aa r X i v : . [ m a t h . AG ] D ec On the Picard number of divisors in Fano manifolds
Cinzia CasagrandeMay 24, 2011
Abstract
Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisorin X . We consider the image N ( D, X ) of N ( D ) in N ( X ) under the natural push-forward of 1-cycles. We show that ρ X − ρ D ≤ codim N ( D, X ) ≤
8. Moreover ifcodim N ( D, X ) ≥
3, then either X ∼ = S × T where S is a Del Pezzo surface, orcodim N ( D, X ) = 3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that ρ X − ρ T = 4. R´esum´e
Soit X une vari´et´e de Fano lisse et complexe de dimension arbitraire, et D un divi-seur premier dans X . Nous consid´erons l’image N ( D, X ) de N ( D ) dans N ( X ) parl’application naturelle de “push-forward” de 1-cycles. Nous d´emontrons que ρ X − ρ D ≤ codim N ( D, X ) ≤
8. De plus, si codim N ( D, X ) ≥
3, alors soit X ∼ = S × T o`u S estune surface de Del Pezzo, soit codim N ( D, X ) = 3 et X a une fibration en surfaces deDel Pezzo sur une vari´et´e de Fano lisse T , telle que ρ X − ρ T = 4. Contents − D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 X is a product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 The case of codimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Primary 14J45; Secondary 14E30. Introduction
Let X be a complex Fano manifold of arbitrary dimension n , and consider a prime divisor D ⊂ X . We denote by N ( X ) the R -vector space of one-cycles in X , with real coefficients,modulo numerical equivalence; its dimension is the Picard number ρ X of X , and similarlyfor D . The inclusion i : D ֒ → X induces a push-forward of one-cycles i ∗ : N ( D ) → N ( X ),that does not need to be injective nor surjective. We are interested in the image N ( D, X ) := i ∗ ( N ( D )) ⊆ N ( X ) , which is the linear subspace of N ( X ) spanned by numerical classes of curves contained in D . The codimension of N ( D, X ) in N ( X ) is equal to the dimension of the kernel of therestriction H ( X, R ) → H ( D, R ).If X is a surface, then it follows from the classification of Del Pezzo surfaces thatcodim N ( D, X ) = ρ X − ≤
8. Our main result is that the same holds in any dimension.
Theorem 1.1.
Let X be a Fano manifold of dimension n . For every prime divisor D ⊂ X ,we have ρ X − ρ D ≤ codim N ( D, X ) ≤ . Moreover, suppose that there exists a prime divisor D with codim N ( D, X ) ≥ . Then oneof the following holds: ( i ) X ∼ = S × T , where S is a Del Pezzo surface with ρ S ≥ codim N ( D, X ) + 1 , and D dominates T under the projection; ( ii ) codim N ( D, X ) = 3 and there exists a flat surjective morphism ϕ : X → T , withconnected fibers, where T is an ( n − -dimensional Fano manifold, and ρ X − ρ T = 4 . When n ≥ D is ample, one has N ( D, X ) = N ( X ) and also dim N ( D, X ) = ρ D by Lefschetz Theorems on hyperplane sections, see [Laz04, Example 3.1.25]. However ingeneral dim N ( D, X ) can be smaller than ρ X : for instance, if D ∼ = P n − is the excep-tional divisor of the blow-up X of any projective manifold at a point, we have ρ D =dim N ( D, X ) = 1 < ρ X .In case ( ii ) of Theorem 1.1 the variety X does not need to be a product of lowerdimensional varieties, see Example 3.4.Theorem 1.1 generalizes an analogous result in [Cas03] for toric Fano varieties, obtainedin a completely different way, using combinatorial techniques.We recall that the pseudo-index of a Fano manifold X is ι X = min {− K X · C | C is a rational curve in X } , and is a multiple of the index of X ; one expects that Fano manifolds with large pseudo-indexare simpler. When ι X > i.e. when X does not contain rational curves of anticanonicaldegree one), we show a stronger version of Theorem 1.1. Theorem 1.2.
Let X be a Fano manifold with pseudo-index ι X > . For every primedivisor D ⊂ X , we have codim N ( D, X ) ≤ . More precisely, one of the following holds: i ) ι X = 2 and there exists a smooth morphism ϕ : X → Y with fibers isomorphic to P ,where Y is a Fano manifold with ι Y > ; ( ii ) for every prime divisor D ⊂ X , we have N ( D, X ) = N ( X ) , ρ X ≤ ρ D , and therestriction H ( X, R ) → H ( D, R ) is injective. Moreover for every pair of primedivisors D , D in X , we have D ∩ D = ∅ . The author was led to this subject by the study of Fano manifolds with large Picardnumber (see [Cas08] for an account of this problem). Let us mention two straightforwardconsequences of Theorem 1.1, which give bounds on ρ X in some good situations. The firstconcerns the case dim X ≤
5, while the second is about Fano manifolds having a morphismonto a curve.
Corollary 1.3.
Let X be a Fano manifold, and suppose that there exists a prime divisor D ⊂ X such that codim N ( D, X ) ≥ .If dim X = 4 then either ρ X ≤ , or X is a product of Del Pezzo surfaces and ρ X ≤ .If dim X = 5 then either ρ X ≤ , or X is a product and ρ X ≤ . Corollary 1.4.
Let X be a Fano manifold, ϕ : X → P a surjective morphism with con-nected fibers, and F ⊂ X a general fiber. Then ρ X ≤ ρ F + 8 .Moreover if ρ X ≥ ρ F + 4 , then X ∼ = S × T where S is a Del Pezzo surface, ϕ factorsthrough the projection X → S , and F ∼ = P × T . Finally, we notice that some of the properties given by Theorem 1.1 are inherited byvarieties dominated by a Fano manifold. We give two applications, and refer the reader toLemma 4.1 for a more general statement.
Corollary 1.5.
Let X be a Fano manifold and ϕ : X → Y a surjective morphism. Supposethat there exists a prime divisor D ⊂ X such that dim ϕ ( D ) ≤ (this always holds if dim Y = 2 ). Then ρ Y ≤ .Moreover if ρ Y ≥ then dim Y ≤ and X ∼ = S × T , where S is a Del Pezzo surface. Corollary 1.6.
Let X be a Fano manifold and ϕ : X → Y a surjective morphism with dim Y = 3 . Then ρ Y ≤ .Moreover if ρ Y ≥ then X ∼ = S × T where S is a Del Pezzo surface, T has a contractiononto P , and ϕ factors through X → S × P . Outline of the paper.
The idea that a special divisor should affect the geometry of X isclassical. In [BCW02] Fano manifolds containing a divisor D ∼ = P n − with normal bundle N D/X ∼ = O P n − ( −
1) are classified. This classification has been extended in [Tsu06] to thecase N D/X ∼ = O P n − ( − a ) with a >
0; moreover [Tsu06, Proposition 5] shows that if X contains a divisor D with ρ D = 1, then ρ X ≤
3. More generally, divisors D ⊂ X withdim N ( D, X ) = 1 or 2 play an important role in [Cas08, Cas09].In section 2 we treat the main construction that will be used in the paper, based on theanalysis of a Mori program for − D , where D ⊂ X is a prime divisor; this is a developmentof a technique used in [Cas09]. Let us give an idea of our approach, referring the reader tosection 2 for more details. 3fter [BCHM10, HK00], we know that we can run a Mori program for any divisor ina Fano manifold X . In fact we need to consider special Mori programs , where all involvedextremal rays have positive intersection with the anticanonical divisor (see section 2.1).Then, given a prime divisor D ⊂ X , we consider a special Mori program for − D , whichroughly means that we contract or flip extremal rays having positive intersection with D ,until we get a fiber type contraction such that (the transform of) D dominates the target.If c := codim N ( D, X ) >
0, by studying how the codimension of N ( D, X ) varies underthe birational maps and the related properties of the extremal rays, we obtain c − pairwisedisjoint prime divisors E , . . . , E c − ⊂ X , all intersecting D , such that each E i is a smooth P -bundle with E i · f i = −
1, where f i ⊂ E i is a fiber (see Proposition 2.5 and Lemma 2.7).We call E , . . . , E c − the P -bundles determined by the special Mori program for − D thatwe are considering; they play an essential role throughout the paper.We conclude section 2 proving Theorem 1.2 about the case with pseudo-index ι X > X : c X := max { codim N ( D, X ) | D is a prime divisor in X } . In terms of this invariant, our main result is that c X ≤
8, and if c X ≥
3, then either X is a product, or c X = 3 and X has a flat fibration onto an ( n − i.e. with codim N ( D, X ) = c X . We show that there exists a prime divisor E with codim N ( E , X ) = c X , such that E is a smooth P -bundle with E · f = −
1, where f ⊂ E is a fiber. Applying the previous results to E , we obtain a bunch of disjointdivisors with a P -bundle structure, and we use them to show that X is a product, or toconstruct a fibration in Del Pezzo surfaces.Finally in section 4 we use this result (Theorem 3.3) to prove the remaining resultsstated above: Theorem 1.1 and its Corollaries 1.3 to 1.6. Acknowledgements.
I am grateful to Tommaso de Fernex for an important suggestionconcerning Proposition 2.4.This paper was written mainly during a visit to the Mathematical Sciences ResearchInstitute in Berkeley, for the program in Algebraic Geometry in spring 2009. I would liketo thank MSRI for the kind hospitality, and GNSAGA-INdAM and the Research Project“Geometria delle variet`a algebriche e dei loro spazi di moduli” (PRIN 2006) for financialsupport.
Notation and terminology
We work over the field of complex numbers.A manifold is a smooth variety.A P -bundle is a projectivization of a rank 2 vector bundle.Let X be a projective variety. N ( X ) (respectively, N ( X )) is the R -vector space of one-cycles (respectively, Cartier di-visors) with real coefficients, modulo numerical equivalence.4 C ] is the numerical equivalence class in N ( X ) of a curve C ⊂ X ; [ D ] is the numericalequivalence class in N ( X ) of a Q -Cartier divisor D in X .If E ⊂ X is an irreducible closed subset and C ⊂ E is a curve, [ C ] E is the numericalequivalence class of C in N ( E ). ≡ stands for numerical equivalence (for both 1-cycles and Q -Cartier divisors).For any Q -Cartier divisor D in X , D ⊥ := { γ ∈ N ( X ) | D · γ = 0 } .NE( X ) ⊂ N ( X ) is the convex cone generated by classes of effective curves, and NE( X ) isits closure.An extremal ray R of X is a one-dimensional face of NE( X ); Locus( R ) ⊆ X is the unionof all curves whose class is in R .If R is an extremal ray of X and D is a Q -Cartier divisor in X , we say that D · R > D · R = 0, etc. if for γ ∈ R r { } we have D · γ >
0, respectively D · γ = 0, etc.Assume that X is normal.A contraction of X is a surjective morphism with connected fibers ϕ : X → Y , where Y isnormal and projective; NE( ϕ ) is the face of NE( X ) generated by classes of curves contractedby ϕ .A contraction ϕ : X → Y is elementary if ρ X − ρ Y = 1.We say that an elementary contraction ϕ : X → Y (or the extremal ray NE( ϕ )) is of type( n − , n − sm if it is the blow-up of a smooth codimension 2 subvariety contained in thesmooth locus of Y (here n = dim X ).If Z ⊆ X is a closed subset and i : Z ֒ → X is the inclusion, we set N ( Z, X ) := i ∗ ( N ( Z )) ⊆ N ( X ) and NE( Z, X ) := i ∗ (NE( Z )) ⊆ NE( X ) ⊂ N ( X ) . In this section we recall what a Mori program is, and explain that by [HK00] and [BCHM10]we can run a Mori program for any divisor on a Fano manifold. We also introduce and showthe existence of “special Mori programs”, where all involved extremal rays have positiveintersection with the anticanonical divisor.We begin by recalling the following fundamental result.
Theorem 2.1 ([BCHM10], Corollary 1.3.2) . Any Fano manifold is a Mori dream space.
We refer the reader to [HK00] for the definition and properties of a Mori dream spaces; inparticular, a Mori dream space is always a normal and Q -factorial projective variety. Wealso need the following. Proposition 2.2 ([HK00], Proposition 1.11(1)) . Let X be a Mori dream space and B adivisor in X . Then there exists a finite sequence (2.3) X = X σ X · · · X k − σ k − X k such that: every X i is a normal and Q -factorial projective variety; • for every i = 0 , . . . , k − there exists an extremal ray Q i of X i such that B i · Q i < ,where B i ⊂ X i is the transform of B , Locus( Q i ) ( X i , and σ i is either the contractionof Q i (if Q i is divisorial), or its flip (if Q i is small); • either B k is nef, or there exists an extremal ray Q k in X k , with a fiber type contraction ϕ : X k → Y , such that B k · Q k < .Moreover, the choice of the extremal rays Q i is arbitrary among those that have negativeintersection with B i . A sequence as above is called a
Mori program for the divisor B . We refer the reader to[KM98, Definition 6.5] for the definition of flip.An important remark is that when X is Fano, there is always a suitable choice of a Moriprogram where all involved extremal rays have positive intersection with the anticanonicaldivisor. Proposition 2.4.
Let X be a Fano manifold and B a divisor on X . Then there exists aMori program for B as (2.3) , such that − K X i · Q i > for every i = 0 , . . . , k . We call sucha sequence a special Mori program for B . This is a very special case of the MMP with scaling, see [BCHM10, Remark 3.10.9]. Forthe reader’s convenience, we give a proof. The idea is to choose a facet of the cone of nefdivisors Nef( X ) ⊂ N ( X ) met by moving from [ B ] to [ − K X ] along a line in N ( X ), andto repeat the same at each step. Proof of Proposition 2.4.
