sheaf of the rank 3 on S, V ↪ P(E), and each fiber π^{−1}(t) is a conic in P^{2} = P(E_{t}^{∗}), and the following condition is satisfied:

Pic V = ZK_{V} ⊕ π^{∗} Pic S.

(iii) Del Pezzo fibrations

π: V → P^{1},

where V is smooth, Pic V = ZK_{V} ⊕π^{∗} Pic P^{1} and the generic fiber F_{η} is a Del Pezzo surface of degree d over the non-closed functional field C(P^{1}).

One more example of a Fano fibration will be given in Section 2 below. It makes the subject of the present paper.

The general task of the our theory is to describe birational correspondences between Fano fibration.

Definition 3. A pair (V′, H′), where V′ is a Fano fibration, π′: V′ → S′, and H′ is a Weil divisor on V′, is said to be a test pair, if the linear system |H′| is free in codimension 1. The threshold of the divisor H′ is said to be the threshold of the pair (V′, H′), and denoted by c(V′, H′).

Examples of test pairs. (i) Let V′ be a Fano variety, |H′| ⊂ | − rK_{V} | be a linear system with no fixed components, r ∈ Q_{+}^{∗}. Obviously, c(V′, H′) = r.

(ii) Let π′: V′ → S′ be a Fano fibration over a positive-dimensional base, |H′| be the inverse image of a linear system on the base, which is free in codimension 1. Obviously, c(V′, H′) = 0.

Assume that there is a birational map

χ: V − − → V′,

where V is a Fano fibration and (V′, H′) is a test pair. The dimensions dim V and dim V′ are, of course, assumed to be equal. Denote by |χ| the proper inverse image of the linear system |H′| on V with respect to χ. It is a linear system of divisors on V with no fixed components. Let D ∈ |χ| be a general member, so that |χ| ⊂ |D|. The threshold n(χ) = c(V, D) is an important characteristic of the birational map χ.

The group of birational automorphisms of the variety V is denoted by Bir V, the group of biregular automorphisms is denoted by Aut V.

Definition 4. For a Fano fibration π: V → S the group of birational automorphisms of the generic fiber

Bir F_{η} ⊂ Bir V

is said to be the group of proper birational automorphisms of the Fano fibration.

The thresholds of canonical adjunction accumulate the most essential, from the viewpoint of birational geometry, information about linear systems. Respectively, if we know how the thresholds change when we apply a birational map, then we know about this map practically everything. Now we introduce the key concept of the theory.

Definition 5. A Fano fibration V is said to be birationally rigid, if for any test pair (V′, H′) and any birational map

χ: V − − → V′,

there is a proper birational automorphism

χ^{∗} ∈ Bir F_{η} ⊂ Bir V

such that the “twisted” birational map

χ ∘ χ^{∗}: V − − → V′

satisfies the following monotonicity condition:

n(χ ∘ χ^{∗}) ≤ c(V′, H′).

Provided this condition is always satisfied:

n(χ) ≤ c(V′, H′),

i.e. we can take χ^{∗} = id_{V}, the Fano fibration V is said to be birationally superrigid.

Proposition 1.1 Assume that the Fano fibration V satisfies the condition Pic F_{η} ≅ Z, where the generic fiber F_{η} is a smooth algebraic variety over the field C(S) (which is non-closed if dim S ≥ 1), and that for any linear system |χ| ⊂ |D|, free from fixed components, the following equality is true:

c(V, D) = c(F_{η}, D|_{F_{η}})

(obviously, in any case c(V, D) ≤ c(F_{η}, D|_{F_{η}})). Assume that V is birationally rigid. Then V can not be birationally equivalent to any Fano fibration π′: V′ → S′ over a base of a higher dimension: dim S′ > dim S. For any birational equivalence

χ: V − − → V′
↓  ↓
S  S′

where dim S′ ≤ dim S, there is a rational map of the base

α: S − − → S′,

which makes a commutative diagram:

π′ ∘ χ = α ∘ π.

Proof. Let

χ: V − − → V′
↓  ↓
S  S′

be a birational equivalence of Fano fibrations. Take a linear system |H′| on V′, which is the inverse image with respect to π′ of a very ample linear system on the base S′. Obviously, c(V′, H′) = 0, so that, twisting by a proper birational automorphism χ^{∗} ∈ Bir F_{η}, we may assume that n(χ) = 0. From the condition Pic F_{η} ≅ Z we derive that the divisors in the linear system |χ| are inverse images of divisors on the base with respect to π. Therefore, χ is a fiber-wise map, with respect to the given structures of Fano fibrations. Q.E.D. for Proposition 1.1.

