BIRATIONAL PROPERTIES OF PENCILS OF DEL PEZZO SURFACES OF DEGREE 1 AND 2.

MIKHAIL GRINENKO

Abstract. In this paper we study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and non-rigid cases.

1. Notions and results.

Due to progress in the Minimal Model Program, the classification problem in the modern birational geometry for varieties of negative Kodaira dimension can be formulated as follows: given a class of birational equivalency, to describe all Mori fibrations and birational maps between them.

Recall that a normal variety V with at most ℚ-factorial terminal singularities is called a Mori fibration if there exists an extremal contraction φ : V → S of fibering type, i.e.

1) φ is a morphism with connected fibers onto a normal variety S and dim S < dim V;

2) −K_{V} is φ-ample and the relative Picard number is equal to 1: ρ(V/S) = ρ(V) − ρ(S) = 1.

Note that often comparing birational classes of varieties, we suffice to know Mori structures rather than Mori fibrations themselves (roughly speaking, Mori fibrations modulo birational maps over the base). From this viewpoint, the following class of varieties is rather important (and simple for describing):

Definition 1.1. A Mori fibration V/S is said to be birationally rigid, if any birational map χ : V −−→ V′ onto another Mori fibration V′/S′ is birational over the base (”square”), i.e., there exists a birational map ψ : S −−→ S′ making the following diagram to be commutative:

This work was partially supported by the grants RFBR no. 99–01–01132, Grant of Leading Scientific Schools no. 96–15–96146, and INTAS-OPEN 97/2072.

\begin{matrix}V & −−→^{χ} & V′ \\ ↓ & & ↓ \\ S & −−→^{ψ} & S′ \end{matrix}

Now let ρ : V → S be a Mori fibration. Then any non-empty linear system D on V is a subsystem of |a(−K_{V}) + ρ^{∗}(A)|, where a = μ(V, D) ≥ 0 is the so-called quasi-effective threshold, and A is a divisor on S. The behaviour of linear systems under birational maps between Mori fibrations is given by the following theorem ([1], theorem 4.2):

Proposition 1.2. Let ρ : V → S and ρ′ : V′ → S′ be Mori fibrations, χ : V −−→ V′ a birational map, D′ = |n′(−K_{V′}) + ρ′^{∗}(A′)| a very ample linear system on V′, where A′ is an ample divisor on S′. Denote D = χ_{∗}^{−1}D′ ⊂ |n(−K_{V}) + ρ^{∗}(A)|. Then:

(i) n ≥ n′ always, and if n = n′ then χ is square; (ii) if a log pair K_{V} + \frac{1}{n}D is canonical and numerically effective, then

χ is an isomorphism inducing also an isomorphism between the bases.

The statement (i) of the proposition allows us to describe various Mori structures on V, while (ii) can be used to get Mori models (i.e., various Mori fibrations birational to V).

There is another important number arising from the maximal singularities method (now playing the key role in the birational classification). This is so-called the adjunction threshold.

Definition 1.3. A non-negative number α(V, D) is called the adjunction threshold of a pair (V, D), where D is a non-empty linear system without fixed components, V is projective and non-singular in codimension 1, if α(V, D) is the smallest number such that for any positive integers m and n with \frac{m}{n} > α(V, D) the linear system |nD + mK_{V} | is empty, where D is an element of D.

The following statement clarifies the role of the adjunction threshold ([10], 2.1.):

Proposition 1.4. Let χ : V → V′ be a birational map between terminal ℚ-factorial varieties, D′ a non-empty linear system on V′ without fixed components, and D = χ_{∗}^{−1}D′ its strict transform. If α(V, D) > α(V′, D′), then a pair K_{V} + \frac{1}{α(V,D)}D is not canonical (in other words, D has a maximal singularity)

This condition ”to be not canonical” is strong enough, and we can often either prove its impossibility, or decrease the adjunction threshold applying an ”untwisting” birational automorphism or jumping to another Mori model (as in the Sarkisov program). A relation between both the thresholds is given by the following obvious lemma:

Lemma 1.5. Let D ⊂ |n(−K_{V}) + ρ^{∗}(A)| be a non-empty linear system without fixed components on a Mori fibration ρ : V → S. Always n = μ(V, D) ≥ α(V, D), and if A is effective, then the equality holds; if C ∘ ρ^{∗}(A) < 0 for a general curve C covering the base, then μ(V, D) > α(V, D).

The key idea is that the relation between these thresholds determines whether a variety is rigid. For Mori fibrations that are pencils of del Pezzo surfaces, it looks as follows:

Conjecture 1.6. Let V/ℙ^{1} be a pencil of del Pezzo surfaces of degree 1,2 or 3 (we assume it to be a Mori fibration). V/ℙ^{1} is birationally rigid if and only if α(V, D) = μ(V, D) for any linear system without fixed components (i.e., a linear system |n(−K_{V}) − F | either is empty, or has a base divisor; here F is the class of a fiber).

Let us note that ”nearly all” (smooth) pencils of del Pezzos of degree 1, 2 or 3 are sufficiently ”twisted” along the base, and in [9] their rigidity is proved. In this paper, we deal with all cases of degree 1 (theorem 2.6) and advance in degree 2 (theorem 3.1). In particular, conjecture 1.6 turns out to be true for smooth cases of degree 1.

The paper is based on using the maximal singularities method ([4], [6]) in its most perfect kind [10].

Everywhere in this paper the characteristic of the ground field is assumed to be 0.

The author would like to thank the Directors and the stuff of the Max-Planck Institute for Mathematics in Bonn for hospitality and excellent conditions during the work.

2. Smooth varieties with a pencil of del Pezzo surfaces of degree 1.

2.1. The essential construction. We follow to the idea given in [5].

Let ρ : V → ℙ^{1} be a smooth Mori fibration on del Pezzo surfaces of degree 1, Pic V = ℤ[−K_{V}] ⊕ ℤ[F], where F is the class of a fiber. By the Grauert theorem, ρ_{∗}O(−2K_{V} + mF) is a vector bundle of rank 4 on ℙ^{1}. We may choose m such that

E = ρ_{∗}O(−2K_{V} + mF) ≃ O ⊕ O(n_{1}) ⊕ O(n_{2}) ⊕ O(n_{3}),

where 0 ≤ n_{1} ≤ n_{2} ≤ n_{3}. Suppose b = n_{1} + n_{2} + n_{3}.

Note that there exists a special section s_{B}:

s_{B} = Bas| − K_{V} + kF |

for k ≫ 0. Obviously, any fiber S ∈ |F | is smooth at the point s_{B} ∩ S, which is also a unique base point of | − K_{S}|.

Let π : X → ℙ^{1} be a natural projection for X =^{def} Proj E, φ : V → X a morphism defined by | − 2K_{V} + mF |. Then ρ = π ∘ φ. It is easy to see that if Q ⊂ X is the image of φ, then φ : V → Q is a degree 2 finite morphism branched along a smooth divisor R_{Q} = R ∩ Q that does not intersect t_{B} =^{def} φ(s_{B}), where R ⊂ X is a divisor.

Q is restricted to any fiber T of X/ℙ^{1} as a non-degenerated quadratic cone with T ∩ t_{B} as its vertex, and R|_{T} is a cubic surface. So if M is the class of the tautological bundle on X and L is the class of a fiber, we have

Q = 2M + aL; R = 3M + cL.