By Theorem 2.1 X is a Mori dream space, therefore Proposition2.2 applies to X , and there exists a Mori program for B . We have to prove that we canchoose Q , . . . , Q k with B i · Q i < − K X i · Q i > i = 0 , . . . , k .We can assume that B is not nef. Set λ := sup { λ ∈ R | (1 − λ )( − K X ) + λB is nef } , so that λ ∈ Q , 0 < λ <
1, and H := (1 − λ )( − K X ) + λ B is nef but not ample.Then there exists an extremal ray Q of NE( X ) such that H · Q = 0 and B · Q <
0; inparticular, − K X · Q > Q is of fiber type, we are done. Otherwise, let σ : X X be either the contractionof Q (if divisorial), or its flip (if small), and let B be the transform of B . Then (1 − λ )( − K X ) + λ B is nef in X .If B is nef we are done. If not, we set λ := sup { λ ∈ R | (1 − λ )( − K X ) + λB is nef } , so that λ ∈ Q , λ ≤ λ <
1, and H := (1 − λ )( − K X ) + λ B is nef but not ample.There exists an extremal ray Q of NE( X ) such that H · Q = 0 and B · Q <
0, hence − K X · Q >
0. Now we iterate the procedure. (cid:4) More precisely, B i is the transform of B i − if σ i − is a flip, and B i = ( σ i − ) ∗ ( B i − ) if σ i − is a divisorialcontraction. .2 Running a Mori program for − D In this section we study in detail what happens when we run a Mori program for − D ,where D is a prime divisor. This point of view has already been considered in [Cas09],and is somehow opposite to the classical one: we consider extremal rays having positive intersection with D . In particular, we are interested in how the number codim N ( D, X )varies under the Mori program.We first describe the general situation for a prime divisor D in a Mori dream space(Lemma 2.6), and then consider the case of a special Mori program for − D where D is aprime divisor in a Fano manifold (Lemma 2.7). In particular, we will show the following. Proposition 2.5.
Let X be a Fano manifold and D ⊂ X a prime divisor. Suppose that codim N ( D, X ) > .Then there exist pairwise disjoint smooth prime divisors E , . . . , E s ⊂ X , with s =codim N ( D, X ) − or s = codim N ( D, X ) , such that every E j is a P -bundle with E j · f j = − , where f j ⊂ E j is a fiber; moreover D · f j > and [ f j ]
6∈ N ( D, X ) . In particular E j ∩ D = ∅ and E j = D . It is important to point out that the P -bundles E , . . . , E s are determined not only by D ,but by the choice of a special Mori program for − D (see Lemma 2.7). In fact the divisors E j are the transforms of the loci of some of the extremal rays of the Mori program, theones where codim N ( D, X ) drops.Finally we study in more detail the case where s = codim N ( D, X ) − X which has a conic bundlestructure (see Lemma 2.8).We conclude the section with the proof of Theorem 1.2. Lemma 2.6.
Let X be a Mori dream space and D ⊂ X a prime divisor. Consider a Moriprogram for − D : X = X σ X · · · X k − σ k − X k . Let D i ⊂ X i be the transform of D , for i = 1 , . . . , k , and set D := D , so that D i · Q i > for i = 0 , . . . , k . We have the following. (1) Every D i is a prime divisor in X i , and the program ends with an elementary contractionof fiber type ϕ : X k → Y such that NE( ϕ ) = Q k and ϕ ( D k ) = Y . (2) { i ∈ { , . . . , k } | Q i
6⊂ N ( D i , X i ) } = codim N ( D, X ) . (3) Set c i := codim N ( D i , X i ) for i = 0 , . . . , k . For every i = 0 , . . . , k − we have c i +1 = ( c i if Q i ⊂ N ( D i , X i ) c i − if Q i
6⊂ N ( D i , X i ) , and c k = ( if Q k ⊂ N ( D k , X k )1 if Q k
6⊂ N ( D k , X k ) . (4) Suppose that X is smooth. Let A ⊂ X be the indeterminacy locus of σ − , and for i = 2 , . . . , k , if σ i − is a divisorial contraction (respectively, if σ i − is a flip), let A i ⊂ X i e the union of σ i − ( A i − ) (respectively, the transform of A i − ) and the indeterminacylocus of σ − i − .Then for all i = 1 , . . . , k we have Sing( X i ) ⊆ A i ⊂ D i , and the birational map X i X is an isomorphism over X i r A i .Proof. Most of the statements are shown in [Cas09] (see in particular Remarks 2.5 and 2.6,and Lemma 3.6); for the reader’s convenience we give a proof. We have D i · Q i > i = 0 , . . . , k , just by the definition of Mori program for − D .Let i ∈ { , . . . , k − } be such that σ i is a divisorial contraction. Then D i = Exc( σ i )(for otherwise D i · Q i < D i +1 = σ i ( D i ) ⊂ X i +1 is a prime divisor. On theother hand D i intersects every non-trivial fiber of σ i (because D i · Q i > D i ∩ Exc( σ i ) = ∅ and D i +1 ⊃ σ i (Exc( σ i )). Notice that σ i (Exc( σ i )) is the indeterminacylocus of σ − i .Consider the push-forward ( σ i ) ∗ : N ( X i ) → N ( X i +1 ). We have ker( σ i ) ∗ = R Q i and N ( D i +1 , X i +1 ) = ( σ i ) ∗ ( N ( D i , X i )), therefore c i +1 = c i if Q i ⊂ N ( D i , X i ), and c i +1 = c i − i ∈ { , . . . , k − } be such that σ i is a flip, and consider the standard flipdiagram: X i ϕ i ❅❅❅❅❅❅❅❅ σ i / / ❴❴❴❴❴❴❴ X i +1 ϕ ′ i } } ③③③③③③③③ Y i where ϕ i is the contraction of Q i , and ϕ ′ i is the corresponding small elementary contrac-tion of X i +1 . We have D i +1 · NE( ϕ ′ i ) <
0, in particular Exc( ϕ ′ i ) ⊂ D i +1 and NE( ϕ ′ i ) ⊂N ( D i +1 , X i +1 ). Notice that Exc( ϕ ′ i ) is the indeterminacy locus of σ − i .Moreover ϕ i ( D i ) = ϕ ′ i ( D i +1 ), so that( ϕ i ) ∗ ( N ( D i , X i )) = N ( ϕ i ( D i ) , Y i ) = ( ϕ ′ i ) ∗ ( N ( D i +1 , X i +1 )) . Since ker( ϕ ′ i ) ∗ ⊆ N ( D i +1 , X i +1 ), we have c i +1 = codim N ( ϕ i ( D i ) , Y i ). We deduce againthat c i +1 = c i if Q i ⊂ N ( D i , X i ), and c i +1 = c i − i = 1 , . . . , k the divisor D i contains the indeterminacy locus of σ − i , so that A i ⊂ D i . By definition, A i contains theindeterminacy locus of the birational map ( σ i − ◦· · ·◦ σ ) − : X i X ; in particular X i r A i is isomorphic to an open subset of X , thus it is smooth if X is smooth. This shows (4).Consider now the prime divisor D k ⊂ X k . Clearly − D k cannot be nef, therefore theprogram ends with a fiber type contraction ϕ : X k → Y . Since D k · Q k > D k intersectsevery fiber of ϕ , namely ϕ ( D k ) = Y , and we have (1).In particular ϕ ∗ ( N ( D k , X k )) = N ( Y ), hence either c k = 0 ( i.e. N ( D k , X k ) = N ( X k )),or c k = 1 and Q k
6⊂ N ( D k , X k ). Thus we have (3), which implies directly (2). (cid:4) emma 2.7. Let X be a Fano manifold and D ⊂ X a prime divisor. Consider a specialMori program for − D : X = X σ X · · · X k − σ k − X k . Then we have the following (we keep the notation of Lemma 2.6). (1)
Let i ∈ { , . . . , k − } be such that Q i
6⊂ N ( D i , X i ) .Then Q i is of type ( n − , n − sm , i.e. σ i : X i → X i +1 is the blow-up of a smoothsubvariety of codimension , contained in the smooth locus of X i +1 . Moreover Exc( σ i ) ∩ A i = ∅ , hence Exc( σ i ) does not intersect the loci of the birational maps σ l for l < i . (2) Set s := { i ∈ { , . . . , k − } | Q i
6⊂ N ( D i , X i ) } . We have two possibilities:either s = codim N ( D, X ) and N ( D k , X k ) = N ( X k ) ,or s = codim N ( D, X ) − , Q k
6⊂ N ( D k , X k ) , and codim N ( D k , X k ) = 1 . (3) Set { i , . . . , i s } := { i ∈ { , . . . , k − } | Q i
6⊂ N ( D i , X i ) } , and let E j ⊂ X be the trans-form of Exc( σ i j ) ⊂ X i j for every j = 1 , . . . , s .Then E j is a smooth P -bundle, with fiber f j ⊂ E j , such that E j · f j = − , D · f j > ,and [ f j ]
6∈ N ( D, X ) . In particular E j ∩ D = ∅ and E j = D . (4) The prime divisors E , . . . , E s are pairwise disjoint. We call E , . . . , E s the P -bundles determined by the special Mori program for − D that we are considering. These divisors will play a key role throughout the paper.Notice that Proposition 2.5 is a straightforward consequence of Proposition 2.4 and ofLemma 2.7, more precisely of 2.7(3) and 2.7(4). Proof.
Statement (1) follows from [Cas09, Lemma 3.9].By 2.6(2) we have s = ( codim N ( D, X ) if Q k ⊂ N ( D k , X k ),codim N ( D, X ) − Q k
6⊂ N ( D k , X k ).Together with 2.6(3) this yields (2).Let j ∈ { , . . . , s } . By (1) we have E j ∼ = Exc( σ i j ), thus E j is a smooth P -bundlewith E j · f j = −
1, where f j ⊂ E j is a fiber, and D · f j > D i j · Q i j > X i j . In particular E j ∩ D = ∅ and E j = D . Moreover [ f j ] ⊂ N ( D, X ) would yield Q i j ⊂ N ( D i j , X i j ), which is excluded by definition. Therefore we have (3).Finally E , . . . , E s are pairwise disjoint, because for j = 1 , . . . , s the divisor Exc( σ i j )does not intersect the loci of the previous birational maps. (cid:4) Here is a more detailed description of the case where s = codim N ( D, X ) − emma 2.8 (Conic bundle case) . Let X be a Fano manifold and D ⊂ X a prime divisor.Consider a special Mori program for − D ; we keep the same notation as in Lemmas 2.6and 2.7. Set c := codim N ( D, X ) , σ := σ k − ◦· · ·◦ σ : X X k , and ψ := ϕ ◦ σ : X Y . X = X σ , , ❡ ❞ ❝ ❜ ❛ ❵ ❴ ❫ ❪ ❭ ❬ ❩ ❨ ❳ ψ / / ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ❬ ❭ ❪ ❫ ❫ ❴ σ / / ❴❴❴ X / / ❴❴❴ · · · / / ❴❴❴ X k − σ k − / / ❴❴❴ X kϕ (cid:15) (cid:15) Y We assume that Q k
6⊂ N ( D k , X k ) , equivalently that s = c − (see 2.7(2)). Then we havethe following. (1) Every fiber of ϕ has dimension , dim Y = n − , and ϕ is finite on D k . (2) Let j ∈ { , . . . , c − } and consider σ i j (Exc( σ i j )) ⊂ X i j +1 . For every m = i j +1 , . . . , k − the set Locus( Q m ) ⊂ X m is disjoint from the image of σ i j (Exc( σ i j )) in X m , so thatthe birational map X i j +1 X k is an isomorphism on σ i j (Exc( σ i j )) , and σ is regularon E j ⊂ X . (3) There exist open subsets U ⊆ X and V ⊆ Y , with E , . . . , E c − ⊂ U , such that V and ϕ − ( V ) are smooth, ϕ | ϕ − ( V ) : ϕ − ( V ) → V and ψ : U → V are conic bundles, and σ | U is the blow-up of pairwise disjoint smooth subvarieties T , . . . , T c − ⊂ ϕ − ( V ) , ofdimension n − , with exceptional divisors E , . . . , E c − . U ψ ) ) σ | U / / ϕ − ( V ) ϕ / / V In particular we have
Locus( Q m ) ⊆ X m r ( σ m − ◦ · · · ◦ σ )( U ) for every m ∈ { , . . . , k − } r { i , . . . , i c − } . (4) Set Z j := ψ ( E j ) ⊂ V for every j ∈ { , . . . , c − } . Then Z , . . . , Z c − ⊂ Y are pairwisedisjoint smooth prime divisors, and ψ ∗ ( Z j ) = E j + b E j , where b E j ⊂ U is a smooth P -bundle with fiber b f j ⊂ b E j , f j + b f j is numerically equivalent to a general fiber of ψ ,and b E j · b f j = − , E j · b f j = b E j · f j = 1 , and [ b f j ]
6∈ N ( E j , X ) , for every j ∈ { , . . . , c − } . In particular the divisors D, E , . . . , E c − , b E , . . . , b E c − areall distinct, and E ∪ b E , . . . , E c − ∪ b E c − are pairwise disjoint. We refer the reader to [Cas03, p. 1478-1479] for an explicit description of the rationalconic bundle ψ in the toric case. Proof of Lemma 2.8.