Corollary 1.1 Birationally rigid Fano fibrations, satisfying the hypothesis of Proposition 1.1, are non-rational.

Corollary 1.2 Birational automorphisms of a rigid Fano fibration are fiber-wise.

Now we can give a precise formulation of what we intend to do in this paper: we are going to prove that certain families of Fano fibrations (over a one-dimensional base) are birationally superrigid.

2 Varieties with a pencil of double quadrics

2.1 Construction of the family V_{k}(λ)

We study a certain family of smooth projective (k + 1)-dimensional varieties V ∈ V_{k}(λ), fibered into double quadrics of index 1. Let Q ⊂ P^{k+1} be a non-singular quadric,

X = P^{1} × Q ⊂ P^{1} × P^{k+1},
W(λ) ∋ W ⊂ P^{1} × P^{k+1}

a divisor of the type (2λ, 2k − 2), λ ∈ Z_{+}, where the scheme-theoretic intersection X ∩ W is a smooth divisor on X. This data determines a variety V, which is a smooth double cover

σ: V → X,

branched over X ∩ W. Denote the family just constructed by the symbol V(λ). Projections of V and X onto P^{1} we denote by the same symbol π.

Let L = X ∩ P^{1} × {hyperplane ⊂ P^{k+1}}, G_{t} = π^{−1}(t) ⊂ X. Obviously, Pic X = ZL ⊕ ZG and K_{X} = −kL − 2G. Furthermore, W ∈ |2(k − 1)L + 2λG|, so that

K_{V} = σ^{∗}(−L + (λ − 2)G).

Obviously, Pic V = ZK_{V} ⊕ ZF, where F is the class of a fiber π^{−1}(t), t ∈ P^{1}. We assume that the following condition of general position is satisfied:

(G) if x ∈ W ∩G_{t} is a singular point, then it is of multiplicity exactly two, the tangent cone T_{x}(W ∩ G_{t}) is reduced at the generic point, and the intersection of projectivized tangent cones T_{x}(T_{x}Q ∩ Q) and T_{x}(W ∩ G_{t}) in P(T_{x}Q) ≅ P^{k−1} is a reduced cycle of dimension (k − 3) and degree 4.

Let y ∈ F = F_{t} be a singular point of the fiber. In other words, x = σ(y) ∈ G_{t} is a singularity of the branch divisor W ∩ G_{t} on the fiber. Let φ: F̃ → F be the blowing up of the point y, Ẽ ⊂ F̃ be the exceptional divisor. Set D = T_{x}Q ∩ Q. It is a quadratic cone with the vertex at x.

Proposition 2.1 σ^{−1}(D) is an irreducible Weil divisor. The multiplicity of its proper inverse image D̃ ⊂ F̃ at any point in D̃ ∩ E is at most 2.

Proof. Let ψ: Q̃ → Q be the blowing up of the point x, E = ψ^{−1}(x) ⊂ Q̃ be the exceptional divisor. Note that there is the following commutative diagram of morphisms:

Ẽ ⊂ F̃ ⟶^{φ} F
σ↓ ↓ σ ↓ σ
E ⊂ Q̃ ⟶^{ψ} Q,

Here the vertical arrows are double covers. It is necessary to prove that the diagram does exist — that is, that the original double cover σ extends to F̃. But this can be done easily by local coordinate computations, taking into consideration the condition (G).

Consider the projectivized tangent cone to the divisor T_{x}Q ∩ Q. It is a nonsingular quadric in E. Moreover, its image on Ẽ is irreducible (since its intersection with the ramification divisor is reduced). Consequently, σ^{−1}(D) is also irreducible.

Finally, D̃ ∩ E is realized as a double cover of the smooth quadric. Therefore multiplicity of any singular point of this irreducible variety cannot be higher than two. Q.E.D. for the Proposition.

As in [P2], we call an effective cycle horizontal, if each of its components covers the base, and vertical, if each of its components lies in a fiber of π.

