Birational automorphisms of algebraic 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 Vivatsgasse 7 53111 Bonn GERMANY e-mail: dost@dost.mccme.rssi.ru

September 27, 1999

Abstract We prove that higher-dimensional algebraic varieties with a pencil of double quadrics of index 1 are birationlly rigid, provided they are sufficiently twisted over the base.

Introduction

0.1. Let π: V → S be a Fano fibration. Here V is a smooth projective variety of dimension ≥ 4, satisfying the condition Pic V = ZK_{V} ⊕ π^{∗} Pic S, S is a smooth projective variety of dimension ≥ 1, and the anticanonical divisor (−K_{V}) is ample on the fibers of π. The generic fiber F_{η} (that is, the very variety V, considered over the non-closed field C(S)) is a Fano variety. The subgroup Bir F_{η} ⊂ Bir V of birational self-maps, preserving the fibers of π, is called the group of proper birational automorphisms of the Fano fibration V/S.

Recall [P2-P5], that V/S is said to be birationally rigid, if for each birational map

χ: V − − → V′

onto a uniruled variety V′ of the same dimension the monotonicity condition

c(D^{∗}, V) ≤ c(D′, V′)

holds. Here D′ ∈ Σ′, Σ′ is a linear system of divisors on V′ with no fixed components, D^{∗} ∈ Σ^{∗}, Σ^{∗} is the proper inverse image of Σ′ on V with respect to the composition

χ^{∗} ∘ χ: V − − → V′,

where χ^{∗} ∈ Bir F_{η} is a certain proper birational self-map of V/S (depending on Σ′), and

c(Y, X) = sup\{\frac{b}{a}|b ≥ 1, a ≥ 1, |aY + bK_{X}| ≠ ∅\}

is the threshold of canonical adjunction. If for each V′, χ, Σ′ we can take χ^{∗} = id, that is,

c(D, V) ≤ c(D′, V′),

then the Fano fibration V/S is said to be birationally superrigid.

0.2. The aim of the present paper is to prove birational superrigidity of higherdimensional algebraic varieties with a pencil of double quadrics. Take S = P^{1} and let π: V → P^{1} be a smooth variety of dimension M + 1 ≥ 5, each fiber of which is a double quadric of index 1. More precisely, there is a commutative diagram

V ⟶^{σ} X ⊂ P(E^{∗})
π ↓  ↓ π
P^{1} ⟷  P^{1},

where σ is a double cover, P(E^{∗}) is a locally trivial P = P^{M+1}-fibration over P^{1}, the fiber X_{t} = π^{−1}(t) ⊂ P is an M-dimensional quadric and the branch divisor of σ is cut out on X_{t} by a hypersurface of degree 2(M − 1) in P (for the details of construction of a pencil of double quadrics, see Sec. 1). The main result of the present paper is given by the following

Theorem 1. Let V be a general variety, satisfying the K^{2}-condition: the numerical class NK_{V}^{2} − f is not effective for any N ∈ Z, where f is the class of an effective divisor in a fiber of the pencil π. Then the Fano fibration π: V → P^{1} is birationally superrigid.

For a Fano fibration π: V → P^{1}, satisfying the K^{2}-condition, superrigidity implies a number of standard properties:

(i) the only non-trivial structure of a Fano fibration on V is the very pencil of fibers of the morphism π; in other words, each birational map

V − − →^{χ} V′
π ↓  ↓ π′
P^{1}  S′

onto an arbitrary Fano fibration of the same dimension is fiber-wise, that is, there is a map α: P^{1} → S′ such that π′ ∘ χ = α ∘ π;

(ii) any birational self-map of V is fiber-wise; for a general V we have Bir V = Aut V = Z/2Z;

(iii) the variety V is not rational.