Let t_{0} be a section corresponding to the surjection E → O → 0, l the class of a line in L ≅ ℙ^{3}; set t_{B} ∼ t_{0} + εl, where ε is obviously nonnegative.

Choose a very ample divisor H ∼ M + βl, β > 0. So D = H ∩ Q is a conic bundle with N = t_{B} ∘ H degenerations. We may suppose that all fibers of D are reduced and all its singular points are du Val A_{1}. It is easy to compute that K_{D}^{2} = 8 − 2b − 2β − 3a. Let us blow up all singular points of D and then contract all (-1)-curves (there will be 2N of them) onto a ruled surface D′. We have

K_{D}^{2} + 2N = K_{D′}^{2} = 8,

hence

N = b + β + \frac{3}{2}a.

Note that a must be even. Set a = 2a′. Then,

N = t ∘ H = (t_{0} + εl) ∘ (M + βl) = β + ε,

and we obtain ε = b + 3a′. Since R|_{Q} ∩ t_{B} = ∅, we have 0 = (t_{0} + εl) ∘ (3M + cL) = 3ε + c. So, we have proved the following lemma:

Lemma 2.1. The following equalities hold:

Q = 2M − \frac{2}{3}(b − ε)L,
t_{B} = t_{0} + εl,
R = 3M − 3εL.

In the sequel we will always assume b > 0. Really, if b = 0, then Q = 2M and ε = 0, whence V ≅ ℙ^{1} × S for some smooth del Pezzo surface S. So V is rational and even not a Mori fibration because of the Picard group rank.

Lemma 2.2. The only following two cases may occur:

1) ε = 0, 2n_{2} = n_{1} + n_{3}, n_{1} and n_{3} are even; so we have

Q = 2M − 2n_{2}L, t_{B} = t_{0}, R = 3M;

2) ε = n_{1} > 0, n_{3} = 2n_{2}, n_{1} is even and n_{2} ≥ 3n_{1}; in this case

Q = 2M − 2n_{2}L, t_{B} = t_{0} + n_{1}l, R = 3M − 3n_{1}L.

Proof. Suppose first ε > 0. Then ε ≥ n_{1} because of irreducibility of t_{B}. Since R is irreducible and curves of the class t_{0} + n_{2}l sweep any effective divisor of the class M − n_{3}L, we have (t_{0} + n_{2}l) ∘ R = 3n_{2} − 3ε ≥ 0, i.e., ε ≤ n_{2}.

Let n_{1} = n_{2} = n_{3}. Since t_{B} ∘ (M − n_{1}L) = 0, the linear system |M − n_{1}L| is two-dimensional, and any of its elements is irreducible, there exists at least one-dimensional subsystem in it that contains t_{B}. Let C ∈ |M − n_{1}L − t_{B}| be a general element, then

Q ∘ C = S + S′ ∼ 2M^{2} − 2 (\frac{1}{3}(b − ε) + n_{1}) ML,

where S and S′ are ruled surfaces covering the base. Note that there exists only a unique curve of the class t_{0}, and t_{B} ∩ t_{0} = ∅. At least one of these surfaces, say, S, containes both t_{0} and t_{B}. It is easy to see that S is of the class M^{2} − 2n_{1}ML, whence S′ ∼ M^{2} − \frac{4}{3}n_{1}ML. Since S ≠ S′, then t_{0} ⊄ S′, and denoting by P the intersection of two general elements of |M − n_{1}L|, we can see that dim(S′ ∩ P) = 0 (note that Bas |M − n_{1}L| = t_{0}). We get a contradiction:

0 ≤ S′ ∘ P = (M^{2} − \frac{4}{3}n_{1}ML) ∘ (M^{2} − 2n_{1}ML) = −\frac{n_{1}}{3} < 0.

So if ε ≥ n_{1} then n_{1} < n_{3}; moreover, n_{1} > 0 because of irreducibility of R_{Q} (otherwise, if n_{1} = 0, there should be at least 1-dimensional family of lines of the class t_{0} sweeping a surface, and R_{Q} should contain it).

Let G be an irreducible ruled surface that is the complete intersection of general elements of |M − n_{2}L| and |M − n_{3}L|. Let us show that t_{B} ⊂ G. Indeed, G has the type F_{n_{1}} with t_{0} as its exceptional section. Assume the converse, i.e., t_{B} ⊄ G. Then, for a general fiber S ∈ |L|, the line G|_{S} does not lie on Q|_{S}. So G ⊄ Q, and we may suppose that G ∘ Q = t_{0} + C, where C is a curve with no t_{0} as a component. It is easy to compute that

C ∼ t_{0} + 2 (b − n_{2} − n_{3} − \frac{b − ε}{3}) l,

and

2 (b − n_{2} − n_{3} − \frac{b − ε}{3}) ≥ n_{1},

which, taking into account ε ≤ n_{2}, gets a contradiction: n_{1} > \frac{4}{3}n_{1}.

So t_{B} ⊂ G, hence n_{2} < n_{3}. In fact, the equality should led dim |M − n_{3}L|, thus Bas |M − n_{3}L| = t_{0}, and we could always find G such that t_{B} ⊄ G.

By T we denote a unique effective element of |M − n_{3}L|. We show that T |_{Q} = 2G. Indeed, assuming the converse, i.e., t|_{Q} = G + G′, G ≠ G′, and taking into account that t_{B} ⊂ G ∩ G′ and t_{0} ⊂ G, we see t_{0} ⊄ G′. Then

G′|_{G} = t_{0} + #(t_{0} ∩ t_{B} = ε − n_{1}) fibers.

Choose a general S ∈ |M − n_{1}L|. It is clear that S|_{T} is a ruled surface of the type F_{n_{2}} with the exceptional section t_{0}, and G ∩ S|_{T} = t_{0}. Let C = G′ ∩ S|_{T} be an irreducible curve. We have C ∘ t_{0} = ε − n_{1}, Q ∘ T ∘ S = t_{0} + C, whence

C ∼ t_{0} + 2 (n_{2} − \frac{b − ε}{3}) l.

This yields C ∘ t_{0} = n_{2} − \frac{2}{3}(b − ε) = ε − n_{1}, and we get a contradiction again:

ε = n_{1} + n_{2} − 2n_{3} < 0.

So we have

T ∘ Q = 2M^{2} − 2 (n_{3} + \frac{b − ε}{3}) ML = 2G = 2(M^{2} − (n_{2} + n_{3})ML),

thus 2n_{2} = n_{1} + n_{3} − ε. If ε were greater than n_{1}, then R_{Q} should contain G, because G is covered by curves of the class t_{0} + n_{1}l. This contradicts to irreducibility of R_{Q}. So ε = n_{1} and then 2n_{2} = n_{3}. In order to show n_{2} ≥ 3n_{1}, we note that S_{Q} = S ∩ Q is a conic bundle without degenerations, i.e., a ruled surface of the type F_{n_{2}}. Then R cuts off an effective curve on S_{Q} that does not contain t_{0}, and t_{0} itself. We get 3n_{3} − 3n_{1} ≥ 5n_{2}, which yields n_{2} ≥ 3n_{1}.