Let F ⊂ X k be a fiber of ϕ . Then F ∩ D k = ∅ because D k · Q k > F ∩ D k ) = 0, because if there exists a curve C ⊂ F ∩ D k , then[ C ] ∈ Q k and [ C ] ∈ N ( D k , X k ), thus Q k ⊂ N ( D k , X k ) against our assumptions. Henceevery fiber of ϕ has dimension 1, dim Y = n −
1, and we have (1).10ecall from 2.6(4) that Sing( X k ) ⊆ A k , and notice that codim A k ≥
2, therefore A k cannot dominate Y . Restricting ϕ we get a contraction X k r ϕ − ( ϕ ( A k )) → Y r ϕ ( A k ) of asmooth variety, with − K X k relatively ample (because − K X k · Q k > Y r ϕ ( A k ) is smooth and that ϕ | X k r ϕ − ( ϕ ( A k )) is a conic bundle(see [AW97, Theorem 4.1(2)]).By 2.6(4), σ : X X k is an isomorphism over X k r A k . If U := σ − ( X k r ϕ − ( ϕ ( A k ))),then ψ : U → Y r ϕ ( A k ) is again a conic bundle; in particular it is flat, and induces aninjective morphism ι : Y r ϕ ( A k ) → Hilb( X ). Let H ⊂ Hilb( X ) be the closure of the imageof ι , and C ⊂ H × X the restriction of the universal family over Hilb( X ). We get a diagram: C π (cid:15) (cid:15) e / / X σ / / ❴❴❴ ψ ❇❇❇❇ X kϕ (cid:15) (cid:15) H Y ι o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ where π : C → H and e : C → X are the projections, and ι is birational. We want tocompare the degenerations in X and in X k of the general fibers the conic bundle ψ | U .Fix j ∈ { , . . . , c − } , and recall from 2.7(1) that Exc( σ i j ) ∩ A i j = ∅ , so that thebirational map X X i j is an isomorphism over Exc( σ i j ). In X i j +1 we have A i j +1 = σ i j (cid:0) Exc( σ i j ) ∪ A i j (cid:1) , hence σ i j (Exc( σ i j )) is a connected component of A i j +1 .Let x ∈ σ i j (Exc( σ i j )) ⊂ X i j +1 and let l ⊆ E j ⊂ X be the transform of the fiber of σ i j over x .Let B ⊆ H be a general irreducible curve which intersects π ( e − ( l )). Since π isequidimensional and the general fiber of π over B is P , the inverse image π − ( B ) ⊆ C isirreducible. Set S := e ( π − ( B )) ⊆ X , then S ∩ l = ∅ by construction.Consider the normalizations B → B and C B → π − ( B ) of B and π − ( B ) respec-tively; we have induced morphisms e B : C B → S and π B : C B → B . C Bπ B (cid:15) (cid:15) / / π − ( B ) ⊆ C π (cid:15) (cid:15) e / / X ⊇ S := e ( π − ( B )) B / / B ⊆ H Because B is general, B ∩ dom( ι − ) = ∅ , and ι − induces a morphism η : B → Y . Set B := η ( B ) ⊂ Y .Again, since ϕ is equidimensional and the general fiber of ϕ over B is P , the inverseimage ϕ − ( B ) ⊂ X k is irreducible; call S k this surface, which is just the transform of S ⊂ X under σ .Recall that ϕ is finite on D k by (1), and A k ⊂ D k by 2.6(4), hence no component of afiber of ϕ can be contained in A k . On the other hand, by the generality of B , the generalfiber of ϕ | S k does not intersect A k . Therefore S k can intersect A k at most in a finite numberof points. 11onsider now σ S := σ | S : S S k . Then σ S is an isomorphism over S k r ( S k ∩ A k ) anddim( S k ∩ A k ) ≤
0, hence by Zariski’s main theorem ξ := σ S ◦ e B : C B → S k is a morphism. C Bπ B (cid:15) (cid:15) e B / / ξ + + S ⊂ X σ S / / ❴❴❴ S k ⊂ X kϕ (cid:15) (cid:15) B η / / B ⊂ Y Let y ∈ B be such that C := e B ( π − B ( y )) ⊂ S intersects l ; in particular C ∩ E j = ∅ ,because l ⊆ E j . Since C is numerically equivalent in X to a general fiber of ψ , we have − K X · C = 2 and E j · C = 0; in particular C has at most two irreducible components,because − K X is ample.Set r := ϕ − ( η ( y )). Since r is numerically equivalent in X k to a general fiber of ϕ , wehave − K X k · r = 2. Recall that no irreducible component of r can be contained in A k ;on the other hand, r must intersect A k , otherwise σ S would be an isomorphism over r , C = σ − S ( r ), and C ∩ E j = ∅ , a contradiction.Let us show that r is an integral fiber of ϕ . Indeed let C be an irreducible componentof r . If C ∩ A k = ∅ , then C is contained in the smooth locus of X k and − K X k · C ≥ C ∩ A k = ∅ , then [Cas09, Lemma 3.8] gives − K X k · C >
1. Since − K X k · r = 2and r must intersect A k , it must be irreducible and reduced.For every i ∈ { , . . . , k − } let e r i ⊂ X i be the transform of r ⊂ X k (where X = X ).Again by [Cas09, Lemma 3.8] we get − K X · e r < − K X k · r = 2, hence − K X · e r = 1.Notice that ξ ( π − B ( y )) ⊂ S k is contained in r ; on the other hand ξ cannot contract toa point a fiber of π B , hence ξ ( π − B ( y )) = r . Then e r ⊆ C , because C = e B ( π − B ( y )), andwe get C = e r ∪ C ′ , where C ′ ⊂ X is an irreducible curve (and possibly C ′ = e r if C isnon-reduced).Since r A k , we have e r E j ; in particular E j · e r ≥
0. If E j · e r = 0, then also E j · C ′ = 0 and C ⊂ E j , which is impossible. Hence E j · e r >
0, and since E j · C = 0, wehave E j · C ′ < C ′ = e r .Consider now the blow-up σ i j : X i j → X i j +1 . We have Exc( σ i j ) · e r i j = E j · e r ≥
1, henceusing the projection formula we get − K X ij +1 · e r i j +1 ≥ − K X ij · e r i j + 1. On the other hand[Cas09, Lemma 3.8] gives1 = − K X · e r ≤ − K X ij · e r i j and − K X ij +1 · e r i j +1 ≤ − K X k · r = 2 . We conclude that Exc( σ i j ) · e r i j = 1, − K X · e r = − K X ij · e r i j , and − K X ij +1 · e r i j +1 = − K X k · r ,and again by [Cas09, Lemma 3.8] this implies that:(2.9) for every m ∈ { , . . . , k − } , m = i j , Locus( Q m ) is disjoint from e r m .We show that C ′ = l (recall that l ⊂ X is the transform of σ − i j ( x ) ⊂ X i j ). Since C ′ intersects e r (because C = e r ∪ C ′ is connected), and e r ∩ Locus( Q ) = ∅ by (2.9),we see that C ′ is not contained in Locus( Q ). Iterating this reasoning for every σ m with12 ∈ { , . . . , i j − } , we see that C ′ intersects the open subset where the birational map X X i j is an isomorphism; let e C ′ ⊂ X i j be its transform.If σ i j ( e C ′ ) were a curve, then by the same reasoning it could not be contained inLocus( Q m ) for any m = i j + 1 , . . . , k −
1, and in the end we would get a curve e C ′ k ⊂ X k ,distinct from r , which should belong to ξ ( π − B ( y )), which is impossible. Thus e C ′ must be afiber of σ i j . On the other hand Exc( σ i j ) · e r i j = 1, thus e r i j intersects a unique fiber of σ i j ,and C ′ = l .In particular this yields that x ∈ e r i j +1 ∩ σ i j (Exc( σ i j )). Since x ∈ σ i j (Exc( σ i j )) wasarbitrary, (2.9) implies statement (2).Let T j ⊂ X k be the image of σ i j (Exc( σ i j )) ⊂ X i j +1 . By (2) the birational map X i j +1 X k yields an isomorphism between σ i j (Exc( σ i j )) and T j , hence T j is smooth of dimension n −
2, and is contained in the smooth locus of X k . Since σ i j (Exc( σ i j )) is a connectedcomponent of A i j +1 , we deduce that T j is a connected component of A k , and A k r T j isclosed in X k .By (2.9) the birational map X i j +1 X k yields also an isomorphism between e r i j +1 and r , and r ∩ ( A k r T j ) = ∅ .Consider the point x ′ ∈ T j corresponding to x ∈ σ i j (Exc( σ i j )). Then x ′ ∈ r ∩ T j because x ∈ e r i j +1 , i.e. r is the fiber of ϕ through x ′ ∈ T j . Again since x was arbitrary in σ i j (Exc( σ i j )), from r ∩ ( A k r T j ) = ∅ we deduce that ϕ − ( ϕ ( T j )) ∩ ( A k r T j ) = ∅ , and hencethat ϕ ( T j ) ∩ ϕ ( A k r T j ) = ∅ in Y .Summing up, we have shown that T , . . . , T c − are connected components of A k (so that A k r ( T ∪ · · · ∪ T c − ) is closed in X k ), and the images ϕ ( T ) , . . . , ϕ ( T c − ) , ϕ ( A k r ( T ∪· · · ∪ T c − )) are pairwise disjoint in Y .Now set(2.10) V := Y r ϕ ( A k r ( T ∪ · · · ∪ T c − )) . Then V is open in Y , ϕ − ( V ) ⊆ σ (dom( σ )), and T ∪ · · · ∪ T c − ⊂ ϕ − ( V ). Set U := σ − ( ϕ − ( V )) ⊆ X . By definition, ϕ − ( V ) ∩ ( A k r ( T ∪ · · · ∪ T c − )) = ∅ ; this means thatfor every m ∈ { , . . . , k − } r { i , . . . , i c − } , Locus( Q m ) is disjoint from the image of U in X m .We have E , . . . , E c − ⊂ U , because E j = σ − ( T j ), and ψ : U → V is regular and proper.More precisely, every fiber of ψ over V is one-dimensional, and as before [AW97, Theorem4.1(2)] shows that this is a conic bundle and that V is smooth. We have a factorization U ψ ) ) σ | U / / ϕ − ( V ) ϕ / / V and σ | U is just the blow-up of T ∪ · · · ∪ T c − , so we get (3). For every j ∈ { , . . . , c − } we have Z j = ψ ( E j ) = ϕ ( T j ), so Z , . . . , Z c − are pairwise disjoint. Now let b E j ⊂ U bethe transform of ϕ − ( Z j ). Then ψ − ( Z j ) = E j ∪ b E j , and the rest of statement (4) followsfrom standard arguments on conic bundles. Just notice that if for some j ∈ { , . . . , c − }
13e have [ b f j ] ∈ N ( E j , X ), then [ σ ( b f j )] ∈ N ( T j , X k ) ⊆ N ( A k , X k ) ⊆ N ( D k , X k ), which isimpossible because σ ( b f j ) is a fiber of ϕ and NE( ϕ )
6⊂ N ( D k , X k ) by assumption. (cid:4) Proof of Theorem 1.2. If N ( D, X ) = N ( X ) for every prime divisor D ⊂ X , then we have( ii ) (just notice that if D , D ⊂ X are two disjoint divisors, then N ( D , X ) ⊆ D ⊥ ( N ( X ), see Remark 3.1.2).Suppose now that there exists a prime divisor D ⊂ X with codim N ( D, X ) >
0, andconsider a special Mori program for − D (which exists by Proposition 2.4). Let E , . . . , E s ⊂ X be the P -bundles determined by the Mori program.If s ≥
1, by 2.7(3) we have − K X · f = 1, where f ⊂ E is a fiber of the P -bundle;this is impossible because ι X > s = 0, and 2.7(2) yields that codim N ( D, X ) = 1 and Q k
6⊂ N ( D k , X k ), sothat Lemma 2.8 applies.We show that k = 0 and X = X k . Indeed if not, we have A k = ∅ in X k (see 2.6(4)).Take r a fiber of ϕ intersecting A k . Then, using [Cas09, Lemma 3.8] as in the proof ofLemma 2.8, we see that r is integral, and that the transform e r ⊂ X of r has anticanonicaldegree 1 in X , a contradiction.Thus X = X k and we get a conic bundle ϕ : X → Y , which is finite on D . Since X contains no curves of anticanonical degree 1, ϕ must be a smooth fibration in P . Then Y is Fano by [Wi´s91, Proposition 4.3], and finally we have ι Y ≥ ι X = 2 by [BCDD03,Lemme 2.5]. (cid:4) Let X be a Fano manifold, and consider c X := max { codim N ( D, X ) | D is a prime divisor in X } . We always have 0 ≤ c X ≤ ρ X −
1. If S is a Del Pezzo surface, then c S = ρ S − ∈ { , . . . , } . Example 3.1.
Consider a Fano manifold X = S × T , where S is a Del Pezzo surface.Then c X = max { ρ S − , c T } . More precisely, for any prime divisor D ⊂ X , we have threepossibilities: • D = C × T where C ⊂ S is a curve, and codim N ( D, X ) = ρ S − • D = S × D T where D T ⊂ T is a divisor, and codim N ( D, X ) = codim N ( D T , T ) ≤ c T ; • D dominates both S and T under the projections, and codim N ( D, X ) ≤ ρ S − D ⊂ X is a prime divisor with codim N ( D, X ) > ρ S −
1. Thendim N ( D, X ) < ρ T +1, so that D cannot dominate T under the projection, and D = S × D T . Example 3.2. If X is a Fano manifold with pseudo-index ι X ≥ X = P n × · · · × P n r with n i ≥ i = 1 , . . . , r ), then c X = 0 by Theorem 1.2.We are going to use the results of section 2.2 to prove the following.14 heorem 3.3. For any Fano manifold X we have c X ≤ . Moreover: • if c X ≥ then X ∼ = S × T where S is a Del Pezzo surface, ρ S = c X + 1 , and c T ≤ c X ; • if c X = 3 then there exists a flat, quasi-elementary contraction X → T where T is an ( n − -dimensional Fano manifold, ρ X − ρ T = 4 , and c T ≤ . A contraction ϕ is quasi-elementary if ker ϕ ∗ is generated by the numerical classes of thecurves contained in a general fiber of ϕ ; we refer the reader to [Cas08] for properties ofquasi-elementary contractions. In particular, in the case where c X = 3 in Theorem 3.3, thegeneral fiber of the contraction X → T is a Del Pezzo surface S with ρ S ≥ Example 3.4 (Codimension 3) . Let n ≥ Z = P P n − ( O ⊕ ⊕ O (1)). Then Z is atoric Fano manifold with ρ Z = 2, and the P -bundle Z → P n − has three pairwise disjointsections T , T , T ⊂ Z which are closed under the torus action. Let X → Z be the blow-upof T , T , T . Then X is Fano with ρ X = 5, and it has a smooth morphism X → P n − such that every fiber is the Del Pezzo surface S with ρ S = 4. If E ⊂ X is one of theexceptional divisors of the blow-up, one easily checks that ρ X − ρ E = codim N ( E, X ) = 3,hence c X ≥
3. However X is not a product, thus c X = 3 by Theorem 3.3. The proof of Theorem 3.3 will take all the rest of section 3; we will proceed in severalsteps. Section 3.1 gathers some preliminary remarks and lemmas. In section 3.2 we treatthe case c X ≥
4, and we show that X ∼ = S × T , where S is a Del Pezzo surface with ρ S = c X + 1, and T a Fano manifold with c T ≤ c X (see Proposition 3.2.1, and 3.2.3 for anoutline of its proof). In particular this implies that c X ≤
8, because ρ S ≤ c X = 3 is more delicate, as we have to treat separately the two following cases:(3.6.a) for every prime divisor D ⊂ X with codim N ( D, X ) = 3, and for every specialMori program for − D , we have N ( D k , X k ) = N ( X k ) (notation as in Lemma 2.6);(3.6.b) there exist a prime divisor D ⊂ X with codim N ( D, X ) = 3, and a special Moriprogram for − D , such that N ( D k , X k ) ( N ( X k ).The first case (3.6.a) is treated together with the case c X ≥
4, in section 3.2. In the endwe reach a contradiction, hence a posteriori we conclude that (3.6.a) never happens (seeCorollary 3.2.2). The second case (3.6.b) is treated in section 3.3, where we show theexistence of a flat, quasi-elementary contraction X → T , where T is an ( n − ρ X − ρ T = 4, and c T ≤ In this section we collect some remarks and lemmas which will be used in the proof ofTheorem 3.3.