For those Fano fibrations V → P^{1}, which cover X = Q × P^{1}, where Q ⊂ P is a smooth quadric hypersurface, our theorem was proved in [P3]. All the arguments of the paper [P3] work for the general case of an arbitrary X, too, but they do not exclude the most difficult case of a maximal singularity, lying over the vertex of the cone Q_{t} ⊂ P. It is exactly to avoid the singular fibers on X, that in [P3] the class of varieties under consideration was restricted by the direct product. However, today the strengthened techniques of hypertangent systems makes it possible to study the case of singular fibers exhaustively, too.The proof of the theorem is given in Sec. 2.

The results of the papers [P1,P3,P4] and Theorem 1 suggest that the implication (K^{2}-condition ⇒ birational rigidity) is true for all varieties with a pencil of Fano double hypersurfaces.

Conjecture. Let π: V → P^{1} be a Fano fibration with V smooth, where each fiber π^{−1}(t) is the double cover of a hypersurface Q_{m} ⊂ P of degree m, branched over W_{2l}^{∗} ∩ Q_{m}, l + m = M + 1. Assume that V satisfies the K^{2}-condition. Then π: V → P^{1} is birationally superrigid.

The technique of hypertangent systems makes it possible to realize some of the principal steps in the proof of Conjecture (following the traditional scheme of the method of maximal singularities). More precisely, one can exclude all the possibilities for a maximal singularity but one exceptional case, when its centre on V is a singular point of a fiber of π, which comes from a singularity on the branch divisor (if the singularity comes from a double point on the hypersurface Q_{t} ⊂ P^{M}, then the arguments of [P5] successfully exclude the case). To complete the proof, we need to strengthen the technique.

0.3. The present paper is a direct continuation of [P3,P4]. The double spaces and double quadrics of index 1 have small degree and thus are most suitable for experiments in studying birational geometry by means of the method of maximal singularities. It was these varieties that made the first examples of birationally (super)rigid Fano varieties of arbitrary dimension [P1].

The new technique of studying Fano fibrations, which was developed in [P4], made it possible at once to prove superrigidity of varieties with a pencil of double quadrics ([P4], Sec. 8). However, the degree 4 turned out to be too high to obtain automatically a similar result for the double quadrics. It was obvious, though, that such a result should be true. The methods of the paper [P4] were strong enough for the double covers of the direct products only [P3], but they failed to exclude a maximal singularity over the vertex of a cone.

The paper [P3] was actually written in 1997 during my stay at the Max-Planck-Institut für Mathematik in Bonn. The technique of hypertangent systems, a modification of which yielded the main theorem of the present paper, was also developed there. I want to use this opportunity to thank the Institute for the marvellous conditions of work.

1 Varieties with a pencil of double quadrics

1.1 Construction

We study smooth projective (M + 1)-dimensional varieties V over the field C of complex numbers, M ≥ 4, which permit a morphism π: V → P^{1}, the fibers of which are double quadrics of index 1. More precisely, let

E = ⨁_{i=0}^{M+1} O_{P^{1}}(a_{i})

be a locally free sheaf of rank M + 2 on P^{1}, where for convenience we assume that 0 = a_{0} ≤ a_{1} ≤ … ≤ a_{M+1}. Its projectivization X = Proj(E) = P(E^{∗}) is a P^{M+1}- bundle over the line. The projection onto the base is denoted by the same symbol π.

Let G be a fiber of the morphism π, then Pic X = ZL ⊕ ZG, where L is the class of the tautological sheaf O_{X/P^{1}}(1), i.e. π_{∗}O_{X}(L) = E. By our assumption the sheaf O_{X}(L) is generated by global sections, i.e. the linear system |L| is free. The canonical class of the variety X is equal to

K_{X} = −(M + 2)L + ( ∑_{i=1}^{M+1} a_{i} − 2) G.

Consider a smooth divisor Q ∈ |2L + μG| on X, μ ∈ Z. Its fibers Q_{t} = Q ∩ π^{−1}(t) over the points t ∈ P^{1} are quadric hypersurfaces. The canonical class of the variety Q is equal to

K_{Q} = (−ML + ( ∑_{i=1}^{M+1} a_{i} + μ − 2) G)|_{Q} .

Now take a prime Weil divisor W ∈ |2(M − 1)L + 2λG| on X such that W ∩ Q is a smooth divisor on Q.