It remains to show that n_{1} is even. Let C ∼ t_{0} + n_{1}l be a general irreducible curve on G, and C_{V} its inverse image on V. Then C_{V} covers C with the ramification divisor of degree t_{B} ∘ C = n_{1}. Clearly, this degree must be even. This completes the case ε > 0.

Now let ε = 0. We may assume n_{1} < n_{3}. Take T ∈ |M − n_{3}L|. It is easy to observe that T = Proj E_{T}, where

E_{T} = O ⊕ O(n_{1}) ⊕ O(n_{2}).

Since T can be covered by curves of the class t_{0}+n_{2}l and Q ∼ 2M − \frac{2}{3}bL is irreducible, then n_{2} ≥ \frac{b}{3}, i.e., 2n_{2} ≥ n_{1} + n_{3}, and n_{2} > n_{1} since n_{3} > n_{1}.

Further, let C ∼ t_{0}+n_{1}l be irreducible. Since b = n_{1}+n_{2}+n_{3} > 3n_{1}, then C ∘ Q = 2(n_{1} − \frac{b}{3}) < 0, and therefore C ⊂ Q. A ruled surface G ∼ (M −n_{2}L)∘(M −n_{3}L) = M^{2} −(n_{2} +n_{3})ML has the type F_{n_{2}} and can be covered by curves of the same class as C, so G ⊂ Q. Therefore, T |_{Q} = G + G′ for some G′.

Let M_{T} = M|_{T} be the tautological divisor on T, L_{T} = L|_{T}. Then G has the class M_{T} − n_{2}L_{T} in T, Q|_{T} ∼ 2M_{T} − \frac{2}{3}bL_{T}, whence G′ ∼ M_{T} − (\frac{2}{3}b − n_{2})L_{T}. Note that \frac{2}{3}b − n_{2} > n_{1} since n_{1} < n_{2}.

But T has only one divisor of the form M_{T} − xL_{T} for x > n_{1}: this is M_{T} − n_{2}L_{T}. So \frac{2}{3}b − n_{2} = n_{2}, hence n_{1} + n_{3} = 2n_{2}.

Now, let C ⊂ Q be a general irreducible curve of the class t_{0} + n_{1}l, C_{V} its inverse image on V. Then C_{V} is the double cover of C branched along a divisor of degree R ∘ C = 3n_{1}, whence n_{1} and n_{3} are even. Lemma 2.2 is proved.

Such a construction of V allows us to deal completely with intersections of cycles on V. Denote G_{V}, F, and H the inverse images of G ∼ (M − n_{2}L) ∘ (M − n_{3}L), L_{Q} = L|_{Q}, and M_{Q} = M|_{Q} on V respectively. Then, let 2s_{0} be the pull back of t_{0}, and 2f the pull back of a line in a fiber of L_{Q}.

Lemma 2.3. \overline{NE}(V) = NE(V) = ℝ_{+}[s_{0}] ⊕ ℝ_{+}[f], and:

1) if ε = 0, then K_{V} = −G_{V} + (\frac{1}{2}n_{1} − 2)F, H = 2G_{V} + n_{3}F, K_{V}^{2} = s_{0} + (4 − n_{2})f, s_{0} ∘ F = f ∘ G_{V} = 1, s_{0} ∘ G_{V} = −\frac{1}{2}n_{3}, f ∘ F = 0.

2) if ε = n_{1} > 0, then K_{V} = −G_{V} − (\frac{1}{2}n_{1} + 2)F, H = 2(G_{V} + n_{2}F), s = s_{0} + \frac{1}{2}n_{1}f, K_{V}^{2} = s_{0} + (4 + \frac{3}{2}n_{1} − n_{2})f, s_{0} ∘ F = f ∘ G_{V} = 1, s_{0} ∘ G_{V} = −\frac{1}{2}n_{3} = −n_{2}, f ∘ F = 0.

Proof. It can be easily checked by the following way. First blow up t_{B} on Q with an exceptional divisor S; then take the double cover branched along a smooth divisor composed from the pre-image of R_{Q} and S; after that, contract the pre-image of S onto s_{B}.

Lemma 2.4. Let ε = 0 (i.e., s_{B} = s_{0}). If |n(−K_{V}) − mF |, m > 0, has no fixed components, then n_{3} = 2.

Proof. Let D ∈ |n(−K_{V}) − mF |. Since

D|_{G_{V}} ∼ ns_{0} − (m + n (\frac{n_{1}}{2} + n_{3} − n_{2} − 2)) f

is an effective curve, we obtain

0 < \frac{m}{n} ≤ 2 + n_{2} − n_{3} − \frac{n_{1}}{2} = 2 − \frac{n_{3}}{2} .

It only remains to take into account that n_{3} is positive and even. The lemma is proved.

Lemma 2.5. Let ε = n_{1} > 0 (so s_{B} ≠ s_{0}). If |n(−K_{V}) + mF | has no fixed components, then m > 0.

Proof. Let D ∈ |n(−K_{V}) + mF |. Since

D|_{G_{V}} ∼ ns_{0} + [m + n(n_{1} − n_{2} + 2)]f

is an effective curve, we have 2 + n_{1} − n_{2} + \frac{m}{n} ≥ 0. Using n_{2} ≥ 3n_{1}, we see that 2 + n_{1} − n_{2} ≤ 2 − 2n_{1} < −2, whence m > 0. The lemma is proved.

2.2. Results about rigidity.

Theorem 2.6. Conjecture 1.6 is true for smooth Mori fibrations on del Pezzo surfaces of degree 1 (over ℙ^{1}).

In order to prove this theorem, we shall use the following proposition:

Proposition 2.7. Let V/ℙ^{1} be a smooth Mori fibration on del Pezzo surfaces of degree 1, V′/S′ another Mori fibration, χ : V −−→ V′ a birational map. Suppose that

D′ = |n′(−K_{V′}) + pull back of an ample divisor from the base |

is a very ample linear system. Suppose aslo

D = χ_{∗}^{−1}D′ ⊂ |n(−K_{V}) + mF |

for some m ≥ 0.

Then n = n′ (so χ is birational over the base), possibly except the case ε = n_{1} = 0, n_{2} = 1, n_{3} = 2.

Proof of the proposition. By lemma 1.5, μ(V′, D′) = n′ = α(V′, D′) and μ(V, D) = n = α(V, D). If n = n′, the assertion follows from proposition 1.2.

Suppose n > n′. Then a log pair K_{V} + \frac{1}{n}D is not canonical, i.e., the linear system D has a maximal singularity over some point. But using [9], we may say a little more.