Remark 3.1.1.
Let X be a projective manifold, ϕ : X → Y a contraction such that − K X is ϕ -ample and dim Y >
0, and D a divisor in X such that ker ϕ ∗ ⊆ D ⊥ . Then we have thefollowing: 151) dim Y = 1 + dim ϕ (Supp D ) and D = ϕ ∗ ( D Y ), D Y a Cartier divisor in Y ;(2) if D is a prime divisor, then ϕ ( D ) is a prime Cartier divisor, and D = ϕ ∗ ( ϕ ( D ));(3) if D is a smooth prime divisor, let ϕ ( D ) ν → ϕ ( D ) be the normalization. Then themorphism ϕ D : D → ϕ ( D ) ν induced by ϕ | D is a contraction, and − K D is ϕ D -ample;(4) if D is a smooth prime divisor and Y is smooth, then ϕ ( D ) is a smooth prime divisor. Proof.
By [KM98, Theorem 3.7(4)] there exists a Cartier divisor D Y on Y such that D = ϕ ∗ ( D Y ). Then Supp D Y = ϕ (Supp D ), so we have (1).If D is a prime divisor, then D Y is a prime divisor supported on ϕ ( D ), namely D Y = ϕ ( D ), and we have (2).For (3), ϕ D is surjective with connected fibers onto a normal projective variety, hencea contraction. Let i : D ֒ → X be the inclusion and take γ ∈ NE( D ) ∩ ker( ϕ D ) ∗ with γ = 0.The restriction ( − K X ) | D is ϕ D -ample, hence ( − K X ) | D · γ >
0. Moreover i ∗ ( γ ) ∈ ker ϕ ∗ , sothat − K D · γ = − ( K X + D ) · i ∗ ( γ ) = − K X · i ∗ ( γ ) > , and − K D is ϕ D -ample.For (4), let y ∈ ϕ ( D ) and let f ∈ O Y,y be a local equation for ϕ ( D ). Then ϕ ∗ ( f ) is alocal equation for D near the fiber over y . Since D is smooth, the differential d x ( ϕ ∗ ( f )) isnon-zero, where x ∈ ϕ − ( y ). Then d y f is non-zero, hence ϕ ( D ) is smooth at y . (cid:4) Remark 3.1.2.
Let X be a projective manifold, Z ⊂ X a closed subset, and D ⊂ X aprime divisor. If Z ∩ D = ∅ , then D · C = 0 for every curve C ⊂ Z , hence N ( Z, X ) ⊆ D ⊥ . Remark 3.1.3.
Let X be a projective manifold, E ⊂ X a smooth prime divisor which is a P -bundle with fiber f ⊂ E , and D ⊂ X a prime divisor with D · f >
0. Then the followingholds:(1) dim N ( D ∩ E, X ) ≥ dim N ( E, X ) − N ( E, X ) = R [ f ] + N ( D ∩ E, X );(2) either [ f ] ∈ N ( D ∩ E, X ) and N ( D ∩ E, X ) = N ( E, X ), or [ f ]
6∈ N ( D ∩ E, X ) and N ( D ∩ E, X ) has codimension 1 in N ( E, X );(3) for every irreducible curve C ⊂ E we have C ≡ λf + µC ′ , where C ′ is an irreduciblecurve contained in D ∩ E , λ, µ ∈ R , and µ ≥ Proof.
Let π : E → F be the P -bundle structure on E , and consider the push-forward π ∗ : N ( E ) → N ( F ). This is a surjective linear map with kernel R [ f ] E .Since D · f >
0, we have π ( D ∩ E ) = F , thus π ∗ ( N ( D ∩ E, E )) = N ( F ). Therefore N ( E ) = R [ f ] E + N ( D ∩ E, E ), and applying i ∗ (where i : E ֒ → X is the inclusion) we get(1) and (2). Statement (3) follows from [Occ06, Lemma 3.2 and Remark 3.3]. (cid:4) Remark 3.1.4.
Let X be a Fano manifold and D, E ⊂ X prime divisors with N ( D ∩ E, X ) ⊆ E ⊥ . E is a smooth P -bundle with fiber f ⊂ E , such that E · f = − D · f > R ≥ [ f ] ⊂ NE( X ) is an extremal ray of type ( n − , n − sm , withcontraction ϕ : X → Y where E = Exc( ϕ ) and Y is Fano. Proof.
Notice first of all that ( − K X + E ) · f = 0.Let C ⊂ X be an irreducible curve. If C E , then ( − K X + E ) · C >
0. If C ⊆ D ∩ E ,then E · C = 0, and again ( − K X + E ) · C > C ⊆ E . By 3.1.3(3) we have C ≡ λf + µC ′ , where C ′ is a curvecontained in D ∩ E , λ, µ ∈ R , and µ ≥
0. Thus( − K X + E ) · C = µ ( − K X + E ) · C ′ ≥ , and ( − K X + E ) · C = 0 if and only if µ = 0, if and only if [ C ] ∈ R ≥ [ f ]. Therefore − K X + E is nef, and ( − K X + E ) ⊥ ∩ NE( X ) = R ≥ [ f ] is an extremal ray.Let ϕ : X → Y be the contraction of R ≥ [ f ]; clearly Exc( ϕ ) = E . Since ( − K X + E ) · C > C ⊂ D ∩ E , ϕ is finite of D ∩ E . Thus if F ⊂ E is a fiber of ϕ , then F ∩ D = ∅ (because D · NE( ϕ ) > F ∩ D ) = 0. This yields that dim F = 1, andby [And85, Theorem 2.3] R ≥ [ f ] is of type ( n − , n − sm and Y is smooth.Finally − K X + E = ϕ ∗ ( − K Y ), thus − K Y is ample and Y is Fano (notice that NE( Y )is closed, because NE( Y ) = ϕ ∗ (NE( X ))). (cid:4) Lemma 3.1.5.
Let X be a Fano manifold and D, E ⊂ X prime divisors with N ( D ∩ E, X ) = N ( E, X ) ∩ D ⊥ ⊆ E ⊥ . Suppose that E is a smooth P -bundle with fiber f ⊂ E , such that E · f = − and D · f > .Then E ∼ = P × F where F is a Fano manifold, and D ∩ E = { pts } × F . Moreover thehalf-line R ≥ [ f ] is an extremal ray of type ( n − , n − sm , it is the unique extremal rayhaving negative intersection with E , and the target of its contraction is Fano.Proof. Consider the divisor D | E in E . We have Supp( D | E ) = D ∩ E , and if C ⊆ D ∩ E is anirreducible curve, then [ C ] ∈ N ( D ∩ E, X ) ⊆ D ⊥ , so that D | E · C = D · C = 0. Therefore D | E is nef.Let i : E ֒ → X be the inclusion and take γ ∈ NE( E ) ∩ ( D | E ) ⊥ with γ = 0. Then i ∗ ( γ ) ∈ N ( E, X ) ∩ D ⊥ ⊆ E ⊥ , hence: − K E · γ = − ( K X + E ) · i ∗ ( γ ) = − K X · i ∗ ( γ ) = ( − K X ) | E · γ > . By the contraction theorem, there exists a contraction g : E → Z such that − K E is g -ampleand NE( g ) = NE( E ) ∩ ( D | E ) ⊥ (see [KM98, Theorem 3.7(3)]). Notice that D | E · f = D · f > g does not contract the fibers of the P -bundle on E , and dim Z ≥
1. On the otherhand g sends D ∩ E to a union of points, so that dim Z = 1 by 3.1.1(1). More precisely,since g ( f ) = Z , we get Z ∼ = P . The general fiber F of g is a Fano manifold of dimension n −
2, because − K E is g -ample.By [Cas09, Lemma 4.9] we conclude that E ∼ = P × F and g is the projection onto P .Since D · f > D ∩ E dominates F under the projection, and is sent by g to a union ofpoints; therefore D ∩ E = { pts } × F . 17sing Remark 3.1.4, we see that R ≥ [ f ] is an extremal ray of type ( n − , n − sm , andthe target of its contraction is Fano.Finally let R be an extremal ray of X with E · R <
0. Then R ⊆ NE(
E, X ) ⊆ NE( X ),thus R must be a one-dimensional face of NE( E, X ). Since E ∼ = P × F , we have NE( E ) = R ≥ [ f ] E + NE( { pt } × F, E ) and NE(
E, X ) = R ≥ [ f ] + NE( { pt } × F, X ). On the other handNE( { pt } × F, X ) ⊂ N ( { pt } × F, X ) = N ( D ∩ E, X ) ⊆ E ⊥ , therefore R = R ≥ [ f ]. (cid:4) Remark 3.1.6.
Let X be a projective manifold and E ⊂ X a smooth prime divisor whichis a P -bundle with fiber f ⊂ E . Let E , . . . , E s ⊂ X be pairwise disjoint prime divisorssuch that E = E i and E ∩ E i = ∅ for every i = 1 , . . . , s . Then either E · f = · · · = E s · f = 0, or E i · f > i = 1 , . . . , s . Proof.
For every i = 1 , . . . , s we have E i · f ≥
0, because E = E i .Suppose that there exists j ∈ { , . . . , s } such that E j · f = 0. Since E ∩ E j = ∅ , thisimplies that E j contains a fiber f of the P -bundle structure on E . If i ∈ { , . . . , s } , i = j ,we have E i ∩ E j = ∅ , in particular E i ∩ f = ∅ and hence E i · f = 0. (cid:4) Lemma 3.1.7.
Let X be a Fano manifold and D ⊂ X a prime divisor with codim N ( D, X ) = c X . Let E , . . . , E s ⊂ X be pairwise disjoint prime divisors such that: D ∩ E i = ∅ , D = E i , and codim N ( D ∩ E i , X ) ≤ c X + 1 , for every i = 1 , . . . , s. If s ≥ , then codim N ( D ∩ E i , X ) = c X + 1 for every i = 1 , . . . , s , and N ( D ∩ E i , X ) = N ( D, X ) ∩ E ⊥ j for every i = j. If s ≥ , then there exists a linear subspace L ⊂ N ( X ) , of codimension c X + 1 , such that L = N ( D ∩ E i , X ) = N ( D, X ) ∩ E ⊥ i for every i = 1 , . . . , s .Proof. Assume that s ≥
2, and let i, j ∈ { , . . . , s } with i = j . Since E i ∩ E j = ∅ , we have N ( D ∩ E i , X ) ⊆ E ⊥ j by Remark 3.1.2. On the other hand, since D ∩ E j = ∅ and D = E j ,there exists some curve C ⊂ D with E j · C >
0, so that N ( D, X ) E ⊥ j . Therefore we get: N ( D ∩ E i , X ) ⊆ N ( D, X ) ∩ E ⊥ j ( N ( D, X ) , hence ρ X − c X − ≤ dim N ( D ∩ E i , X ) ≤ dim N ( D, X ) ∩ E ⊥ j = dim N ( D, X ) − ρ X − c X −
1, and this yields the statement.Assume now that s ≥
3, and set L := N ( D ∩ E , X ); the first part already gives thatcodim L = c X + 1 and that L = N ( D, X ) ∩ E ⊥ i for every i = 2 , . . . , s . If i, j ∈ { , . . . , s } are distinct, again by the first part we get L = N ( D, X ) ∩ E ⊥ i = N ( D ∩ E j , X ) = N ( D, X ) ∩ E ⊥ . (cid:4) Since F and E are Fano, the cones NE( F ), NE( E ), NE( E, X ), etc. are closed and polyhedral. emma 3.1.8. Let X be a Fano manifold and D ⊂ X a prime divisor with codim N ( D, X ) = c X . Let E , . . . , E s ⊂ X be pairwise disjoint smooth prime divisors, and suppose that E i isa P -bundle with fiber f i ⊂ E i , such that E i · f i = − and D · f i > , for every i = 1 , . . . , s .Assume that s ≥ . Then codim N ( E i , X ) = c X and codim N ( D ∩ E i , X ) = c X + 1 for every i = 1 , . . . , s ; moreover N ( D ∩ E i , X ) = N ( D, X ) ∩ E ⊥ j for every i = j .Proof. Let i ∈ { , . . . , s } . We have D ∩ E i = ∅ and D = E i because D · f i > E i · f i = −
1. Since D · f i >
0, by 3.1.3(1) and by the definition of c X we have(3.1.9) codim N ( D ∩ E i , X ) ≤ codim N ( E i , X ) + 1 ≤ c X + 1 . Therefore Lemma 3.1.7 yields that N ( D ∩ E i , X ) = N ( D, X ) ∩ E ⊥ j if i = j , and codim N ( D ∩ E i , X ) = c X + 1. By (3.1.9) we get codim N ( E i , X ) = c X . (cid:4) Lemma 3.1.10.