2.2 K^{2}-condition. Formulation of the principal result

Assume that the variety V ∈ V(λ) satisfies the following K^{2}-condition (see [P2]):

(K^{2}) the numerical class of codimension 2 cycles

MK_{V}^{2} + (K_{V} · F)

is not effective for any M ∈ Z_{+}.

Obviously, the K^{2}-condition is certainly satisfied, if the following inequality

(K_{V}^{2} · σ^{∗}L^{k−1}) ≤ 0,

is true, because the system |L| is free. It is easy to see that the left-hand side is equal to 4(2 − λ). Thus, it is sufficient to assume that λ ≥ 2.

Theorem 1 If the conditions (G) and (K^{2}) are satisfied, then the Fano fibration (V, π) is birationally superrigid.

Remark. If λ = 0, then any variety V ∈ V(λ) is a direct product: V = V^{∗} ×P^{1}, where V^{∗} is a smooth double quadric of index 1. Thus V has a large group of birational automorphisms Bir V.

2.3 Start of the proof. Maximal singularities

Fix a test pair (V′, H′) and a birational map

χ: V − − → V′

(provided there are any). Then there are a nonnegative integer n ∈ Z_{+} and an integer l ∈ Z such that

|χ| ⊂ | − n(χ)K_{V} + lF |.

Lemma 2.1 l ≥ 0

Proof. Indeed, let D_{1}, D_{2} ∈ |χ| be generic divisors, then Z = (D_{1} • D_{2}) is an effective cycle of codimension 2 on V. Its numerical class

z = n^{2}K_{V}^{2} + 2nl(−K_{V} · F)

is effective. Now the K^{2}-condition implies that l ∈ Z_{+}.

Corollary 2.1 The integer n = n(χ) is equal to the threshold of canonical adjunction c(D, V) of the system |χ|, D ∈ |χ|.

Now let us assume that the monotonicity condition is not satisfied:

n > c(V′, H′).

Our aim is to show that this inequality implies a contradiction, proving the theorem.

We use the language of discrete valuations in the form of [P2]. For a quasiprojective variety X the symbol N (X) denotes the oriented graph of geometric discrete valuations ν, which have centres Z(X, ν) ⊄ Sing X of codimension ≥ 2. Recall that a discrete valuation ν ∈ N (V) is said to be a maximal singularity of the linear system |χ|, if the Nöther-Fano inequality is true:

ν(|χ|) > a(V, ν)n(χ).

For a valuation ν ∈ N (V) set

e(ν) = ν(|χ|) − nδ(ν),

where δ(ν) = a(V, ν) is the discrepancy of ν. Thus, ν is a maximal singularity of the system |χ| if and only if e(ν) > 0.

Abusing the notations, we sometimes write T instead of ν, if ν = ν_{T}, where T is a prime Weyl divisor on a model Ṽ of the field C(V), T ⊄ Sing Ṽ, realizing the discrete valuation ν.

Proposition 2.2 A maximal singularity does exist. Moreover, one of the following two cases takes place: (i) there is a maximal singularity ν ∈ N (V) such that its centre on V covers the base: π(Z(V, ν)) = P^{1};

(ii) there is a finite set of maximal singularities M ⊂ N (V), whose centres on V lie in the fibers of π,

π(Z(V, ν)) = t(ν) ∈ P^{1},

and the following inequality is satisfied:

∑_{t∈P^{1}} ( max_{\{ν∈M|t(ν)=t\}} \frac{e(ν)}{ν(F_{t})}) > l.

Proof follows the regular scheme [P2] almost word for word. There exist a smooth quasi-projective variety W and a birational morphism φ : W → V such that for the proper inverse image D̃ ⊂ W of a general divisor D ∈ |χ| we have

|aD̃ + bK_{W} | = ∅,

if b > c(V′, H′)a. Writing down

D̃ = φ^{∗}D − ∑_{T ∈L} ν_{T} (D)T,

K_{W} = φ^{∗}K_{V} + ∑_{T ∈L} δ(T)T,

where L is the set of all those Weil divisors on W, which lose dimension when they are put down onto V, and taking into account that n > c(V′, H′) by hypothesis, we get that the linear system

|lφ^{∗}F − ∑_{T ∈L} e(T)T |

is empty. Setting

M = {T ∈ L|e(T) > 0},

we see that M ≠ ∅. It was shown in [P2], that for any point t ∈ P^{1} the Weil divisor

φ^{∗}F_{t} − ∑_{{T ∈M|t(T)=t}} ν_{T} (F_{t})T

is effective, so that the Q-divisor

max_{t} (\frac{e(T)}{ν_{T} (F_{t})}) φ^{∗}F_{t} − ∑_{t} e(T)T

is effective, too (where the lower index t means that the operation max or ∑ is taken over those valuations ν_{T}, whose centres lie in the fiber F_{t}). Now, if the statement (i) is not true, then obviously (ii) is true. Q.E.D. for Proposition.