Namely, let D_{1}, D_{2} be general elements of D. We observe that

D_{1} ∘ D_{2} = \{ \begin{matrix}n^{2}s_{0} + ((4 − n_{2})n^{2} + 2mn)f, & if ε = 0 \\ n^{2}s_{0} + ((4 + \frac{3}{2}n_{1} − n_{2})n^{2} + 2mn)f, & if ε = n_{1} > 0 \end{matrix}

Put down D_{1} ∘ D_{2} = Z^{h} + Z^{v}, where Z^{h} and Z^{v} are horizontal and vertical cycles respectively. Then there exist a geometric discrete valuation ν centered at a point B_{0} over some point t ∈ ℙ^{1}, and a positive number e = ν(D) − nδ, where δ is the canonical multiplicity with respect to ν, such that for the component Z_{t}^{v} ⊂ F_{t} of the cycle Z^{v} in the fiber F_{t} over the point t we have

deg Z_{t}^{v} < \{ \begin{matrix}(4 − n_{2})n^{2} + 2n\frac{e}{ν(F_{t})}, & if ε = 0 \\ (4 + \frac{3}{2}n_{1} − n_{2})n^{2} + 2n\frac{e}{ν(F_{t})}, & if ε = n_{1} > 0, \end{matrix}

where deg Z_{t}^{v} =^{def} Z_{t}^{v} ∘ (−K_{V}) (this is so-called the supermaximal singularity condition, in the notions of [9]). Besides, there exist positive numbers Σ_{0}, Σ′_{0} ≤ Σ_{0}, and Σ_{1} such that multiplicities m^{h} and m^{v} of the cycles Z^{h} and Z_{t}^{v} at B_{0} satisfy

(Σ_{0} + Σ_{1})(m^{h}Σ_{0} + m^{v}Σ′_{0}) > (2nΣ_{0} + nΣ_{1} + e)^{2}.

Note that m^{h} ≤ deg Z^{h} =^{def} Z^{h} ∘ F = n^{2} and m^{v} ≤ 2 deg Z_{t}^{v}. So, in the case ε = n_{1} > 0 we have

(Σ_{0} + Σ_{1})((9 + 3n_{1} − 2n_{2})n^{2}Σ_{0} + 4ne) > (2nΣ_{0} + nΣ_{1} + e)^{2},

hence

(2n_{2} − 3n_{1} − 5)Σ_{0}(Σ_{0} + Σ_{1}) + (nΣ_{1} − e)^{2} < 0,

i.e., 2n_{2} < 5 + 3n_{1}. Taking into account the condition n_{2} ≥ 3n_{1}, we obtain n_{1} ≤ 1, which is impossible because n_{1} must be even.

Now let ε = 0. Then

(Σ_{0} + Σ_{1})((8 − 2n_{2})n^{2}Σ_{0} + 4ne) > (2nΣ_{0} + nΣ_{1} + e)^{2},

whence

(2n_{2} − 5)Σ_{0}(Σ_{0} + Σ_{1}) + (nΣ_{1} − e)^{2} < 0.

So n_{2} < \frac{5}{2}, so the only three cases are possible:

n_{1} = 0, n_{2} = 2, n_{3} = 4;

n_{1} = 2, n_{2} = 2, n_{3} = 2;
n_{1} = 0, n_{2} = 1, n_{3} = 2.

We show that the first two cases can not occur. Suppose first that there are no sections of the class s_{0} through the point B_{0}. Let Z_{1}^{h} be the sum of all horizontal cycles through B_{0}. So Z_{1}^{h} ∼ As_{0} + Bf for some B ≥ A. Since n_{2} = 2 in the considering cases, we have

∑_{p∈ℙ^{1}} deg Z_{p}^{v} ≤ 2n^{2} + 2mn − B ≤ 2n^{2} + 2mn − A,

which implies the following inquality for the supermaximal singularity:

m^{v} ≤ 2 deg Z_{t}^{v} < 4n^{2} + 4n\frac{e}{ν(F_{t})} − 2A.

But we know that m^{h} ≤ A, and then

(Σ_{0} + Σ_{1})(AΣ_{0} + (4n^{2} − 2A)Σ_{0} + 4ne) > (2nΣ_{0} + nΣ_{1} + e)^{2}, (2.1)

so we get a contradiction:

AΣ_{0}(Σ_{0} + Σ_{1}) + (nΣ_{1} − e)^{2} < 0.

It remains to prove that there are no maximal singularities over points that lie on sections of the class s_{0}.

Consider the case n_{1} = n_{2} = n_{3} = 2. Note that s_{B} ∼ s_{0} is a unique section of such a class in this case. If D has a maximal singularity over the point B_{0} ∈ s_{B}, then for a general D ∈ D it holds ν_{1} = mult_{B_{0}} D > n, and for a general curve C ∼ f in the fiber through B_{0} we get n = D ∘ C ≥ ν_{1} > n, which is impossible.

Now let n_{1} = 0, n_{2} = 2, n_{3} = 4. Proving of this case consists of two lemmas below and an observation that any curve of the class s_{0} lies on the divisor G_{V}. Recall that the valuation ν can be realized as a chain of blow-ups centered in nonsingular centers B_{0}, B_{1}, …; a general D ∈ D has multiplicities ν_{1}, ν_{2}, … in these centers respectively.

Lemma 2.8. Let D ⊂ |n(−K_{V}) + mF |, m ≥ 0, has a maximal singularity over a point B_{0} ∈ S, where S is a fiber of V/ℙ^{1}. By l ∈ | − K_{S}| we denote the curve through B_{0}. Then

a) S is smooth at B_{0};

b) B_{0} is a double point of l;

c) ν defines an infinitely near singularity, and if B_{1} ∩ S^{1} ≠ ∅ (upper indices denote the strict transform on the correspondig floor of the chain of blow-ups realizing ν), then B_{1} ∩ l^{1} ≠ ∅ and ν_{1} + ν_{2} ≤ 3n.

Proof. If S is singular at B_{0}, then a general curve C ∈ |2(−K_{S})−B_{0}| is also singular at this point, so we get a contradiction:

2n = D ∘ C ≥ 2ν_{1} > ν_{1}.

Further, suppose l is nonsingular at B_{0}. Set

D|_{S} ∼ kl + C,

where l ⊄ Supp C. We have

n < ν_{1} ≤ mult_{B_{0}} D|_{S} = k + mult_{B_{0}} C.

But mult_{B_{0}} C ≤ C ∘ l = (n(−K_{S}) − kl) ∘ l = n − k, and then

n < ν_{1} ≤ k + n − k = n.

So l has a double point at B_{0}. Now we show that ν defines an infinitely near singularity. Assuming the converse, we get ν_{1} > 2n. Supposing D|_{S} ∼ kl + C, we see that

2 mult_{B_{0}} C ≤ C ∘ l = n − k,

whence

2n < ν_{1} ≤ 2k + mult_{B_{0}} C ≤ \frac{3}{2}k + \frac{1}{2}n,

i.e., k > n, a contradiction.

Now let B_{1} ∩ S^{1} ≠ ∅, but B_{1} ∩ l^{1} = ∅. Denote E_{1} the exceptional divisor of the blow-up of B_{0}. Then

D^{1}|_{S^{1}} = (D ∘ S)^{1} + mE_{1}|_{S^{1}},

and for the decomposition D|_{S} ∼ kl + C we get

ν̃_{1} + ν̃_{2} ≤ 2k + mult_{B_{0}} C + multB̃_{1} C^{1} + m,

where B̃_{1} = B_{1} ∩ S^{1}, ν̃_{1} = mult_{B_{0}} D|_{S}, ν̃_{2} = multB̃_{1}(D|_{S})^{1}. Then,

multB̃_{1} C^{1} ≤ mult_{B_{0}} C ≤ \frac{1}{2}(n(−K_{S}) − kl) ∘ l = \frac{1}{2}(n − k),

so

ν̃_{1} + ν̃_{2} ≤ 2k + 2\frac{n − k}{2} + m = n + k + m.