Let X be a Fano manifold and D ⊂ X a prime divisor with codim N ( D, X ) = c X . Let E , . . . , E s , b E , . . . , b E s ⊂ X be prime divisors such that E i and b E i are smooth P -bundles, with fibers respectively f i ⊂ E i and b f i ⊂ b E i , and moreover: E i · f i = b E i · b f i = − , D · f i > , E i · b f i > , b E i · f i > , [ b f i ]
6∈ N ( E i , X ) , and no fiber b f i is contained in D , for every i = 1 , . . . , s . We assume also that E ∪ b E , . . . , E s ∪ b E s are pairwise disjoint, and that s ≥ .Then codim N ( E i , X ) = codim N ( b E i , X ) = c X and [ f i ]
6∈ N ( b E i , X ) for every i =1 , . . . , s .Proof. Lemma 3.1.8 (applied to D and E , . . . , E s ) shows that codim N ( E i , X ) = c X forevery i = 1 , . . . , s .Fix i ∈ { , . . . , s } . Since N ( E i ∩ b E i , X ) ⊆ N ( E i , X ), we have [ b f i ]
6∈ N ( E i ∩ b E i , X ).Because E i · b f i >
0, 3.1.3(2) yields that N ( E i ∩ b E i , X ) has codimension 1 in N ( b E i , X ).Recall that by the definition of c X we have codim N ( b E i , X ) ≤ c X , so that codim N ( E i ∩ b E i , X ) ≤ c X + 1.Let us show that(3.1.11) codim N ( E i ∩ b E i , X ) = c X + 1 and codim N ( b E i , X ) = c X . If D ∩ b E i = ∅ , then N ( E i ∩ b E i , X ) ⊆ N ( E i , X ) ∩ D ⊥ (see Remark 3.1.2); on the other hand N ( E i , X ) ∩ D ⊥ ( N ( E i , X ), because D · f i >
0. This yields codim N ( E i ∩ b E i , X ) = c X +1.If instead D ∩ b E i = ∅ , then D · b f i >
0, because D cannot contain any curve b f i . Thus wecan apply Lemma 3.1.8 to the divisors D and E , . . . , E i − , b E i , E i +1 , . . . , E s , and we deducethat codim N ( b E i , X ) = c X . Hence we have (3.1.11).Since b E i · f i > N ( E i , X ) = c X = codim N ( E i ∩ b E i , X ) −
1, again by3.1.3(2) we get [ f i ]
6∈ N ( E i ∩ b E i , X ). For dimensional reasons N ( E i ∩ b E i , X ) = N ( E i , X ) ∩N ( b E i , X ), and we conclude that [ f i ]
6∈ N ( b E i , X ). (cid:4) .2 The case where X is a product The main results of this section are the following.
Proposition 3.2.1.
Let X be a Fano manifold such that either c X ≥ , or c X = 3 and X satisfies (3.6.a).Then X ∼ = S × T , where S is a Del Pezzo surface with ρ S = c X + 1 , and c T ≤ c X . Inparticular, c X ≤ . Corollary 3.2.2.
Let X be a Fano manifold with c X = 3 . Then X satisfies (3.6.b).Proof of Corollary 3.2.2. By contradiction, suppose that X satisfies (3.6.a). Then by Propo-sition 3.2.1 we have X ∼ = S × T and ρ S = 4, i.e. S is the blow-up of P in three non-collinearpoints. Consider the sequence: X ∼ = S × T −→ S × T −→ F × T −→ P × T, where S is the blow-up of P in two distinct points. Let C ⊂ F be the section of the P -bundle containing the two points blown-up under S → F . Let moreover e C ⊂ S beits transform, and D := e C × T ⊂ X . Then codim N ( D, X ) = 3, and the sequenceabove is a special Mori program for − D . The image of D in F × T is C × T , and N ( C × T, F × T ) ( N ( F × T ). Thus we have a contradiction with (3.6.a). (cid:4) There are three preparatory steps,and then the actual proof.The first step is to apply the construction of section 2.2 to a prime divisor D ⊂ X withcodim N ( D, X ) = c X . We consider a special Mori program for − D , and this determinespairwise disjoint P -bundles E , . . . , E s ⊂ X as in Lemma 2.7; we denote by f i ⊂ E i afiber. The crucial property here is that s ≥
3: indeed s ≥ codim N ( D, X ) − c X − s ≥ c X ≥
4. On the other hand if c X = 3 we have s = 3 by (3.6.a). Then for i = 1 , . . . , s we show that codim N ( E i , X ) = c X and that R ≥ [ f i ] is an extremal ray of type( n − , n − sm , such that the target of its contraction is again Fano. This is Lemma 3.2.4.In particular, this shows that X has at least one extremal ray R of type ( n − , n − sm such that if E := Locus( R ), then codim N ( E , X ) = c X , and the target of the contractionof R is Fano.Now we replace D by E , and apply again the same construction. Let p : E → F bethe P -bundle structure. Since E , . . . , E s are pairwise disjoint, either E ∩ E i is a unionof fibers of p for every i = 1 , . . . , s , or p ( E ∩ E i ) = F for every i = 1 , . . . , s . The secondpreparatory step is to show that if E , . . . , E s intersect E horizontally with respect to the P -bundle ( i.e. p ( E ∩ E i ) = F ), the divisors E , . . . , E s have very special properties; inparticular, for every i = 0 , . . . , s , E i ∼ = P × F where F is an ( n − R ,and the special Mori program for − E , in such a way that E , . . . , E s actually intersect E horizontally with respect to the P -bundle, so that the previous result applies. This isLemma 3.2.10. 20hen we are ready for the proof of Proposition 3.2.1. We use the the properties given byLemma 3.2.7 to show that E , . . . , E s are the exceptional divisors of the blow-up σ : X → X s of a Fano manifold X s in s smooth codimension 2 subvarieties. Moreover there is anelementary contraction of fiber type ϕ : X s → Y such that if ψ := ϕ ◦ σ : X → Y , then ψ ( E ) = Y , and ψ is finite on { pt } × F ⊂ E (recall that E ∼ = P × F ). We have thentwo possibilities: either ψ is not finite on E and dim Y = n −
2, or ψ is finite on E anddim Y = n − ψ is not finite on E , in 3.2.21. We use the divisors E , . . . , E s to define a contraction X → S onto a surface, such that the induced morphism π : X → S × Y is finite. Finally we show that in fact π is an isomorphism; here the keyproperty is that E , . . . , E s are products.Then we consider in 3.2.24 the case where ψ is finite on E . In this situation Y issmooth, and both ψ and ϕ are conic bundles. If T , . . . , T s ⊂ X s are the subvarietiesblown-up by σ , the transforms b E , . . . , b E s ⊂ X of ϕ − ( ϕ ( T i )) are smooth P -bundles.Similarly to what previously done for E , . . . , E s , we show that b E i ∼ = P × F for every i = 1 , . . . , s .Since ψ ( E ) = Y , Y is covered by the family of rational curves ψ ( P × { pt } ). We use aresult from [BCD07] to show that in fact these rational curves are the fibers of a smoothmorphism Y → Y ′ , where dim Y ′ = n − X → Y ′ , and we proceed similarly to the previouscase: we use the divisors E , E , . . . , E s , b E , . . . , b E s to define a contraction X → S onto asurface, and show that the induced morphism X → S × Y ′ is an isomorphism.Let us start with the first preparatory result. Lemma 3.2.4.
Let X be a Fano manifold such that either c X ≥ , or c X = 3 and X satisfies (3.6.a).Let D ⊂ X be a prime divisor with codim N ( D, X ) = c X , consider a special Moriprogram for − D , and let E , . . . , E s ⊂ X be the P -bundles determined by the Mori program.For i = 1 , . . . , s let f i ⊂ E i be a fiber of the P -bundle, and set R i := R ≥ [ f i ] . Then wehave the following: (1) s ∈ { c X − , c X } and s ≥ ; (2) R i is an extremal ray of type ( n − , n − sm , the target of the contraction of R i isFano, and codim N ( E i , X ) = c X , for every i = 1 , . . . , s ; (3) there exists a linear subspace L ⊂ N ( X ) , of codimension c X + 1 , such that L = N ( D ∩ E i , X ) = N ( D, X ) ∩ E ⊥ i = N ( E i , X ) ∩ E ⊥ i for every i = 1 , . . . , s. We will call R , . . . , R s the extremal rays determined by the special Mori programfor − D that we are considering. Notice that differently from the case of the P -bundles E , . . . , E s , the extremal rays R , . . . , R s are defined only when X satisfies the assumptionsof Lemma 3.2.4, and D ⊂ X is a prime divisor with codim N ( D, X ) = c X .21 roof. We know by Lemma 2.7 that: E i · f i = − D · f i > i = 1 , . . . , s , E , . . . , E s are pairwise disjoint, and s ∈ { c X − , c X } because codim N ( D, X ) = c X . Moreover, if c X = 3, then s = 3 by (3.6.a), so that in any case s ≥
3, and we get (1).Therefore, by Lemma 3.1.8, we have codim N ( E i , X ) = c X and codim N ( D ∩ E i , X ) = c X + 1 for every i = 1 , . . . , s . In particular, Lemma 3.1.7 applies; let L ⊂ N ( X ) be thelinear subspace such that codim L = c X + 1 and L = N ( D ∩ E i , X ) = N ( D, X ) ∩ E ⊥ i forevery i = 1 , . . . , s .Fix i ∈ { , . . . , s } . Since E i · f i = −
1, we have N ( E i , X ) E ⊥ i , therefore dim N ( E i , X ) ∩ E ⊥ i = dim N ( E i , X ) − ρ X − c X − L . On the other hand we have L ⊆ E ⊥ i and L = N ( D ∩ E i , X ), in particular L ⊆ N ( E i , X ). Thus L ⊆ N ( E i , X ) ∩ E ⊥ i , so the twosubspaces must coincide, and we get (3).Finally, (2) follows from Remark 3.1.4 applied to D and E i . (cid:4) Lemma 3.2.5.
Let X be a Fano manifold such that either c X ≥ , or c X = 3 and X satisfies (3.6.a).Let D ⊂ X be a prime divisor with codim N ( D, X ) = c X , and R an extremal ray oftype ( n − , n − sm such that D · R > , R
6⊂ N ( D, X ) , and the target of the contractionof R is Fano.Set E := Locus( R ) . Then N ( D ∩ E, X ) = N ( D, X ) ∩ E ⊥ = N ( E, X ) ∩ E ⊥ .Proof. Consider the contraction ϕ : X → Y of R , so that by the assumptions Y is a Fanomanifold, and consider the prime divisor ϕ ( D ) ⊂ Y .By Proposition 2.4, there exists a special Mori program for − ϕ ( D ) in Y . Together with ϕ , this gives a special Mori program for − D in X , where the first extremal ray is precisely Q = R : X ϕ −→ Y = Y σ Y · · · Y k − σ k − Y k . We apply Lemmas 2.7 and 3.2.4; since R
6⊂ N ( D, X ), E is one of the P -bundles determinedby this special Mori program for − D . Thus the statement follows from 3.2.4(3). (cid:4) Remark 3.2.6.