Let V be the (smooth) double cover σ: V → Q, branched over W ∩ Q. The just constructed family of smooth varieties V is denoted by V(a_{∗}, μ, λ). The projection of V onto P^{1} we denote by the symbol π, and its fiber — by the symbol F_{t}, t ∈ P^{1}. Obviously, Pic V = ZK_{V} ⊕ ZF and the morphism π: V → P^{1} fibers V into Fano varieties of index 1 and degree 4. The canonical class of the variety V takes the form

K_{V} = −σ^{∗}(L|_{Q}) + ( ∑_{i=1}^{M+1} a_{i} + μ + λ − 2) F.

1.2 The condition of general position

We will assume that the following three conditions of general position hold.

(G1) If Q_{t} ⊂ P^{M+1} is a singular quadric, then Q_{t} has exactly one singular point, that is, Q_{t} is a cone over a non-singular quadric in P^{M}.

(G2) If the variety W ∩ Q_{t} is singular (this may happen for a finite number of points t ∈ P^{1} only), then Q_{t} is a smooth quadric, the set Sing W ∩ Q_{t} is finite and each singularity x ∈ W ∩ Q_{t} is a non-degenerate quadratic singularity. Moreover, the quadric tangent cones

T_{x}(T_{x}Q_{t} ∩ Q_{t}) and T_{x}(W ∩ Q_{t})

are in general position, that is, the intersection of their projectivizations in the projective space P(T_{x}Q_{t}) ≅ P^{M−1} is a smooth variety of dimension M − 3 and degree 4.

(G3) If Q_{t} ⊂ P^{M+1} is a quadric cone with the vertex x, then consider a system of linear coordinates (z_{1}, …, z_{M+1}) on C^{M+1} ⊂ P^{M+1} with the origin at x. Let q = 0 and g = w_{0} + w_{1} + … + w_{2(M−1)} be the equations of the quadric Q_{t} and the hypersurface W_{t} = W ∩ G_{t}, respectively. First of all, w_{0} ≠ 0, i.e. x ∉ W. From this moment on we assume that w_{0} = 1.

Consider the power series (1 + a)^{1/2} = 1 + ∑ γ_{i}a^{i} = 1 + \frac{1}{2}a + …, where a is nothing but a formal variable. Set

\begin{matrix}\sqrt{g} = (1 + w_{1} + … + w_{2(M−1)})^{1/2} = 1 + ∑_{i=1}^{∞} γ_{i}(w_{1} + … + w_{2(M−1)})^{i} = \\ = 1 + ∑_{i=1}^{∞} Φ_{i}(w_{1}, …, w_{2(M−1)}), \end{matrix}

where Φ_{i}(w_{1}(z_{∗}), …, w_{2(M−1)}(z_{∗})) are homogeneous of degree i in z_{∗}. Obviously, Φ_{i}(w_{∗}) = w_{i} + … for i ≤ 2(M − 1), where the dots stand for a polynomial in w_{1}, …, w_{i−1}. For k ≥ 1 set

[\sqrt{g}]_{k} = 1 + ∑_{i=1}^{k} Φ_{i}(w_{∗}), g^{(k)} = g − [\sqrt{g}]_{k}^{2}.

It is easy to see that the first non-zero homogeneous component in g^{(k)} is of degree k + 1 in z_{∗}. This component for k + 1 ≤ 2(M − 1) is of the form w_{k+1} + …, where the dots stand for a polynomial in w_{1}, …, w_{k}.

Now our condition of general position (G3) is formulated as follows. For a certain system of coordinates z_{∗} the set of polynomials

q, g^{(M−1)}, …, g^{(2M−3)}

makes a regular sequence in O_{x,P^{M+1}} = O_{x,C^{M+1}}. Moreover, in a neighborhood of the point x the system of equations

q = g^{(M−1)} = … = g^{(2M−4)} = 0

defines a surface, no one component of which lies in a hyperplane, containing the point x. In other words, if l(z) is an arbitrary non-zero linear form, then the sequence

l, q, g^{(M−1)}, …, g^{(2M−4)}

remains regular in O_{x,P^{M+1}}. For M = 4 we assume in addition that for each linear form l(z) the scheme l = q = g^{(3)} = g = 0 is reduced at the generic point.