But ν̃_{1} ≥ ν_{1} + m and ν̃_{2} ≥ ν_{2}, thus

2n < ν_{1} + ν_{2} ≤ n + k,

i.e., k > n, which is impossible.

So B_{1} ∩ l^{1} ≠ ∅. In order to show that ν_{1} + ν_{2} ≤ 3n, we argue as before, only taking into account that

ν̃_{1} + ν̃_{2} ≤ 3k + mult_{B_{0}} C + multB̃_{1} C^{1} + m.

We have ν_{1} + ν_{2} ≤ 2k + n ≤ 3n. The lemma is proved.

Lemma 2.9. Let ε = 0, n_{1} = 0. Then G_{V} ≅ C × ℙ^{1}, where C is an elliptic curve.

Proof. We use the notation of section 2.1. It is obvious that G_{V} ≅ C × ℙ^{1}. We shall prove the smoothness of C.

Notice that R|_{G} ∼ 3t_{0}. Let B be an irreducible component of Supp R|_{G}. Clearly, B ∼ t_{0}. Let ψ : Ṽ → V be the blow-up of B with the exceptional divisor E. E is a ruled surface of the type F_{n_{2}}. Denote t_{E} and l_{E} the classes of the exceptional section and a fiber of E respectively. Then for strict transforms of the divisors G and R on Ṽ we have (using their smoothness along B)

G̃|_{E} ∼ t_{E}, R̃|_{E} ∼ t_{E} + n_{2}l_{E},

hence G̃|_{E} ∩R̃|_{E} = ∅. So any fiber of G ≅ ℙ^{1}×ℙ^{1} meets R transversally, i.e., at three different points. Thus, C is nonsingular.

The lemma is proved. This completes also the proof of proposition 2.7.

Remark 2.10. In fact, proposition 2.7 works also in the case n_{1} = 0, n_{2} = 1, n_{3} = 2, but the proof is more complicated. Since this case is non-rigid, satisfies theorem 2.6, and can not be advanced to a more complete description yet (see below), we omit proving of it.

Corollary 2.11. For ε = n_{1} > 0, all smooth Mori fibrations on del Pezzo surfaces of degree 1 over ℙ^{1} are rigid over the base; for ε = 0, all cases are also rigid, except the following:

a) n_{1} = n_{2} = n_{3} = 2;

b) n_{1} = 0, n_{2} = 1, n_{3} = 2.

Proof. Rigidity follows from proposition 2.7, using lemma 2.4 in the case ε = 0 or lemma 2.5 when ε = n_{1} > 0.

We will deal with non-rigid cases after the following remark.

Remark 2.12. In [9] it was proved that (smooth) pencils of del Pezzo surfaces (of degree 1, 2 or 3) are all rigid, if the so-called K^{2}-condition holds:

cycles aK^{2} − bf are not effective for any a, b > 0.

For degree 1, this condition is satisfied exactly when n_{2} ≥ 4 (for ε = 0) or n_{2} ≥ 4 + \frac{3}{2}n_{1} (for ε = n_{1} > 0). The second is always true except n_{1} = 2, n_{2} = 6, n_{3} = 12.

Another sufficient condition of rigidity was proposed in [5] (conjecture 1.2):

(−K_{V})^{3} + m_{0} + 1 ≤ 2,

where

m_{0} = min{r ∈ ℚ : (−K_{V}) + rF is nef}.

It holds exactly when n_{2} ≥ 3 (ε = 0) or n_{2} ≥ 2 + \frac{3}{2}n_{1} (ε = n_{1} > 0). Notice that the second is always true.

Thus, the second sufficient condition is more exact. Both these conditions do not include the rigid case n_{1} = 0, n_{2} = 2, n_{3} = 4 for ε = 0.

Now we consider the non-rigid cases. We start with the case n_{1} = 0, n_{2} = 1, n_{3} = 2. It was already considered in some works (see [5], section 2, §2, and [7]). It is easy to see that a unique effective divisor of the class G_{V} is the direct product of ℙ^{1} and an elliptic curve, the linear system H = |2G_{V} +2F | = |2(−K_{V})−2F | is base points free and defines a morphism contracting G_{V} along ℙ^{1} onto a curve l on a Fano variety U of index 2 and degree 5 (this is so-called the double cone over the Veronese surface). Indeed, the linear system |M_{Q}| maps Q onto a cone Q̃ ∈ ℙ^{6} over the Veronese surface in ℙ^{5}, and U is obtained by taking the double cover branched along a cubic section that does not pass through the cone vertex. l covers one of the generators of Q̃. Let

μ_{l} : V = V_{l} → U

be such a contraction. Notice that −K_{V} ∼ G_{V} + 2F, and V is a Fano itself. Thus, V/ℙ^{1} is not rigid, and 1 = α(V, H) < μ(V, H) = 2, i.e., the conditions of conjecture 1.6 hold. Notice that U has a lot of such structures. Unfortunately, the technique of the maximal singularities method is not enough yet to deal completely with this case (for example, it fails when l has a double point; see also remark 2.10). Nevertheless, we can formulate the following conjecture, which seems to be true:

Conjecture 2.13. Any smooth Mori fibration in the class of birational equivalency of U is biregular to either U or V_{l} for some l.

The remaining case is ε = 0, n_{1} = n_{2} = n_{3} = 2. Let V/ℙ^{1} be the corresponding pencil of del Pezzo surfaces of degree 1. Notice that

N_{s_{0}|V} ≃ O(−1) ⊕ O(−1)

and |mH| for m ≫ 0 gives a contraction of s_{0}. So there exists a flop

μ : V −−→ U

centered at s_{0}. Then, dim |G_{V} | = 1:

2 dim |G_{V} | = 2 dim |G| ≤ dim |2G| = dim |(M − 2L)|_{Q}| = 2.

Let G_{U} = \overline{μ(F)}, and F_{U} the class of a fiber of U/ℙ^{1}. Obviously, μ^{−1}|G_{U}| = |F | and μ^{−1}|F_{U}| = |G_{V} |. Moreover, U/ℙ^{1} is a pencil of del Pezzo surfaces of degree 1 with the same structure parameters n_{1} = n_{2} = n_{3} = 2.

Proposition 2.14. Let χ : V −−→ W be a birational map onto a Mori fibration γ : W → S. Then either χ or χ∘μ^{−1} : U −−→ W are birational over the base. Moreover, U and V are unique smooth Mori models in their class of birational equivalency; Bir(V) = Aut(V), and in general case, Bir(V) =< σ >_{2}, where σ is the double cover involution.

Proof. Let D_{W} = |n′(−K_{W}) + γ^{∗}(A)| be a very ample linear system, where A is an ample divisor on S, and D_{V} = χ_{∗}^{−1}D_{W} ⊂ |n(−K_{W}) + mF |.

Suppose m ≥ 0. Then from proposition 1.2 it follows that n = n′ and χ is birational over the base. So we may assume m = −l < 0. Denote D_{U} = μ^{−1}D_{V}. Since −K_{U} = G_{U} + F_{U}, we have

D_{U} ⊂ |(n − l)(−K_{U}) + lF_{U}|,

and χ ∘ μ^{−1} is birational over the base by proposition 2.7.