Let X be a Fano manifold such that either c X ≥
4, or c X = 3 and X satisfies (3.6.a). Recall from Proposition 2.4 that there exists a special Mori program forany divisor in X .The first consequence of Lemma 3.2.4 (applied to any prime divisor D ⊂ X withcodim N ( D, X ) = c X ) is that X has an extremal ray R of type ( n − , n − sm suchthat if E := Locus( R ), then codim N ( E , X ) = c X , and the target of the contraction of R is Fano.In particular, we can consider a special Mori program for − E , and apply again Lemma3.2.4. Let R , . . . , R s be the extremal rays determined by the Mori program, with loci E , . . . , E s . Since, by 2.7(3) and 2.7(4), E , . . . , E s are pairwise disjoint and E = E i , E ∩ E i = ∅ for i = 1 , . . . , s , by Remark 3.1.6 we have two possibilities: either E · R = · · · = E s · R = 0, or E i · R > i = 1 , . . . , s .In the next Lemma we are going to show that in the second case ( i.e. when E · R > R , . . . , R s have very special properties, in particular that the divisors E , . . . , E s are products. 22 emma 3.2.7. Let X be a Fano manifold such that either c X ≥ , or c X = 3 and X satisfies (3.6.a).Let R be an extremal ray of X , of type ( n − , n − sm , such that the target of thecontraction of R is Fano, and codim N ( E , X ) = c X , where E := Locus( R ) .Consider a special Mori program for − E , let R , . . . , R s be the extremal rays determinedby the Mori program, and set E i := Locus( R i ) for i = 1 , . . . , s .Assume that E · R > . Then we have the following: (1) codim N ( E i , X ) = c X , and E i ∼ = P × F with F an ( n − -dimensional Fano manifold,for i = 0 , . . . , s . We set F i := { pt } × F ⊂ E i ; (2) R i is the unique extremal ray of X having negative intersection with E i , and the targetof the contraction of R i is Fano, for every i = 0 , . . . , s ; (3) E , . . . , E s are pairwise disjoint, and E ∩ E i = { pts } × F for every i = 1 , . . . , s ; (4) E i · R > and E · R i > for every i = 1 , . . . , s ; (5) there exists a linear subspace L ⊂ N ( X ) , of codimension c X + 1 , such that L = N ( E ∩ E i , X ) = N ( F j , X ) and N ( E j , X ) = R R j ⊕ L for every i = 1 , . . . , s and j = 0 , . . . , s , and moreover dim( R ( R + · · · + R s ) + L ) = s + 1 + dim L ; (6) L ⊆ E ⊥ ∩ · · · ∩ E ⊥ s , and equality holds if s = c X .Proof. By 2.7(3) and 2.7(4) we know that E · R i > E = E i and E ∩ E i = ∅ )and R i
6⊂ N ( E , X ) for i = 1 , . . . , s , and that E , . . . , E s are pairwise disjoint.Secondly, Lemma 3.2.4 shows that s ∈ { c X − , c X } and s ≥
3, that codim N ( E i , X ) = c X for i = 1 , . . . , s , and that there exists a linear subspace L ⊂ N ( X ), of codimension c X + 1, such that(3.2.8) L = N ( E ∩ E i , X ) = N ( E , X ) ∩ E ⊥ i = N ( E i , X ) ∩ E ⊥ i for every i = 1 , . . . , s. Moreover Remark 3.1.6 yields E i · R > i = 1 , . . . , s ,because E · R >
0, so we get (4).Fix i ∈ { , . . . , s } . We have dim N ( E ∩ E i , X ) = dim L = ρ X − c X − < ρ X − c X =dim N ( E , X ), and since E i · R >
0, 3.1.3(2) gives R
6⊂ N ( E ∩ E i , X ). Moreover N ( E ∩ E i , X ) ⊆ N ( E , X ) ∩ N ( E i , X ) ( N ( E , X )(because R i
6⊂ N ( E , X )), and since N ( E ∩ E i , X ) has codimension 1 in N ( E , X ), wededuce that N ( E ∩ E i , X ) = N ( E , X ) ∩ N ( E i , X ). This yields that R
6⊂ N ( E i , X ).Now we can apply Lemma 3.2.5 to E i and R , and deduce that(3.2.9) L = N ( E ∩ E i , X ) = N ( E i , X ) ∩ E ⊥ . D = E i and E = E , and we deduce that E ∼ = P × F where F is an ( n − E ∩ E i = { pts } × F ⊂ E . Moreover we get(2) for R .Then we apply Lemma 3.1.5 again, with D = E and E = E i , and we get E i ∼ = P × F i and E ∩ E i = { pts } × F i ⊂ E i ; in particular, F i = F , and we have (3). Moreover we get(2) for R i .We have L ⊆ E ⊥ ∩ · · · ∩ E ⊥ s by (3.2.8) and (3.2.9). To get (5), it is enough to show that[ f ] , . . . , [ f s ] ∈ N ( X ) are linearly independent and that R ([ f ] + · · · + [ f s ]) ∩ L = { } . Sosuppose that there exist λ , . . . , λ s ∈ R such that s X i =0 λ i f i ∈ L. Intersecting with E j for j ∈ { , . . . , s } we get λ j = λ E j · f , and intersecting with E weget λ ( P si =1 ( E i · f )( E · f i ) −
1) = 0. Since E i · f and E · f i are positive integers by (4),and s ≥
3, we get λ = 0 and hence λ i = 0 for i = 1 , . . . , s , and we are done.We are left to show (6). Similarly to what we have done for [ f ] , . . . , [ f s ], one checksthat [ E ] , . . . , [ E s ] are linearly independent in N ( X ), so that codim( E ⊥ ∩ · · · ∩ E ⊥ s ) = s + 1.Since L ⊆ E ⊥ ∩ · · · ∩ E ⊥ s and codim L = c X + 1, if s = c X the two subspaces coincide. (cid:4) Lemma 3.2.10.
Let X be a Fano manifold such that either c X ≥ , or c X = 3 and X satisfies (3.6.a). Then X has an extremal ray R with the following properties: • R is of type ( n − , n − sm , the target of the contraction of R is Fano, and codim N ( E , X ) = c X , where E := Locus( R ) ; • there exists a special Mori program for − E such that, if R , . . . , R s are the extremal raysdetermined by the Mori program, we have have Locus( R i ) · R > for every i = 1 , . . . , s .Proof. Let S = { S , . . . , S h } be an ordered set of extremal rays of X , and set E i :=Locus( S i ). Consider the following properties:(P1) S i is of type ( n − , n − sm , the target of the contraction of S i is Fano, andcodim N ( E i , X ) = c X , for every i = 1 , . . . , h ;(P2) E i − · S i > S i
6⊂ N ( E i − , X ), for every i = 2 , . . . , h ;(P3) for every 1 ≤ j < i ≤ h we have E i · S j = 0 and E i ∩ E j = ∅ .We notice first of all that by Remark 3.2.6, there exists an extremal ray S of X , oftype ( n − , n − sm , such that codim Locus( S ) = c X , and the target of the contractionof S is Fano. Then S = { S } satisfies properties (P1), (P2), and (P3).Consider now an arbitrary ordered set of extremal rays S = { S , . . . , S h } satisfyingproperties (P1), (P2), and (P3). We show that h ≤ ρ X .Let γ i ∈ S i a non-zero element, for i = 1 , . . . , h . We have E i · γ i = 0 for every i = 1 , . . . , h ,and E i · γ j = 0 for every 1 ≤ j < i ≤ h by (P3). This shows that γ , . . . , γ h are linearly24ndependent in N ( X ): indeed if there exists a , . . . , a h ∈ R such that P hi =1 a i γ i = 0, thenintersecting with E h we get a h = 0, and so on. Thus h ≤ ρ X .Then Lemma 3.2.10 is a consequence of the following claim. (cid:4) Claim 3.2.11.
Assume that S = { S , . . . , S h } is an ordered set of extremal rays hav-ing properties (P1), (P2), and (P3). Then either R := S h satisfies the statement ofLemma 3.2.10, or there exists an extremal ray S h +1 such that S ′ := { S , . . . , S h , S h +1 } stillhas properties (P1), (P2), and (P3).Proof of Claim 3.2.11. By (P1) the ray S h is of type ( n − , n − sm , the target of itscontraction is Fano, and codim N ( E h , X ) = c X . Consider a special Mori program for − E h (which exists by Proposition 2.4), and let S h +11 , . . . , S h +1 s be the extremal rays determinedby the Mori program, as in Lemma 3.2.4. Notice that s ≥ E h +1 l :=Locus( S h +1 l ) for l = 1 , . . . , s , so that E h +11 , . . . , E h +1 s are the P -bundles determined by theMori program. By 2.7(3) we have(3.2.12) E h · S h +1 l > S h +1 l
6⊂ N ( E h , X ) for every l = 1 , . . . , s, and E h +11 , . . . , E h +1 s are pairwise disjoint by 2.7(4).Remark 3.2.6 shows that the intersections E h +1 l · S h (for l = 1 , . . . , s ) are either all zero,or all positive. In the latter case, S h satisfies the statement of Lemma 3.2.10.Thus let us assume that E h +11 · S h = · · · = E h +1 s · S h = 0, and set S h +1 := S h +11 and E h +1 := E h +11 .Since by assumption S has properties (P1) and (P2), in order to show that S ′ stillsatisfies (P1) and (P2), we just have to consider the case i = h + 1. Then (P2) is given by(3.2.12), and (P1) follows from 3.2.4(2).Now let us show the following:(3.2.13) E h +1 l · S j = 0 and E h +1 l ∩ E j = ∅ for every j = 1 , . . . , h and l = 1 , . . . , s. In particular, for l = 1, (3.2.13) implies that S ′ satisfies (P3).Let l ∈ { , . . . , s } . Since E h · S h +1 l > E h ∩ E h +1 l = ∅ ; moreoverwe have assumed that E h +1 l · S h = 0. Therefore (3.2.13) holds for j = h and l = 1 , . . . , s .We proceed by decreasing induction on j : we assume that (3.2.13) holds for some j ∈{ , . . . , h } and for every l = 1 , . . . , s , and we show that E h +1 l · S j − = 0 and E h +1 l ∩ E j − = ∅ for every l = 1 , . . . , s .Fix l ∈ { , . . . , s } . Since E h +1 l · S j = 0 and E h +1 l ∩ E j = ∅ by the induction assumption, E h +1 l contains a curve C with class in S j , in particular(3.2.14) S j ⊂ N ( E h +1 l , X ) . Since E j − · S j > E j − ∩ C = ∅ and hence E h +1 l ∩ E j − = ∅ . Moreover E h +1 l · S j = 0 implies that E h +1 l = E j − , thus E h +1 l · S j − ≥ E j − is the locus of the extremal ray S j − , of type ( n − , n − sm ;in particular E j − is a P -bundle. Since E h +11 , . . . , E h +1 s are pairwise disjoint, by Remark3.1.6 the intersections E h +1 l · S j − (for l = 1 , . . . , s ) are either all zero or all positive.25y contradiction, suppose that E h +1 l · S j − > l = 1 . . . , s . We havecodim N ( E j − , X ) = c X by (P1), hence 3.1.3(1) givescodim N ( E j − ∩ E h +1 l , X ) ≤ codim N ( E j − , X ) + 1 = c X + 1 for every l = 1 , . . . , s. Since s ≥
3, we can apply Lemma 3.1.7 to E j − and E h +11 , . . . , E h +1 s , and deduce thatcodim N ( E j − ∩ E h +1 , X ) = c X + 1 and N ( E j − ∩ E h +1 , X ) ⊆ ( E h +1 ) ⊥ . In particular N ( E j − ∩ E h +1 , X ) ⊆ N ( E h +1 , X ) ∩ ( E h +1 ) ⊥ . On the other hand N ( E h +1 , X ) ( E h +1 ) ⊥ because E h +1 · S h +1 <
0, thereforecodim (cid:16) N ( E h +1 , X ) ∩ ( E h +1 ) ⊥ (cid:17) = c X + 1 = codim N ( E j − ∩ E h +1 , X ) , and the two subspaces coincide.By (3.2.14) and by the induction assumption we have S j ⊂ N ( E h +1 , X ) ∩ ( E h +1 ) ⊥ ,therefore S j ⊂ N ( E j − , X ), and this contradicts property (P2). (cid:4) Proof of Proposition 3.2.1.