Proposition 2.3 The case (i) of Proposition 2.2 does not realize.

Proof. Assume the converse: the centre Z(V, ν) of a maximal singularity ν ∈ N (V) covers the base of our Fano fibration. It is easy to see that on a general fiber F_{t} the valuation ν induces a maximal singularity ν_{t} ∈ N (F_{t}) of the linear system |χ|_{t} = |χ||_{F_{t}}. However, in [P1] it was proved that this is impossible. Q.E.D.

Remark. We have actually proved the following fact. If (V, π) is an arbitrary Fano fibration, whose generic fiber F_{η} (or a general fiber F_{s}, where s ∈ S is a general point) is a birationally superrigid variety (that is, linear systems cannot have maximal singularities), then linear systems |χ| on V do not permit maximal singularities ν ∈ N (V), whose centre Z(V, ν) covers the base S of the Fano fibration. Similarly, in the general context all the maximal singularities of this type come from maximal singularities on the general fiber.

Proposition 2.4 The centre of any maximal singularity is of codimension ≥ 2 with respect to the fiber, in which it lies.

Proof. Let B = Z(V, ν) ⊂ F be a Weil divisor. Then for a general divisor D ∈ |χ| we get σ_{∗}D_{F} ∈ |2nH|, where H is a hyperplane section. Q = G_{t}σ(F), and moreover σ_{∗}B comes into σ_{∗}D_{F} with multiplicity > n. But Q is a factorial variety with Pic Q = ZH, so that σ_{∗}B is a hyperplane section.

Lemma 2.2 Let Y = R_{1} ∩ … ∩ R_{k} ⊂ P^{N} be a complete intersection, H ⊂ P^{N} be a hyperplane. Then

dim Sing(Y ∩ H) ≤ dim Sing Y + 1.

Proof. See [IP], also [P1]. This fact seems to be very old and classical (as Professor F.L.Zak told the author), but I have no idea where and when it first appeared.

Corollary 2.2 B ∈ | − K_{F} |

Indeed, the intersection of σ_{∗}(B) with W cannot be non-reduced everywhere. But |χ|_{F} ⊂ | − nK_{F} |, so that B can come into this linear system with multiplicity at most n. A contradiction. Q.E.D.

To complete the proof, we have to exclude the infinitely near case, when the centres of all the maximal singularities are of codimension ≥ 2 in the fibers.

This will be done in the next section.

3 The infinitely near case

3.1 Supermaximal singularity

For a pair of general divisors D_{i} ∈ |χ|, i = 1, 2, consider the effective codimension 2 cycle Z = (D_{1} • D_{2}) of their scheme-theoretic intersection. We have the decomposition

Z = Z^{v} + Z^{h} = Z^{h} + ∑_{t∈P^{1}} Z_{t}^{v},

where Z^{h} is a horizontal cycle, Z^{v} is a vertical cycle, and the support of the cycle Z_{t}^{v} is contained in the fiber F_{t}. For a vertical cycle Y ⊂ F_{t} of codimension k (with respect to V) set

deg Y = (Y · L_{V}^{k} ) ∈ Z.

Now we get the following fact.

Lemma 3.1

∑_{t∈P^{1}} deg Z_{t}^{v} ≤ 8nl.

Proof: just repeat word for word the argument of Section 4, Lemma 4.1 in [P2].

Corollary 3.1 There exists a maximal singularity T ∈ M, t(T) = t ∈ P^{1}, such that

8ne(T) > ν_{T} (F_{t}) deg Z_{t}^{v}.

This singularity is said to be supermaximal. Fix it. We write F instead of F_{t}, Z^{v} instead of Z_{t}^{v} (now this will cause no misunderstanding), e instead of e(T), ν instead of ν_{T}, δ instead of δ(T).