Now let χ : V −−→ W be a birational map onto a Mori fibration W/ℙ^{1} with −K_{W} to be nef. We may assume χ to be birational over the base. Suppose Y = |n(−K_{W}) + m′F_{W} | is a very ample linear system, m′ > 0, and D = χ_{∗}^{−1}Y ⊂ |n(−K_{V}) + mF | for some m ≥ 0. If K_{V} + \frac{1}{n}D were canonical, i.e., D had no maximal singularities, then χ would be an isomorphism by proposition 1.2.

So let D have maximal singularities. We may assume that they are all infinitely near (see [9]). By Y and D we denote general elements of Y and D respectively. Choose a resolution of singularities of χ:

\begin{aligned}& Z \\ & ^{φ}↙ & ↘ ^{ψ} \\ V & −−→^{χ} & W \end{aligned}

Notice that

nK_{Z} + ψ^{−1}Y = m′ψ^{∗}(F_{W}) + ∑ a_{i}E′_{i}

and

nK_{Z} + φ^{−1}D = mφ^{∗}(F) + ∑ b_{i}E_{i},

where E′_{i}, E_{i} are exceptional divisors, a_{i} are all positive. Since D has maximal singularities, then

M = {i : b_{i} < 0}

is not empty. For a point t ∈ ℙ^{1} we denote M_{t} = {i : φ(E_{i}) ∈ F_{t}}. Obviously, dim |nK_{Z} + ψ^{−1}Y | = m′, so

dim |mφ^{∗}(F) + ∑ b_{i}E_{i}| = dim |(m′ + m − m′)φ^{∗}(F) + ∑ b_{i}E_{i}| = m′.

Denote I = {t ∈ ℙ^{1} : M_{t} ≠ ∅}. Then there exists a split

m − m′ = ∑_{t∈I} k_{t},

where k_{t} are all positive, such that

dim | ∑_{t∈I} k_{t}φ^{−1}(F_{t}) + ∑_{t∈I} ∑_{i∈M_{t}}(k_{t}c_{i} − b_{i})E_{i}| = 0,

where for t ∈ I positive numbers c_{i} are defined by

φ^{∗}(F_{t}) = φ_{∗}^{−1}(F_{t}) + ∑_{i∈M_{t}} c_{i}E_{i}.

So for any t ∈ I and i ∈ M_{t} we have k_{t}c_{i} − b_{i} ≥ 0, and, for every i, k_{i} is the smallest positive number with such a property. Thus

k_{t} = max_{i∈M_{t}}⌜b_{i}c_{i}^{−1}⌝,

where ⌜ ⌝ is the round up.

Let D_{1}, D_{2} ∈ D be general. We can put down

D_{1} ∘ D_{2} = Z^{h} + Z^{v} ∼ n^{2}s + (2n^{2} + 2mn)f,

where Z^{v} and Z^{h} are vertical and horizontal cycles. Since s_{0} is unique in its class and centers of maximal singularities can not lie on s_{0}, then horizontal cycles being caught by at least one of these centers can put down as As_{0} + Bf for some B ≥ A. Taking into account that

m ≤ m′ + ∑_{t∈I} k_{t},

we get an estimation

deg Z^{v} = ∑_{t∈ℙ^{1}}deg Z_{t}^{v} ≤ 2n^{2} + 2mn − A ≤ 2n^{2} + 2m′n + 2n ∑_{t∈I} k_{t} − A

(the degrees of vertical cycles are estimated by intersection with −K_{V}). Thus there exist p ∈ I and j ∈ M_{p} such that

deg Z_{p}^{v} ≤ 2n^{2} + 2m′n + 2n⌜b_{j}c_{j}^{−1}⌝ − A.

As in [9], §4, there exist positive numbers Σ_{0}, Σ_{1}, and Σ′_{0}, where Σ_{0} ≥ Σ′_{0} and c_{j} ≥ Σ′_{0}, such that the following inequality for multiplicities m^{h} = mult_{B_{0}} Z^{h} and m^{v} = mult_{B_{0}} Z_{j}^{v} of horizontal and vertical cycles at B_{0} = φ(E_{j}) holds:

(Σ_{0} + Σ_{1})(Σ_{0}m^{h} + Σ′_{0}m^{v}) ≥ (2nΣ_{0} + nΣ_{1} + b_{j})^{2}. (2.2)

Notice that ⌜b_{j}c_{j}^{−1}⌝Σ_{0} ≤ b_{j} + Σ_{0}. By lemma 2.8, D has an infinitely near singularity over the point B_{0} ∈ F_{p}. Consider two cases.

1. B_{1} ∩ F_{p}^{1} = ∅, i.e., the center of the second blow up in the chain realizing the maximal singularity, does not intersect the strict transform of the fiber. In this case, since B_{1} is a point, we have Σ′_{0} ≤ \frac{1}{2}Σ_{0}. Then

Σ′_{0}m^{v} ≤ 2 deg Z_{p}^{v} \frac{Σ_{0}}{2} ≤ (2n^{2} + 2m′n + 2n − A)Σ_{0} + 2nb_{j},

and, using (2.2) and an estimation m^{h} ≤ A, we get

n^{2} − 2m′n − 2n < 0,

i.e.,

\frac{m′}{n} > \frac{1}{2} − \frac{1}{n}. (2.3)

2. Now let B_{1} ∩ S^{1} ≠ ∅. Here we can not state that Σ′_{0} ≤ \frac{1}{2}Σ_{0}, but lemma 2.8 gives us a good estimation ν_{1} + ν_{2} ≤ 3n.

We know that m^{h} ≤ A and

Σ′_{0}m^{v} ≤ 2 deg Z_{p}^{v}Σ_{0} ≤ (4n^{2} + 4m′n + 4n − 2A)Σ_{0} + 4nb_{j}.

Substituting this in (2.2), we obtain

(A − 4m′n − 4n)Σ_{0}(Σ_{0} + Σ_{1}) + (nΣ_{1} − b_{j})^{2} < 0.

Notice that a condition b_{j} > 0 is nothing but the Noether-Fano inequality ([10]): for some set of non-increasing numbers {r_{i}}, where ∑ r_{i} = Σ_{0} + Σ_{1}, it holds

b_{j} = ∑ r_{i}ν_{i} − 2nΣ_{0} − nΣ_{1} > 0.

Applying ν_{1} + ν_{2} ≤ 3n, we get

(nΣ_{1} − b_{j})^{2} ≥ \frac{1}{2}n^{2}Σ_{0}(Σ_{0} + Σ_{1}).

Thus

\frac{1}{2}n^{2} + A − 4m′n − 4n < 0,

and since A ≥ 0, than

\frac{m′}{n} > \frac{1}{8} − \frac{1}{n}. (2.4)

Comparing (2.3) and (2.4), we may assume that (2.4) holds always. It only remains to use the condition that −K_{W} is nef: we can choose Y such that n is big enough but \frac{m′}{n} is as small as we want, and then get a contradiction because of (2.4).

Now, substituting U for W, we obtain the statment about the birational automorphisms group. The uniqueness of smooth models of U and V follows from the description of all smooth (rigid and non-rigid) Mori fibrations on del Pezzo surfaces of degree 1 given above.

Proposition 2.14 is proved. This also completes corollary 2.11 and theorem 2.6.