Let R be the extremal ray of X given by Lemma 3.2.10, andset E := Locus( R ). Then codim N ( E , X ) = c X , and there exists a special Mori programfor − E which determines extremal rays R , . . . , R s such that E i · R > i = 1 , . . . , s ,where E i := Locus( R i ). Thus Lemma 3.2.7 applies.If R is an extremal ray of X different from R , . . . , R s , by 3.2.7(2) we have E i · R ≥ i = 1 , . . . , s , hence ( − K X + E + · · · + E s ) · R >
0. On the other hand( − K X + E + · · · + E s ) · R i = 0 for every i = 1 , . . . , s (recall from 3.2.7(3) that E , . . . , E s are pairwise disjoint), therefore − K X + E + · · · + E s is nef and( − K X + E + · · · + E s ) ⊥ ∩ NE( X ) = R + · · · + R s is a face of NE( X ), of dimension s by 3.2.7(5).Let σ : X → X s be the associated contraction, so that ker σ ∗ = R ( R + · · · + R s ). Since E , . . . , E s are pairwise disjoint, we see that Exc( σ ) = E ∪ · · · ∪ E s , X s is smooth, and σ is the blow-up of s smooth, pairwise disjoint, irreducible subvarieties T , . . . , T s ⊂ X s ofcodimension 2, where T i := σ ( E i ) for i = 1 , . . . , s . Moreover X s is again Fano, because − K X + E + · · · + E s = σ ∗ ( − K X s ). Recall from 3.2.7(1) that E i ∼ = P × F , and notice that σ | E i is the projection onto F ∼ = T i .Set ( E ) s := σ ( E ) ⊂ X s . Since E ∼ = P × F and E ∩ E i = { pts } × F for i = 1 , . . . , s by 3.2.7(1) and 3.2.7(3), the morphism σ | E : E → ( E ) s is birational and finite, i.e. it isthe normalization. Moreover for i = 1 , . . . , s we have T i = σ ( E ∩ E i ) ⊂ ( E ) s , so that(3.2.15) N ( T i , X s ) = σ ∗ ( N ( E ∩ E i , X )) = σ ∗ ( L ) , where L ⊂ N ( X ) is the linear subspace defined in 3.2.7(5). Again by 3.2.7(5) we know that N ( E , X ) = R R ⊕ L , and that dim(ker σ ∗ + N ( E , X )) = dim ker σ ∗ + dim N ( E , X ),therefore:(3.2.16) ker σ ∗ ∩ N ( E , X ) = { } and N (( E ) s , X s ) = R σ ∗ ( R ) ⊕ σ ∗ ( L ) . σ ∗ (( E ) s ) = E + P si =1 ( E · f i ) E i (as usual we denote by f i ⊆ E i a fiberof the P -bundle), by 3.2.7(4) and 3.2.7(6) we see that(3.2.17) ( E ) s · σ ( f ) = s X i =1 ( E · f i )( E i · f ) − > σ ∗ ( L ) ⊆ ( E ) ⊥ s (recall that s ≥ s ∈ { c X − , c X } by 3.2.4(1)).Factoring σ as a sequence of s blow-ups, we can view σ : X → X s as a part of a specialMori program for − E in X , with s steps, and by (3.2.16) at each step we have Q i (( E ) i , X i ). In particular 2.6(3) yields that codim N (( E ) s , X s ) = codim N ( E , X ) − s = c X − s , hence either s = c X and N (( E ) s , X s ) = N ( X s ), or s = c X − N (( E ) s , X s ) = 1. Suppose that there exists an extremal ray R of X s with ( E ) s · R > R ) ( X s . Then s = c X − R
6⊂ N (( E ) s , X s ).Since we have shown that N (( E ) s , X s ) = N ( X s ) when s = c X , it is enough to showthat R
6⊂ N (( E ) s , X s ).We first show that R NE(( E ) s , X s ). Otherwise, since NE(( E ) s , X s ) ⊆ NE( X s ), R should be a one-dimensional face of NE(( E ) s , X s ). We have NE( E , X ) = R + NE( F , X )and NE(( E ) s , X s ) = σ ∗ ( R ) + σ ∗ (NE( F , X )). On the other hand 3.2.7(5) and (3.2.17)give σ ∗ (NE( F , X )) ⊂ σ ∗ ( N ( F , X )) = σ ∗ ( L ) ⊆ ( E ) ⊥ s , while ( E ) s · R >
0, therefore we get R = σ ∗ ( R ). But ( E ) s is covered by the curves σ ( f ),so that Locus( R ) ⊇ ( E ) s , which is impossible.Therefore R NE(( E ) s , X s ), and in particular the contraction of R is finite on ( E ) s .Since ( E ) s · R >
0, this means that the contraction of R has fibers of dimension ≤ R is of type ( n − , n − sm by [And85, Theorem 2.3] and [Wi´s91, Theorem 1.2].In particular, E R := Locus( R ) is a prime divisor covered by curves of anticanonicaldegree 1. Moreover these curves have class in R , thus they cannot be contained in T ∪· · · ∪ T s , because T ∪ · · · ∪ T s ⊂ ( E ) s . By a standard argument (see for instance [Cas08,Remark 2.3]) we deduce that E R ∩ ( T ∪ · · · ∪ T s ) = ∅ , hence by (3.2.15) and Remark 3.1.2we have σ ∗ ( L ) = N ( T , X s ) ⊆ E ⊥ R . Moreover E R · σ ( f ) ≥
0, because E R = ( E ) s (as ( E ) s · R > R
6⊂ N (( E ) s , X s ). By contradiction, suppose that R ⊂ N (( E ) s , X s ),and let C be an irreducible curve with class in R . Then by (3.2.16) we have [ C ] = λ [ σ ( f )]+ γ , with λ ∈ R and γ ∈ σ ∗ ( L ). Using (3.2.17) we get 0 < ( E ) s · C = λ ( E ) s · σ ( f ) and( E ) s · σ ( f ) >
0, thus λ >
0. On the other hand − E R · C = λE R · σ ( f ), which givesa contradiction. Thus R
6⊂ N (( E ) s , X s ). We show that we can assume that there exists an extremal ray R of X s such that( E ) s · R > R ) = X s . 27his is clear if s = c X , by 3.2.18. Suppose that s = c X −
1, and consider an extremalray R of X s with ( E ) c X − · R >
0. If Locus( R ) = X c X − , we are done; otherwise, by3.2.18, we have R
6⊂ N (( E ) c X − , X c X − ).Let σ c X − : X c X − → X c X be the contraction of R , and consider the sequence X σ −→ X c X − σ cX − −→ X c X . Again, factoring σ as a sequence of c X − − E in X , with c X steps, and at each step Q i
6⊂ N (( E ) i , X i ).The P -bundles determined by this special Mori program are E , . . . , E c X − , and thetransform of E R in X ; the associated extremal rays (see Lemma 3.2.4) are R , . . . , R c X − ,and an additional extremal ray R c X .Since E · R >
0, Lemma 3.2.7 still applies, thus we can just replace R , . . . , R c X − with R , . . . , R c X , and restart. Since now the extremal rays are c X (instead of c X − By 3.2.19 there exists an elementary contraction of fiber type ϕ : X s → Y suchthat ( E ) s · NE( ϕ ) >
0; set ψ := ϕ ◦ σ : X → Y , and notice that ϕ (( E ) s ) = ψ ( E ) = Y . X ψ ( ( σ / / X s ϕ / / Y The sequence above is a Mori program for − E , with s steps, and at each step Q i (( E ) i , X i ). By 2.7(2) we have two possibilities: either N (( E ) s , X s ) = N ( X s ) and s = c X , or NE( ϕ )
6⊂ N (( E ) s , X s ) and s = c X − N ( T , X s ) ⊆ ( E ) ⊥ s by (3.2.15) and (3.2.17), ϕ must be finite on T , so thatdim Y ≥ n − ϕ is not finite on ( E ) s . In this case NE( ϕ ) ⊂ N (( E ) s , X s ),therefore N (( E ) s , X s ) = N ( X s ) and s = c X . This also shows that L = E ⊥ ∩ · · · ∩ E ⊥ c X ,by 3.2.7(6).Recall that Y = ψ ( E ), and that E ∼ = P × F is smooth and Fano by 3.2.7(1). Moreoverif F := { pt }× F ⊂ E , then N ( F , X ) = L by 3.2.7(5). Finally N ( σ ( F ) , X c X ) = σ ∗ ( L ) ⊆ ( E ) ⊥ c X by (3.2.17), so that ϕ is finite on σ ( F ). Since σ is finite on E , we deduce that ψ is finite on F .Let E α → e Y → Y be the Stein factorization of ψ | E . Since ϕ is not finite on ( E ) c X , α is a non-trivial contraction of E . On the other hand α is finite on F : the only possibilityis that e Y ∼ = F and α is the projection.We deduce that dim Y = n − ψ contracts f , hence NE( ϕ ) = σ ∗ ( R ).Thus NE( ψ ) is a ( c X + 1)-dimensional face of NE( X ) containing R , . . . , R c X ; in particular ρ Y = ρ X − c X − H := 2 E + c X X i =1 E i X . By 3.2.7(4) we have H · R i > i = 0 , . . . , c X , and L = E ⊥ ∩ · · · ∩ E ⊥ c X ⊆ H ⊥ . Recall from 3.2.7(1) and 3.2.7(5) that for every i = 0 , . . . , c X we have E i ∼ = P × F ,and if F i := { pt } × F ⊂ E i , then N ( F i , X ) = L ⊂ H ⊥ . In particular NE( E i , X ) = R i + NE( F i , X ) ⊂ R i + L .Let C ⊂ X be an irreducible curve with C ⊂ Supp H = E ∪ · · · ∪ E c X . Then C ⊆ E i for some i ∈ { , . . . , c X } , hence [ C ] ∈ R i + L and H · C ≥ H is effective, we have H · C ′ ≥ C ′ not contained in Supp H . Therefore H is nef and defines a contraction ξ : X → S suchthat NE( ξ ) = H ⊥ ∩ NE( X ). X ξ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ ψ ! ! ❈❈❈❈❈❈❈❈❈ σ / / X c X ϕ (cid:15) (cid:15) S Y
Let i ∈ { , . . . , c X } . Since N ( F i , X ) ⊂ H ⊥ , the image ξ ( F i ) is a point, and ξ ( E i ) = ξ ( f i )is an irreducible rational curve (because H · f i > ξ | E i : E i → ξ ( f i ) factorsthrough the projection E i → P . In particular dim ξ (Supp H ) = 1, hence S is a surface by3.1.1(1).Let us show that(3.2.22) NE( ξ ) = L ∩ NE( X ) . We already have NE( ξ ) = H ⊥ ∩ NE( X ) ⊇ L ∩ NE( X ). Conversely, let C ⊂ X be anirreducible curve such that ξ ( C ) = { pt } , i.e. H · C = 0.If C is disjoint from Supp H = E ∪ · · · ∪ E c X , then C · E i = 0 for i = 0 , . . . , c X , hence[ C ] ∈ L .If instead C intersects E ∪ · · · ∪ E c X , then it must be contained in it, and we have C ⊂ E i for some i . Since ξ | E i factors as the projection onto P followed by a finite map,we get C ⊂ F i , and again [ C ] ∈ N ( F i , X ) = L . Therefore we have (3.2.22).In particular, for every i = 0 , . . . , c X we have NE( ξ ) ⊆ E ⊥ i , therefore E i = ξ ∗ ( ξ ( E i )) by3.1.1(2).Let π : X → S × Y be the morphism induced by ξ and ψ . We have ker ψ ∗ = R ( R + · · · + R c X ), and ker ψ ∗ ∩ L = { } by 3.2.7(5). Moreover ker ξ ∗ ⊆ L by (3.2.22), therefore π is finite.In particular, ξ must be equidimensional, hence S is smooth by [ABW92, Proposition1.4.1] and [Cas08, Lemma 3.10]. We need the following remark. Remark 3.2.23.
Let W be a smooth Fano variety and suppose we have two contractions W π } } ④④④④④④④④ π ! ! ❈❈❈❈❈❈❈❈ W W W is smooth and the induced morphism π : W → W × W is finite. Considerthe relative canonical divisor K W/W := K W − π ∗ K W . If ker( π ) ∗ ⊆ ( K W/W ) ⊥ in N ( W ),then π is an isomorphism.This is rather standard, we give a proof for the reader’s convenience. Let d be thedegree of π , and F ⊂ W a general fiber of π ; the restriction f := ( π ) | F : F → W is finiteof degree d . We observe that F is Fano, hence numerical and linear equivalence for divisorsin F coincide, and by assumption ( K W/W ) | F ≡
0. Then K F = ( K W ) | F = ( π ∗ K W ) | F = f ∗ K W , so that f is ´etale. Therefore W is Fano too, in particular it is simply connected, thus f isan isomorphism and d = 1.We carry on with the proof of Proposition 3.2.1. We want to apply Remark 3.2.23to deduce that π : X → S × Y is an isomorphism; for this we just need to show that K X/S · R i = 0 for i = 0 , . . . , c X , because ker ψ ∗ = R ( R + · · · + R c X ). But this follows easilybecause E i are products.Indeed since both S and E i are smooth, 3.1.1(4) yields that ξ ( E i ) is a smooth curve.Therefore ξ ( E i ) ∼ = P and ξ | E i is the projection, hence K X/S · f i = ( K X/S ) | E i · f i = K E i /ξ ( E i ) · f i = 0 . Thus we conclude that π is an isomorphism and X ∼ = S × Y . Moreover since ρ Y = ρ X − c X −
1, we have ρ S = c X + 1. ϕ is finite on ( E ) s . Then dim Y = n − ϕ isone-dimensional; moreover every fiber of ψ has an irreducible component of dimension 1.Since X and X s are Fano, [AW97, Lemma 2.12 and Theorem 4.1] show that Y is smoothand that ϕ and ψ are conic bundles. X ψ ( ( σ / / X s ϕ / / Y Set Z i := ϕ ( T i ) = ψ ( E i ) ⊂ Y for i = 1 , . . . , s . By standard arguments on conic bundles(as at the end of the proof of Lemma 2.8), we see that Z , . . . , Z s ⊂ Y are pairwise disjointsmooth prime divisors, and that ϕ is smooth over Z ∪ · · · ∪ Z s . For i = 1 , . . . , s let b E i ⊂ X be the transform of ϕ − ( Z i ) ⊂ X s , so that ψ − ( Z i ) = E i ∪ b E i . Then b E i is a smooth P -bundle with fiber b f i ⊂ b E i , such that b E i · b f i = −
1. Moreover f i + b f i is numerically equivalentto a general fiber of ψ , and E i · b f i = b E i · f i = 1.In particular, the divisors E , E , . . . , E s , b E , . . . , b E s are all distinct (recall that ψ ( E ) = Y ), and E ∪ b E , . . . , E s ∪ b E s are pairwise disjoint.Let us show that [ E ] , [ E ] , . . . , [ E s ] , [ b E ] are linearly independent in N ( X ). Indeedsuppose that aE + s X i =1 b i E i + d b E ≡ , a, b i , d ∈ R . Intersecting with a general fiber of ψ : X → Y , we get a = 0. Intersectingwith f , . . . , f s , we get b = · · · = b s = 0. Finally intersecting with f we get d = b , thatis, d ( E + b E ) ≡
0, which yields d = 0, and we are done.If i, j ∈ { , . . . , s } with i = j , we have E i ∩ b E j = ∅ , and hence L ⊆ N ( E i , X ) ⊆ b E ⊥ j (see Remark 3.1.2). Therefore by 3.2.7(6) L ⊆ E ⊥ ∩ E ⊥ ∩ · · · ∩ E ⊥ s ∩ b E ⊥ ∩ · · · ∩ b E ⊥ s ⊆ E ⊥ ∩ E ⊥ ∩ · · · ∩ E ⊥ s ∩ b E ⊥ . Since the classes of E , . . . , E s , b E in N ( X ) are linearly independent and s ≥ c X −
1, weget c X + 1 = codim L ≥ s + 2 ≥ c X + 1 , which yields s = c X − L = E ⊥ ∩ E ⊥ ∩ · · · ∩ E ⊥ c X − ∩ b E ⊥ = E ⊥ ∩ E ⊥ ∩ · · · ∩ E ⊥ c X − ∩ b E ⊥ ∩ · · · ∩ b E ⊥ c X − . Let i ∈ { , . . . , c X − } . Observe that [ b f i ]
6∈ N ( E i , X ): otherwise by 3.2.7(5) we wouldhave b f i ≡ λf i + γ , with λ ∈ R and γ ∈ L ⊂ E ⊥ ∩ E ⊥ i . Intersecting with E i we get λ = −
1, hence E · b f i = − E · f i <
0, which is impossible because E = b E i . We alsonotice that E cannot contain any curve b f i , because σ ( b f i ) is a fiber of ϕ , and ϕ is finite on( E ) c X − = σ ( E ).Therefore we can apply Lemma 3.1.10 to E and E , . . . , E c X − , b E , . . . , b E c X − , and weget: codim N ( b E i , X ) = c X and R i
6⊂ N ( b E i , X ) for every i = 1 , . . . , c X − i ∈ { , . . . , c X − } . Lemma 3.2.5, applied to b E i and R i , yields that N ( E i ∩ b E i , X ) = N ( b E i , X ) ∩ E ⊥ i = N ( E i , X ) ∩ E ⊥ i = L (see (3.2.8) for the last equality). Finally we apply Lemma 3.1.5 to D = E i and E = b E i ,and we deduce that b R i := R ≥ [ b f i ] is an extremal ray of type ( n − , n − sm , b E i ∼ = P × b F i ,and E i ∩ b E i = { pts } × b F i ⊂ b E i . On the other hand again Lemma 3.1.5, applied now to D = b E i and E = E i , shows that E i ∩ b E i = { pts } × F ⊂ E i ∼ = P × F , hence b F i = F .Observe that NE( ψ ) = R + b R + · · · + R c X − + b R c X − has dimension c X , and that ψ | E : E ∼ = P × F → Y is finite. We need the following lemma. Lemma 3.2.25.