It is easy to see that for a general variety V ∈ V(a_{∗}, μ, λ) the conditions (G1-G3) are true: each of them deals with a finite set of points.

1.3 The main result

Now let us give a precise formulation of our main result.

Theorem 1^{∗}. Let V ∈ V(a_{∗}, μ, λ) be a variety, satisfying the conditions (G1-G3) and the K^{2}-condition. Then the Fano fibration π: V → P^{1} is birationally superrigid.

Note that since the linear system |L| is free, the K^{2}-condition is guaranteed by the inequality

(K_{V}^{2} · σ^{∗}(L|_{Q})^{M−1}) ≤ 0.

Explicitly, this inequality takes the form

3μ + 4λ + 2 ∑_{i=1}^{M+1} a_{i} ≥ 8.

For instance, if X = P^{1} × P^{M+1}, i.e. a_{1} = … = a_{M+1} = 0, then for μ ≥ 3 we have no restrictions for the hypersurface W except for the conditions of general position. If μ = 0, i.e. Q = P^{1} × Q_{t}, then we get the case, which was studied in [P3]. In any case, however, we get one more confirmation of the general principle: “twistedness” over the base implies rigidity.

2 Proof of Theorem 1^{∗}

2.1 The method of maximal singularities

The principal concepts, constructions and theorems of the method have been published many times, starting from the fundamental paper of V.A.Iskovskikh and Yu.I.Manin on the quartic [IM], see [IP,P2,P4]. Therefore there is no need to reproduce the details here. For Fano fibrations over P^{1} the technique of the method of maximal singularities was developed in [P4] and later used in [P3,P5]. In the present paper we formulate a sufficient condition of superrigidity in terms of degrees and multiplicities of algebraic cycles and verify it for our Fano fibrations. The details of the proof of this sufficient condition the reader can find in [P3-P5].

Let π: Y → P^{1} be a smooth Fano fibration with Pic Y = ZK_{Y} ⊕ π^{∗} Pic P^{1}, satisfying the K^{2}-condition, i.e. the numerical class NK_{Y}^{2} − f, where f ∈ A^{2}Y is the class of a hyperplane section of a fiber F_{t} = π^{−1}(t), does not represent an effective cycle of codimension 2 for any N ∈ Z. Following [P3,P4], we call an irreducible cycle B ⊂ Y horizontal, if it covers the base, π(B) = P^{1}, and vertical, if it lies in a fiber of π. The degree of a vertical cycle B is defined to be the integer

deg B = (B · (−K_{Y})^{dim B}) > 0.

The degree of a horizontal cycle B is defined to be the integer deg B = deg(B ∩ F_{t}), where F_{t} is a fiber of general position.

Proposition 2.1. (a sufficient condition of birational superrigidity) Assume that for each moving linear system Σ ⊂ | − nK_{Y} + lF |, each irreducible vertical cycle A ⊂ F_{t} of codimension 2 (with respect to Y), each irreducible horizontal cycle B ⊂ Y of codimension 2, each irreducible vertical cycle C ⊂ F_{t} of codimension 3 (with respect to Y) and each point x ∈ F_{t} the following conditions are satisfied:

(i) mult_{A} Σ ≤ n, mult_{B} Σ ≤ n;

(ii) mult_{x} C ≤ (4/ deg Y) mult_{x} F_{t} deg C;

(iii) mult_{x} A ≤ (2 mult_{x} F_{t}/ deg Y) deg A, mult_{y} Ã ≤ (2/ deg Y) deg A, where y ∈ E_{x} ⊂ F̃_{t} is an arbitrary infinitely near point over x ∈ F_{t} and Ã is the proper inverse image of A with respect to the blow-up of the point x, φ: F̃_{t} → F_{t}, E_{x} = φ^{−1}(x).