3. Smooth varieties with a pencil of del Pezzo surfaces of degree 2.

3.1. The essential construction. In this section we study smooth Mori fibrations on del Pezzo surfaces of degree 2. Let ρ : V → ℙ^{1} be such a fibration. Denoting F the class of a fiber of ρ, we have

Pic(V) = ℤ[−K_{V}] ⊕ ℤ[F]

and (−K_{V})^{2} ∘ F = 2. Since ρ is flat and −K_{V} is ρ-ample, for some integer m we have

ρ_{∗}O(−K_{V} + mF) = E,

where E = O ⊕ O(n_{1}) ⊕ O(n_{2}) is a vector bundle of rank 3 over ℙ^{1}, 0 ≤ n_{1} ≤ n_{2}. Set

X = Proj E

with a natural projection π : X → ℙ^{1}. By M we denote the class of the tautological bundle on X, L the class of a fiber. Then

Pic(X) = ℤ[M] ⊕ ℤ[L].

It is easy to see that there exists a double cover φ : V → X branched along a smooth divisor R of the class 4M + 2aL such that

ρ = π ∘ φ.

Further, let t_{0} be the class of a section that corresponds to E → O → 0, l the class of a line in a fiber of π, s_{0} = \frac{1}{2}φ^{∗}(t_{0}), and f = \frac{1}{2}φ^{∗}(l). Then

\overline{NE}(X) = NE(X) = ℝ_{+}[t_{0}] ⊕ ℝ_{+}[l],
\overline{NE}(V) = NE(V) = ℝ_{+}[s_{0}] ⊕ ℝ_{+}[f].

Denote b = n_{1} +n_{2}, H = φ^{∗}(M); clearly, F = φ^{∗}(L). We have M^{3} = b, M^{2} = t_{0} + bl, K_{X} = −3M + (b − 2)L, M ∘ t_{0} = L ∘ l = 0, M ∘ l = L ∘ t_{0} = 1, V: K_{V} = −H + (a + b − 2)F, H^{2} = 2s_{0} + 2bf, H ∘ F = 2f, H ∘ s_{0} = F ∘ f = 0, H ∘ f = F ∘ s_{0} = 1, (−K_{V})^{2} = 2s_{0} + (8 − 4a − 2b)f, H^{3} = 2b, (−K_{V})^{3} = 12 − 6a − 4b (see [5]).

Such varieties with K_{V}^{2} = 2s_{0} + βf for β ≤ 0 were first studied in [9]. That was an exclusively important step in studying of geometry of Fano-fibered varieties.

3.2. Results about rigidity.

Theorem 3.1. Let V/ℙ^{1} be a smooth Mori fibration on del Pezzo surfaces of degree 2 over ℙ^{1} with K_{V}^{2} = 2s_{0} + βf, where β ≤ 2 (i.e., 2a + b ≥ 3). Then V/ℙ^{1} is birationally rigid.

Proof. Let χ : V −−→ V′ be a birational map onto a Mori-fibration ρ′ : V′ → S′, D′ = | − n′K_{V′} + ρ′^{∗}A′| a linear system as in proposition 1.2, and D = χ_{∗}^{−1}D′ ⊂ | − nK_{V} + mF |. We will denote D a general element of D. Obviously, D has no fixed components. If n = n′, the assertion follows from proposition 1.2. So let n > n′.

Lemma 3.2. m ≥ 0, i.e. μ(D) = α(D).

Proof of the lemma. The cases β ≤ 0 are proved in [9] (the socalled K^{2}-condition holds in these cases). Let β = 2. Then 2a + b = 3, b is odd, so n_{1} < n_{2}. Clearly, μ(D) = n.

Assume the converse, i.e., m < 0. Consider a general curve C ∼ 2s_{0} + 2n_{1}f on V (there is at least 1-dimensional family of such curves). Then for general D we have D ∘ C ≥ 0, so

(nH − (\frac{n}{2}(b − 1) − m)F) ∘ (s_{0} + n_{1}f) ≥ 0,

and we get a contradiction:

0 < \frac{−m}{n} ≤ n_{2} − \frac{b}{2} + \frac{1}{2} = \frac{n_{1} − n_{2} + 1}{2} ≤ 0.

The lemma is proved.

By the lemma, we may assume m ≥ 0. The following argumentation proving the theorem is nearly the same as in [9].

Obviously, D is not canonical, so it has maximal singularities. Let a curve B be the center of a maximal singularity. Then B is a section of ρ not lying on the ramification divisor, and we can ”untwist” such a maximal singularity using the Bertini involution centered at B.

So we assume that the center of any maximal singularity is a point. Set D_{1} ∘ D_{2} = Z^{h} + Z^{v}, where Z^{h} and Z^{v} are horizontal and vertical (effective) cycles. We see that

Z^{h} + Z^{v} = 2s_{0} + (βn^{2} + 4mn)f.

Denote deg Z^{h} =^{def} Z^{h}∘F and deg Z^{v} =^{def} Z^{v}∘(−K_{V}). Then, there exist a discrete valuation ν centered over a point B_{0} in a fiber F_{t} (over a point t ∈ ℙ^{1}), a positive number e = ν(D) − nδ, where δ is the canonical multiplicity with respect to ν, such that a component Z_{t}^{v} of Z^{v} lying in F_{t} has the degree

deg Z_{t}^{v} < \{ \begin{matrix}βn^{2} + 4n\frac{e}{ν(F_{t})}, & if β > 0; \\ 4n\frac{e}{ν(F_{t})}, & if β ≤ 0. \end{matrix}

Notice that m^{h} =^{def} mult_{B_{0}} Z^{h} ≤ deg Z^{h} = 2n^{2} and m^{v} =^{def} mult_{B_{0}} Z_{t}^{v} ≤ deg Z_{t}^{v}. Then, there exist positive numbers Σ_{0} ≥ Σ′_{0} and Σ_{1} suct that the following inequality holds:

(Σ_{0} + Σ_{1})(m^{h}Σ_{0} + m^{v}Σ′_{0}) > (2nΣ_{0} + nΣ_{1} + e)^{2}.

It only remains to substitute the estimations of m^{h} and m^{v}:

(Σ_{0} + Σ_{1})((2 + βn^{2})Σ_{0} + 4ne) > (2nΣ_{0} + nΣ_{1} + e)^{2},

i.e.,

(2 − β)n^{2}Σ_{0}(Σ_{0} + Σ_{1}) + (nΣ_{1} − e)^{2} < 0,

which is impossible, if β ≤ 2. Theorem 3.1 is proved.

Now we shall consider smooth Mori fibrations on del Pezzo surfaces of degree 2 for 2a + b ≤ 2. First, let 2a + b = 2, i.e., K_{V}^{2} = 2s_{0} + 4f.

Lemma 3.3. For 2a + b = 2, the only following cases may occur: 1) a = −1, n_{1} = n_{2} = 2;

2) a = 1, n_{1} = n_{2} = 0;

3) a = 0, n_{1} = n_{2} = 1;

4) a = 0, n_{1} = 0, n_{2} = 2;

5) a = −2, n_{1} = 2, n_{2} = 4;

6) a = −3, n_{1} = 2, n_{2} = 6;

7) a = −4, n_{1} = 2, n_{2} = 8; The first three cases are non-rigid.

Proof. Since b ≥ 0, the only cases 2), 3), and 4) are possible if a ≥ 0.