3.2 Resolution of the supermaximal singularity.

Let

φ_{i,i−1} : V_{i} → V_{i−1}
⋃  ⋃
E_{i} → B_{i−1}

be the resolution of ν: φ_{i,i−1} blows up the irreducible cycle B_{i−1}, which is the centre Z(V_{i−1}, ν) of the valuation ν on V_{i−1}, E_{i} = φ_{i,i−1}^{−1}(B_{i−1}) is the exceptional divisor, 1 ≤ i ≤ K and the triplet (V_{K}, φ_{K,0}, E_{K}) is a realization of ν, i.e. ν_{E_{K}} = ν. Recall that for i > j the symbol φ_{i,j} denotes the composition

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

φ_{i,i} = id_{V_{i}} and V_{0} = V. On the set of indices {1, …, K} we introduce a natural oriented graph structure, joining the vertices i and j by an oriented edge (an arrow), if i > j and

B_{i−1} ⊂ E_{j}^{i−1}

(the upper index means, as usually, the operation of taking the proper inverse image on V_{i−1}). Notation: i → j. We set

p(i, j) = ( number of paths from i to j ),

if i ≠ j, and p(i, i) = 1, otherwise. Finally, instead of p(K, i) we write p_{i}.

Since φ_{i,i−1}(B_{i}) = B_{i−1}, the dimensions dim B_{i} do not decrease. Following [P2], we decompose the succession of blowing ups into the lower part, when codim B_{i−1} ≥ 3, corresponding to the indices 1 ≤ i ≤ L, L ≤ K, and the upper part, when codim B_{i−1} = 2, i ≥ L + 1, if L ≠ K. We have got the following presentations:

δ ≥ 2 ∑_{i=1}^{L} p_{i} + ∑_{i=L+1}^{K} p_{i},

ν(|χ|) = ∑_{i=1}^{K} p_{i} mult_{B_{i−1}} |χ|^{i−1}.

Set

N = max{i|1 ≤ i ≤ L, B_{i−1} ⊂ F^{i−1}},
μ_{i} = mult_{B_{i−1}} F^{i−1}

for i = 1, …, N. In terms of these multiplicities we get the following inequality:

ν(F) ≥ ∑_{i=1}^{N} μ_{i}p_{i}.

To make our formulae more compact, set

Σ_{l} = ∑_{i=1}^{L} p_{i}, Σ_{u} = ∑_{i=L+1}^{K} p_{i}.

Finally, for i = 1, …, L set

m_{i}^{h} = mult_{B_{i−1}}(Z^{h})^{i−1},
m_{i}^{v} = mult_{B_{i−1}}(Z^{v})^{i−1},
m_{i} = mult_{B_{i−1}} Z^{i−1} = m_{i}^{h} + m_{i}^{v}.

Note at once that since the support of the 1-cycle Z^{v} is contained in F, all the integers m_{i}^{v} vanish for L ≥ i ≥ N + 1.

3.3 Counting multiplicities.

In accordance with the standard techniques of counting multiplicities [P2], which goes back to the test class techniques of V.A.Iskovskikh and Yu.I.Manin [IM], we get the inequality

∑_{i=1}^{L} p_{i}m_{i} ≥ \frac{(2Σ_{l}n + Σ_{u}n + e)^{2}}{Σ_{l} + Σ_{u}} .

Rewrite this inequality in the expanded form:

(Σ_{l} + Σ_{u}) (∑_{i=1}^{L} p_{i}m_{i}^{h} + ∑_{i=1}^{N} p_{i}m_{i}^{v}) ≥

≥ (2Σ_{l} + Σ_{u})^{2}n^{2} + 2(2Σ_{l} + Σ_{u})ne + e^{2}.

Note, first of all, that

m_{1}^{h} ≤ deg(Z^{h} ∩ F) = 4n^{2}.

Since the multiplicities m_{i}^{h,v} do not increase as functions of the index i, our inequality will be true when we replace m_{i}^{h} by m_{1}^{h}. In this way we get the following inequality (*):

(Σ_{l} + Σ_{u}) ∑_{i=1}^{N} p_{i}m_{i}^{v} ≥ Σ_{u}^{2}n^{2} + 2(2Σ_{l} + Σ_{u})ne + e^{2}.

To get a contradiction and complete the proof, we need to study two cases. Set B = B_{0} = Z(V, ν) ⊂ F.

We say that our case is non-singular, if B ⊄ Sing F, and singular, otherwise. In accordance with the condition (G), in the singular case B is a point, σ(B) ∈ W ∩ G_{t} is a singularity of the branch divisor.