Then the Fano fibration π: Y → P^{1} is birationally superrigid.

Here deg Y = (F · (−K_{Y})^{dim Y −1}), in accordance with the definition of degree, given above.

For the proof, see [P3-P5]. The scheme of arguments is as follows. If Y → P^{1} is not birationally superrigid, then there is a moving linear system Σ ⊂ | − nK_{Y} + lF | on Y with a maximal singularity ν. If the centre of the maximal singularity is of codimension 2, we get a contradiction with the condition (i). Let x ∈ F_{t} be a point of the centre. Consider the effective cycle Z = (D_{1} • D_{2}), D_{i} ∈ Σ, and decompose it into the vertical and horizontal components

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

where the support of the cycle Z_{s}^{v} is contained in the fiber F_{s}. Note that by the condition (ii)

mult_{x} Z^{h} ≤ 4n^{2}.  (1)

Indeed, take the effective cycle C = (Z^{h}•F_{t}). Obviously, mult_{x} C ≥ mult_{x} Z^{h} mult_{x} F_{t}, whereas deg Z^{h} = deg C = n^{2} deg Y. Applying (ii), we get the estimate (1). Furthermore, by the condition (iii) we get

mult_{x} Z_{t}^{v} ≤ 4nl.  (2)

We may assume that the maximal singularity ν is supermaximal [P3,P4]. Now arguing as in [P3,P4], we get a contradiction.

Thus to prove Theorem 1^{∗} it is sufficient to check that the conditions (i)-(iii) are satisfied for our Fano fibration π: V → P^{1}.

2.2 Maximal cycles

The condition (i) has been actually verified in [P3]. Since a linear system on a smooth double quadric of dimension ≥ 4 can not have a maximal singularity [P1], a moving linear system Σ on V can not have maximal singularities covering the base, i.e. mult_{B} Σ ≤ n for each horizontal cycle B ⊂ V of codimension 2. Furthermore, for each Weil divisor A ⊂ F_{t} we have: σ(A) ⊂ Q_{t} is a prime Weil divisor on the quadric Q_{t}. Divisors of the family σ(Σ|_{F_{t}}) are cut out on Q_{t} by hypersurfaces of degree 2n, so that if mult_{A} Σ > n, then σ(A) = H ∩ Q_{t} is a hyperplane section and, moreover, the morphism σ: A → σ(A) is birational. Therefore the scheme-theoretic intersection H ∩ Q_{t} ∩ W is non-reduced at every point. But this is impossible: by Lemma 3.4.2 from [IP] dim Sing H ∩ Q_{t} ∩ W ≤ 1. Contradiction.

Q.E.D. for the condition (i).

2.3 Proof of condition (ii)

Obviously, deg V = 4, so that (ii) takes the form of the inequality

mult_{x} C ≤ mult_{x} F_{t} deg C,

which is certainly true, since a stronger inequality is true: mult_{x} C ≤ deg C.

2.4 Proof of condition (iii)

If z = σ(x) is a smooth point of the quadric Q_{t}, then we prove (iii), repeating the arguments of the paper [P3] word for word. The scheme of the proof is as follows. There are two possibilities: σ(A) = T_{z}Q_{t} ∩ Q_{t} and σ(A) ≠ T_{z}Q_{t} ∩ Q_{t}. In the first case (iii) can be verified directly, taking into account (G1), (G2) and the above-mentioned Lemma 3.4.2 from [IP]. In the second case we obtain (iii), applying the estimate (ii), which has already been proved, to the effective cycle (A • σ^{−1}(T_{z}Q_{y} ∩ Q_{t})) of codimension 2.

Assume that z = σ(x) is the vertex of the cone Q_{t}. First of all, note that of the two inequalities (iii) the first one in this case is trivial: it asserts that mult_{x} A ≤ deg A. Assume that for a prime Weil divisor A ⊂ F_{t} the second inequality is not true:

mult_{y} Ã > \frac{1}{2} deg A,

where y ∈ E_{x} = φ^{−1}(x) ⊂ F̃_{t} is an infinitely near point of the first order over x, φ: F̃_{t} → F_{t} is the blowing up of x. Let us show that this assumption leads to a contradiction. We do it in three steps.