Let a < 0. Then b > 0, and since R ∘ t_{0} < 0, any curve of the class s_{0} lies on R. So n_{1} > 0 because of irreducibility of R, and such a curve is unique on X. Further, let ψ : X̃ → X be the blow-up of t_{0} with E as an exceptional divisor. E is a ruled surface of the type F_{n_{2}−n_{1}}. By t_{E} and l_{E} we denote the classes of an exceptional section and a fiber of E. Suppose R̃ is the strict transform of R, then

R̃|_{E} ∼ t_{E} + (2a + n_{2})l_{E}

is an irreducible curve on V because of the smoothness of R. So either 2a+n_{2} = 0, or 2a+n_{2} ≥ n_{2}−n_{1}. It is easy to check that the second case is possible only if n_{1} = n_{2} = 2 and a = −1. Further, if 2a + n_{2} = 0, then n_{1} = 2. Moreover, curves of the class t_{0} + n_{1}l lie on a unique effective divisor of the class M − n_{2}L, and since R is irreducible, we have R ∘ (t_{0} + n_{1}l) ≥ 0, whence a ≥ −4.

Now we show that the first three cases are non-rigid. Let a = 1, n_{1} = n_{2} = 0. We see that X ≅ ℙ^{2} × ℙ^{1}, and V is a conic bundle with respect to the projection onto ℙ^{2}: double covers of curves of the class t_{0} are conics since R ∘ t_{0} = 2. So V is non-rigid. Notice that this projection is given by a linear system | − K_{V} − F |, which is free from base points.

Let a = −1, n_{1} = n_{2} = 2. A unique curve of the class t_{0} on X lies on the ramification divisor R, so s_{0} is unique on V, too. Note that

N_{s_{0}|V} = O(−1) ⊕ O(−2)

and |nH| for n ≥ 2 gives a birational morphism contracting s_{0}. Then, the base set of the pencil |M − L| is exactly s_{0}; all elements of this pencil are smooth and isomorphic to F_{2}. For a general S ∈ |M − L| the restriction R|_{S} is composed from s_{0} and some three-section that does not intersect s_{0}, so after taking the double cover and contracting the pre-image of s_{0} the surface S becomes a del Pezzo surface of degree 1. This means that the anti-flip V/ℙ^{1} −−→ V′/ℙ^{1} (centered at s_{0}) gives us a Mori fibration on del Pezzo surfaces of degree 1 with a terminal singular point lying on the exceptional curve of V′. Thus, V/ℙ^{1} is not rigid. Note that the pencil | − K_{V} − F | has no fixed components.

Finally, for a = 0, n_{1} = n_{2} = 1 the linear system | − 2K_{V} | gives a small contraction onto the canonical model of V, which can be realized as double covering of a non-degenerated quadratic cone in ℙ^{4} branched along a quartic section. If a curve of the class s_{0} is unique on V (this means that t_{0} lies on the ramification divisor), then s_{0} is -2-curve of the width 2 (in the notions of [11]). Otherwise, there are two curves of the class s_{0}, which are disjoint and -2-curves of the width 1. In both the cases we obtain another structure of a smooth Mori fibration on del Pezzo surfaces of degree 2 after making a flop centered at these curves. Notice again that | − K_{V} − F | has no fixed components. The case of two curves was studied in detail in [3].

Lemma 3.3 is proved.

Remark 3.4. It is highly likely that cases 4) – 7) are all rigid. Unfortunately, the author can not prove it yet. As it often occur in the practice of the maximal singularity method, the problems are related to excluding of infinitely near singularities over points on some rational curves of a special kind.

Now we consider cases when 2a + b = 1, i.e., K_{V}^{2} = 2s_{0} + 6f.

Lemma 3.5. If 2a + b = 1, the only following three cases may occur: 1) a = 0, n_{1} = 0, n_{2} = 1;

2) a = −1, n_{1} = 1, n_{2} = 2;

3) a = −2, n_{1} = 1, n_{2} = 4. The first two are non-rigid.

Proof. Arguing as before, it is easy to show that a ≥ −2, so the only 1) – 3) are possible.

In order to show that the case 1) is non-rigid, note that V can be obtained by blowing up of an elliptic curve of degree 2 on a smooth Fano variety U of genus 9 and index 2 (this is so-called the double space of index 2, i.e., the double cover of ℙ^{3} branched along a quartic). Observe that the birational morphism V → U is defined by the linear system | − 2K_{V} − 2F |. Some results (but very incomplete) about these varieties are contained in [8].

Consider the case a = −1, n_{1} = 1, n_{2} = 2. There exists a unique curve of the class s_{0}, and it is a -2-curve of the width 1. Let V −−→ V^{+} be a flop centered at this curve. A linear system |H − 2F | contains a unique element, which we denote G_{V}. Its strict transform G_{V}^{+} on V^{+} is a surface that is isomorphic to the double cover of ℙ^{2} branched along either a smooth conic or a couple of different lines. So G_{V}^{+} is either ℙ^{1} × ℙ^{1}, or a quadratic cone in ℙ^{3}. Then, there exists an extremal contraction V^{+} → U of G_{V}^{+}. U is a double cone over the Veronese surface, but with a quadratic singularity (arising from a quadratic singularity of the ramification divisor). We see also that U has (birationally) structures of fibrations on del Pezzo surfaces of degree 1 (see section 2, the case n_{1} = 0, n_{2} = 1, n_{3} = 2). Notice that | − K_{V} − F | has no fixed components.

The lemma is proved.

As after lemma 3.3, the author can say the same thing about the case 3): it should be rigid, but I can not prove it yet.

We complete this survey of del Pezzo fibrations by the following lemma:

Lemma 3.6. Cases 2a + b ≤ 0 can not occur.

Proof. The same reasons as above, except the case a = 0. If a = 0, then b = 0, so V is isomorphic to the direct product of ℙ^{1} and a smooth del Pezzo surface of degree 2. But in this case V is not a Mori fibration because of the relative Picard number (it is equal to 8). The lemma is proved.

References

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[2] Corti, A., Del Pezzo surfaces over Dedekind schemes. Ann. Math., II. Ser. 144, 3 (1996), 641-683 [3] Grinenko, M.M. Birational automorphisms of a three-dimensional double cone.

Sb. Math. 189 (1998), No.7, 991-1007

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[5] Iskovskikh, V.A. On the rationality of three-dimensional Del Pezzo fibrations. Proc. Steklov Inst. Math. 208 (1995), 115-123

[6] Iskovskikh, V.A. Birational automorphisms of three-dimensional algebraic varieties. J. Sov. Math. 13 (1980), 815-868

[7] Khashin, S.I., Birational automorphisms of a double Veronese cone of dimension three. Mosc. Univ. Math. Bull. 39 (1984), No.1, 15-20

[8] Khashin, S.I., Birational automorphisms of a index 2 Fano variety of degree 2. MGU, M., 1985, pp. 41 (in russian)

[9] Pukhlikov, A.V. Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces. Izv. Math. 62 (1998), No.1, 115-155

[10] Pukhlikov, A.V., Essentials of the method of maximal singularities. Warwick Preprint 31/1996

[11] Reid, M., Minimal models of canonical 3-folds. Algebraic varieties and analytic varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1 (1983), 131-180

Steklov Mathematical Institute

Max-Planck Institut für Mathematik

E-mail address: grin@mi.ras.ru / grinenko@mpim-bonn.mpg.de