3.4 Exclusion of the non-singular case

Let x ∈ F \ Sing F be a smooth point of the fiber F.

Lemma 3.2 For any cycle T ⊂ F of codimension 1 the following inequality holds:

mult_{x} T ≤ \frac{1}{2} deg T.

Proof. Consider the image Y = σ(T) of the Weil divisor T on Q. Obviously,

\frac{mult_{z} Y}{deg Y} ≥ \frac{mult_{x} T}{deg T} .

Let us assume that the assertion of the lemma is not true for T. Consequently,

mult_{z} Y > \frac{1}{2} deg Y.

This implies that Y lies entirely in the tangent hyperplane T_{z}Q. Since Q is nonsingular, Y = Q ∩ T_{z}Q. If z ∉ W, then we come to an obvious contradiction:

mult_{x} T = 2, deg T = 4.

If z ∈ W, then we take a general 3-plane P in P^{k+1} ⊃ Q, containing z. Obviously, P ∩ Y is a pair of (distinct) lines, meeting W transversally at the point z. Consequently,

mult_{x} T ≤ mult_{x}(T ∩ σ^{−1}(P ∩ Q)) = 2,

and we get a contradiction again. Q.E.D. for the Lemma.

By the Lemma we get the following estimate:

mult_{x} Z^{v} < \frac{4ne(T)}{ν(F)}.

Now we complete the proof in the usual way [P2]:

∑_{i=1}^{N} p_{i}m_{i}^{v} < \frac{4ne}{ν(F)} ∑_{i=1}^{N} p_{i} ≤ 4ne,

so that

0 > Σ_{u}^{2}n^{2} − 2Σ_{u}ne + e^{2},

which is impossible. The non-singular case has been excluded.

3.5 Exclusion of the singular case

Let B = x be a singular point of the fiber F, z = σ(x) be its image on the quadric G_{t} = Q.

Set T^{∗} = σ^{−1}(T_{z}Q ∩ Q). We know (Section 2), that T^{∗} ⊂ F is an irreducible Weil divisor. Set

Z^{v} = aT^{∗} + Z_{1}^{v},

where Z_{1}^{v} does not come into T^{∗} as a component, a ∈ Z_{+}. As above (subsection 4), for any irreducible cycle T ≠ T^{∗} of codimension 1 with respect to F we have the estimate

mult_{x} T ≤ \frac{1}{2} deg T.

By Proposition 2.1 we get for T^{∗} the following inequality

mult_{y} T̃^{∗} ≤ 2 = \frac{1}{2} deg T^{∗},

where y ∈ E_{1} is an arbitrary infinitely near point over x. Using these two inequalities, applied to the two components, into which the cycle Z^{v} was decomposed, we get

∑_{i=1}^{N} p_{i}m_{i}^{v} ≤ (2p_{1} + ∑_{i=2}^{N} p_{i}) (\frac{1}{2} deg Z^{v}).

But since x is a singularity of the fiber, we have the following estimate:

ν(F) ≥ 2p_{1} + ∑_{i=2}^{N} p_{i}.

Combining these inequalities, we get what we need:

∑_{i=1}^{N} p_{i}m_{i}^{v} < 4ne.

Now we complete the proof, repeating the “non-singular” arguments above word for word.

Q.E.D. for the Theorem.

References

[IM] Iskovskikh V.A. and Manin Yu.I. Three-dimensional quartics and counterexamples to the Lüroth problem. // Math. USSR Sb. 1971. V. 86. No. 1. P. 140-166.

[IP] Iskovskikh V.A. and Pukhlikov A.V. Birational automorphisms of higher-dimensional algebraic varieties // Algebraic Geometry. 1. J. Math. Sci. 1996. V. 82. No. 4. P. 3528-3613.

[P1] Pukhlikov A.V. Birational automorphisms of a double space and a double quadric // Math. USSR Izv. 1989. V. 32. P. 233-243.

[P2] Pukhlikov A.V. Birational automorphisms of three-dimensional algebraic varieties with a pencil of Del Pezzo surfaces // To appear in: Izvestiya. 1998. V. 62. No. 1. P. 123-164 (in Russian)

[S] Sarkisov V.G. On conic bundles structures // Math. USSR Izv. 1982. V. 20. No. 2. P. 355-390.