\underline{Step 1.} The point z = σ(x) does not lie on the branch divisor, so that σ(y) is a well-defined infinitely near point of the first order over z, defining a line (z, σ(y)) ⊂ Q_{t}. Denote by T_{y}Q_{t} the hyperplane in P, which is tangent to Q_{t} along this line. Set T = σ^{−1}(T_{y}Q_{t} ∩ Q_{t}) ⊂ F_{t}. Obviously, mult_{y} T̃ = 2, deg T = 4, so that the condition (iii) holds for T. In particular, A ≠ T.

\underline{Step 2.} Set B = E_{x} ∩ T̃ (if T_{y}Q_{t} ∩ Q_{t} is looked at as an affine cone with the vertex x, then B is its projectivization).

Lemma 2.1. The following estimate holds:

mult_{B} Ã ≥ \frac{1}{4}(2 mult_{y} Ã − mult_{x} A).

Proof: see [P5], Sec. 3.6.

Now consider an algebraic cycle (A • T) = C. It is easy to see that C satisfies the inequality

mult_{x} C > \frac{3}{4} deg C.

\underline{Step 3.} Let us show that this is impossible. Let us assume that M ≥ 5. Obviously, the cycle C can be assumed to be irreducible. Let H = T_{y}Q_{t} ≅ P^{M} be the tangent hyperplane, σ: R → H the double cover, branched over the hypersurface W_{H} = W ∩ H. The variety R is irreducible and possibly singular. The cycle T naturally identifies with a divisor in R. Thus C can be identified with a codimension 2 cycle in R, too. Meaning this identification, we write x ∈ R etc.

Let P ∈ Pic R be the class of a hyperplane section, i.e. O_{R}(P) ≅ σ^{∗}O_{H}(1), H ↪ P. We define the k-th hypertangent system at the point x ∈ R by the condition

Λ_{k} = { D ∈ |kP | | mult_{x} D ≥ k + 1}.

Lemma 2.2. The hypertangent linear systems Λ_{k} are non-empty for k = M − 1, …, 2M − 3. Moreover,

codim_{T} Bs(Λ_{M−2+i}|_{T}) ≥ i

for i = 1, …, M − 2.

Proof. It is immediately implied by the condition (G3), where we take H as that hyperplane, containing the point p = σ(x) ∈ Q_{t} ∩ H. Indeed, let z_{1}, …, z_{M+1} be a system of linear coordinates with the origin at p, where P ⊃ H = {z_{M+1} = 0}. Take U = C_{(z_{1},...,z_{M})}^{M}. Over the set U the variety R is defined by the equation y^{2} = g(z_{1}, …, z_{M}, 0) in C_{(y)} × U. The coordinate y defines a section of the sheaf O_{R}((M − 1)P). Therefore, the functions

f_{k} = y − [\sqrt{g}]_{k},

regular on C×U, for k ≥ M −1 define sections of the sheaves O_{R}(kP), respectively. Moreover, for k ≥ M − 1 set

Δ_{k} = \overline{\{f_{i} = 0\}} ⊂ R.

It is easy to see that Δ_{k} ∈ Λ_{k}. The image of the divisor Δ_{k} on U = C^{M} is given by the equation g^{(k)}(z_{1}, …, z_{M}, 0) = 0. Now applying (G3), we get the assertion of our lemma.

Let D_{i} ∈ Λ_{M−2+i}, i = 1, …, M − 1, be general divisors. Let us construct a sequence of irreducible cycles C_{2}, C_{3}, …, C_{M−1} ⊂ R, codim C_{j} = j, in the following way. Set C_{2} = C. Assume that C_{j}, j = 2, …, l, have already been constructed and l ≤ M − 2. By Lemma 2.2 a general divisor D_{l} does not contain C_{l}. By the definition of hypertangent systems there exists an irreducible component Z of the cycle (C_{l} • D_{l}), satisfying the inequality

\frac{mult_{x}}{deg} Z ≥ \frac{mult_{x}}{deg} C_{l} · \frac{M − 1 + l}{M − 2 + l};

set C_{l+1} = Z. The cycle C_{M−1} ⊂ R is an irreducible curve. We get the estimate

\frac{mult_{x}}{deg} C_{M−1} > \frac{3}{4} · \frac{M + 1}{M} · \frac{M + 2}{M + 1} · · · · · \frac{2M − 3}{2M − 4}

∥ \frac{3(2M − 3)}{4M},

and it is easy to see that 3(2M − 3)/4M ≥ 1, which is impossible.

Now assume that M = 4. Here the linear system Λ_{3} contains the single effective divisor D_{1}. There are two possible cases: either C ⊂ D_{1}, or C ⊄ D_{1}. In the first case (iii) is implied by the condition of general position, since C = T ∩ D_{1}. In the second case the cycle C_{3} = (C • D_{1}) is effective. However, it satisfies the inequality (mult_{x} / deg)C_{3} > 1, which is impossible. Q.E.D. for the property (iii) and thus for Theorem 1^{∗}.

3 Three-dimensional case

Most of the arguments above work for M = 3 without any modifications. There are, however, two points of possible complications. First of all, the dimension no more allows to use Lemma 3.4.2 from [IP]. However, imposing additional conditions of general position on V, we can avoid using this lemma. The reader will easily reconstruct the necessary details.

The second complication is more essential: in dimension three maximal cycles can appear on V! The reason is that on a smooth three-dimensional double quadric of index 1 each line outside the branch divisor can be a maximal singularity, see [IP]. Accordingly, the group of birational self-maps of this variety is rather big (it was described by V.A.Iskovskikh in the mid-70s).

Let L(V) be the (quasi-projective) variety of lines in the fibers of the morphism π: V → P^{1}, which lie outside the branch divisor, i.e.

L(V) = {L ⊂ F_{t} | (L · K_{V}) = −1, σ(L) ⊄ W }.

The natural projection L(V) → P^{1} will be denoted by the same symbol π. Each section (defined, generally speaking, outside a finite set of points) s: P^{1} → L(V), π ∘ s = id, defines a fiber-wise birational self-map τ_{s} ∈ Bir F_{η} ⊂ Bir V, where F_{η} is the fiber over the generic point, that is, a Fano variety over the non-closed field C(P^{1}). The involutions τ_{s} freely generate in Bir V a normal subgroup

B(V) = ∗ ; \begin{matrix}s: P^{1} → L(V) \\ π ∘ s = id \end{matrix} ⟨τ_{s}⟩,

see [IP]. The following exact sequence

1 → B(V) → Bir V → Aut V → 1  (3)

holds, where the group Aut V is finite, generically isomorphic to Z/2Z.

Repeating the arguments of Sec.2 word for word and taking into account these remarks, we get the following result.

Theorem 2. A general four-dimensional Fano fibration V, satisfying the K^{2}- condition, is birationally rigid. Its group of birational self-maps is described by the exact sequence (3).

One may ask: do the sections s: P^{1} → L(V) exist for a general Fano fibration V? The author does not know the answer, but it seems to be negative. If it is negative, then for M = 3 we also get birational superrigidity.

References

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

[IP] Iskovskikh V.A. and Pukhlikov A.V., Birational automorphisms of multi-dimensional algebraic varieties. J. Math. Sci. 82 (1996), 3528-3613.

[CPR] Corti A., Pukhlikov A. and Reid M., Fano 3-fold hypersurfaces, Warwick Preprint 10/1999, to appear in “Birational Geometry of Threefolds”

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[K] Kollár J., et al., Flips and Abundance for Algebraic Threefolds, Asterisque 211, 1993.

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

[P2] Pukhlikov A.V., Essentials of the method of maximal singularities, Warwick Preprint 31/1996, to appear in “Birational Geometry of Threefolds”

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