Let E be a projective manifold and π : E → W a P -bundle with fiber f ⊂ E . Moreover let ψ : E → Y be a morphism onto a projective manifold Y , such that dim ψ ( f ) = 1 . Suppose that there exists a prime divisor Z ⊂ Y such that N ( Z , Y ) ( N ( Y ) and ψ ∗ ( Z ) · f > . Then there is a commutative diagram: E ψ / / π (cid:15) (cid:15) Y ζ (cid:15) (cid:15) W / / Y ′ where Y ′ is smooth and ζ is a smooth morphism with fibers isomorphic to P . roof of Lemma 3.2.25. Consider the morphism φ : E → W × Y induced by π and ψ , set E ′ := φ ( E ) ⊂ W × Y , and let π ′ : E ′ → W be the projection. For every p ∈ W we have π − ( p ) = φ − (( π ′ ) − ( p )), hence ( π ′ ) − ( p ) = ψ ( π − ( p )) ⊂ Y is an irreducible and reducedrational curve in Y .Now π ′ : E ′ → W is a well defined family of algebraic one-cycles on Y over W (see[Kol96, Definition I.3.11 and Theorem I.3.17]), and induces a morphism ι : W → Chow( Y ).Set V := ι ( W ) ⊂ Chow( Y ). Then V is a proper, covering family of irreducible and reducedrational curves on Y , so that V is an unsplit family (see [Kol96, Definition IV.2.1]).The family V induces an equivalence relation on Y as a set, called V -equivalence;we refer the reader to [Deb01, §
5] and references therein for the related definitions andproperties.We have Z · ψ ( f ) >
0; in particular Z intersects every V -equivalence class in Y . Thisimplies that N ( Y ) = R [ ψ ( f )] + N ( Z , Y )(see for instance [Occ06, Lemma 3.2]). On the other hand by assumption N ( Z , Y ) ( N ( Y ), therefore [ ψ ( f )]
6∈ N ( Z , Y ).Let T ⊆ Y be a V -equivalence class; notice that T is either a closed subset, or a countableunion of closed subsets. Let T ⊆ T be an irreducible closed subset with dim T = dim T .We have N ( T , Y ) = R [ ψ ( f )] by [Kol96, Proposition IV.3.13.3], and T ∩ Z = ∅ . Thisimplies that dim( T ∩ Z ) = 0 and dim T = dim T = 1, that is: every V -equivalence classhas dimension
1. Then by [BCD07, Proposition 1] there exists a contraction ζ : Y → Y ′ whose fibers coincide with V -equivalence classes.Since Y is smooth, Y ′ is irreducible, and ζ has connected fibers, the general fiber l of ζ is irreducible and smooth. But l is a V -equivalence class and dim l = 1, hence l ∼ = P and − K Y · l = 2. Moreover NE( ζ ) = R ≥ [ l ], so − K Y is ζ -ample; this implies that ζ is anelementary contraction and a conic bundle, and that Y ′ is smooth (see [And85, Theorem3.1]). Finally ζ cannot have singular fibers, because the family V is unsplit. (cid:4) Let us carry on with the proof of Proposition 3.2.1. We have ψ ∗ ( Z ) · f = ( E + b E ) · f >
0, and N ( Z , Y ) ⊆ Z ⊥ ( N ( Y ) because Z ∩ Z = ∅ (see Remark 3.1.2). Therefore wecan apply Lemma 3.2.25 to E and ψ := ( ψ ) | E : E → Y . This shows that [ ψ ( f )] belongsto an extremal ray of Y , whose contraction is a smooth conic bundle ζ : Y → Y ′ .We consider the composition ψ ′ := ζ ◦ ψ : X → Y ′ ; the cone NE( ψ ′ ) is a ( c X + 1)-dimensional face of NE( X ) containing R , R , . . . , R c X − , b R , . . . , b R c X − , and ρ Y ′ = ρ X − c X − H ′ := 2 E + 2 c X − X i =1 E i + c X − X i =1 b E i on X . We have H ′ · R > H ′ · R i > H ′ · b R i > i = 1 , . . . , c X − H ′ ) ⊥ ⊇ L . As before, H ′ is nef and defines a contraction onto a surface ξ ′ : X → S ,32uch that ξ ′ ( E ), ξ ′ ( E i ), and ξ ′ ( b E i ) are irreducible rational curves and E = ( ξ ′ ) ∗ ( ξ ′ ( E )), E i = ( ξ ′ ) ∗ ( ξ ′ ( E i )), b E i = ( ξ ′ ) ∗ ( ξ ′ ( b E i )) for all i = 1 , . . . , c X − X ψ ′ (cid:15) (cid:15) ξ ′ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ ψ ❋❋❋❋❋❋❋❋❋❋ σ / / X c X − ϕ (cid:15) (cid:15) S Y ′ Y ζ o o Then we consider the morphism π ′ : X → S × Y ′ induced by ξ ′ and ψ ′ . As in theprevious case, one sees first that π ′ is finite, and then that it is an isomorphism, applyingRemark 3.2.23. Finally we have ρ S = c X + 1, because ρ Y ′ = ρ X − c X − We have shown in 3.2.21 and 3.2.24 that X ∼ = S × T , where S is a Del Pezzo surfacewith ρ S = c X + 1 (and T = Y in 3.2.21, while T = Y ′ in 3.2.24). In particular c X ≤
8, as ρ S ≤
9. Finally c T ≤ c X by Example 3.1, and this concludes the proof of Proposition 3.2.1. (cid:4) In this section we show the following.
Proposition 3.3.1.
Let X be a Fano manifold with c X = 3 . Then there exists a flat,quasi-elementary contraction X → T where T is an ( n − -dimensional Fano manifold, ρ X − ρ T = 4 , and c T ≤ .Proof. By Corollary 3.2.2, there exist a prime divisor D ⊂ X with codim N ( D, X ) = 3,and a special Mori program for − D , such that Q k
6⊂ N ( D k , X k ).(3.3.2) X = X σ , , ❡ ❞ ❝ ❜ ❛ ❵ ❴ ❫ ❪ ❭ ❬ ❩ ❨ ❳ ψ / / ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ❬ ❭ ❪ ❫ ❫ ❴ σ / / ❴❴❴ X / / ❴❴❴ · · · / / ❴❴❴ X k − σ k − / / ❴❴❴ X kϕ (cid:15) (cid:15) Y We apply Lemmas 2.7 and 2.8. By 2.7(2) and 2.7(3), there exist exactly two indices i , i ∈ { , . . . , k − } such that Q i j
6⊂ N ( D i j , X i j ); the P -bundles E , E ⊂ X determinedby the Mori program are the transforms of Exc( σ i ) , Exc( σ i ) respectively. Let moreover b E , b E ⊂ X be as in 2.8(4). Recall that for i = 1 , E i (respectively, b E i ) is a smooth P -bundle with fiber f i ⊂ E i (respectively, b f i ⊂ b E i ), such that E i · f i = b E i · b f i = − E i · b f i > b E i · f i >
0. Moreover ( E ∪ b E ) ∩ ( E ∪ b E ) = ∅ . Before going on, let us give an outline of what we are going to do.Our goal is to show that k = 2 and σ is just the composition of two smooth blow-upswith exceptional divisors E and E . The proof of this fact is quite technical, and will beachieved in several steps.We first show in 3.3.4 some properties of N ( E i , X ) and N ( b E i , X ) which are needed inthe sequel. 33n 3.3.6 we prove that if F ⊂ X is a prime divisor whose class in N ( X ) spans a one-dimensional face of the cone of effective divisors Eff( X ) ⊂ N ( X ) (see 3.3.5), then F mustintersect both E ∪ b E and E ∪ b E .Then we show in 3.3.7 that the Mori program (3.3.2) contains only two divisorial con-tractions, the ones with exceptional divisors E and E . We proceed by contradiction,applying 3.3.6 to the exceptional divisor of a divisorial contraction (different from σ i and σ i ) in the Mori program.In 3.3.9 and 3.3.10 we prove the existence of two disjoint prime divisors F, b F ⊂ X ,which are smooth P -bundles with fibers l ⊂ F , b l ⊂ b F such that F · l = b F · b l = − ψ : X Y , and intersect the divisors E , E , b E , b E in a suitable way.Finally in 3.3.11 and 3.3.13 we use F and b F to show that the Mori program (3.3.2)contains no flips. This means that k = 2, X and Y are smooth, σ is just a smooth blow-upwith exceptional divisors E and E , and ϕ and ψ are conic bundles.The situation is now analogous to the one in 3.2.24, and similarly to that case we provethat there is a smooth conic bundle Y → Y ′ , where dim Y ′ = n − ρ X − ρ Y ′ = 4, and the contraction X → Y ′ is flat and quasi-elementary.To conclude, in 3.3.16 we show that the conic bundle ϕ : X → Y is smooth. Thisimplies that every fiber of the conic bundle ψ : X → Y is reduced, and hence by a result in[Wi´s91] both Y and Y ′ are Fano. For i = 1 , N ( E i , X ) = codim N ( b E i , X ) = 3 , [ b f i ]
6∈ N ( E i , X ) , and [ f i ]
6∈ N ( b E i , X );in particular N ( E i , X ) = N ( b E i , X ).Indeed [ b f i ]
6∈ N ( E i , X ) by 2.8(4). Moreover D cannot contain any curve b f i , because σ ( b f i ) is a fiber of ϕ , and ϕ is finite on D k ⊂ X k . Therefore Lemma 3.1.10 yields thestatement. Let Z be a Mori dream space, and Eff( Z ) ⊂ N ( Z ) the convex cone spanned byclasses of effective divisors. By [HK00, Proposition 1.11(2)] Eff( Z ) is a closed, convexpolyhedral cone. If F ⊂ Z is a prime divisor covered by a family of curves with which F has negative intersection, then it is easy to see that [ F ] ∈ N ( Z ) spans a one-dimensionalface of Eff( Z ), and that the only prime divisor whose class belongs to this face is F itself.In particular, this is true for E , E , b E , b E ⊂ X (recall that X is a Mori dream space byTheorem 2.1). Consider a prime divisor F ⊂ X such that [ F ] spans a one-dimensional face ofEff( X ). We show that if F is different from E , E , b E , b E , then F must intersect both E ∪ b E and E ∪ b E .Indeed if for instance F is disjoint from E ∪ b E , then N ( E , X ) ∪N ( b E , X ) ⊆ E ⊥ ∩ b E ⊥ ∩ F ⊥ (see Remark 3.1.2). However this is impossible, because since [ E ] , [ b E ] , [ F ] ∈ N ( X )span three distinct one-dimensional faces of Eff( X ), they must be linearly independent, thus E ⊥ ∩ b E ⊥ ∩ F ⊥ has codimension 3, while N ( E , X ) and N ( b E , X ) are distinct subspacesof codimension 3 by 3.3.4. 34 .3.7. Let us show that σ i is a flip for every i ∈ { , . . . , k − } r { i , i } , namely that σ i and σ i are the unique divisorial contractions in the Mori program (3.3.2).By contradiction, suppose that there exists i ∈ { , . . . , k − } r { i , i } such that σ i is a divisorial contraction. By 3.3.5 Exc( σ i ) ⊂ X i is a prime divisor whose class spansa one-dimensional face of Eff( X i ), and it is the unique prime divisor in X i with class in R ≥ [Exc( σ i )]. Let G ⊂ X be the transform of Exc( σ i ). By 2.8(3) and 2.8(4) there exists an open subset U ⊆ X , containing E , E , b E , b E , such that σ is regular on U , and Exc( σ i ) is disjoint fromthe image of U in X i . Therefore G ∩ U = ∅ , in particular the divisor G is disjoint from E , E , b E , b E .Then 3.3.6 shows that [ G ] ∈ N ( X ) cannot span an extremal ray of Eff( X ). This meansthat [ G ] = P j λ j [ G j ] with λ j ∈ R > and G j ⊂ X prime divisors such that [ G ] R ≥ [ G j ];in particular G j = G .On the other hand, the map ξ := σ i − ◦ · · · ◦ σ : X X i induces a surjective linearmap ξ ∗ : N ( X ) → N ( X i ) such that ξ ∗ (Eff( X )) = Eff( X i ). Then in N ( X i ) we get[Exc( σ i )] = [ ξ ∗ ( G )] = X j λ j [ ξ ∗ ( G j )] , hence [ ξ ∗ ( G j )] ∈ R ≥ [Exc( σ i )] for every j . If ξ ∗ ( G j ) = 0 for some j , then ξ ∗ ( G j ) is a primedivisor, and we get ξ ∗ ( G j ) = Exc( σ i ) and hence G j = G , a contradiction. Thus ξ ∗ ( G j ) = 0for every j , therefore [Exc( σ i )] = 0, again a contradiction. Let F ⊂ X be a smooth prime divisor which is a P -bundle with F · l = −
1, where l ⊂ F is a fiber. Suppose that F is different from E , E , b E , b E . Then: • F must intersect both E ∪ b E and E ∪ b E ; • either E · l = b E · l = E · l = b E · l = 0, or ( E + b E ) · l > E + b E ) · l > F ] spans a one-dimensional face of Eff( X ), so that 3.3.6 gives the first state-ment.Recall that ( E ∪ b E ) ∩ ( E ∪ b E ) = ∅ . If ( E + b E ) · l = 0, since F intersects E ∪ b E ,there exists a fiber l of the P -bundle structure of F which is contained in E ∪ b E . Thus l ∩ ( E ∪ b E ) = ∅ , and we get ( E + b E ) · l = 0. In this way we see that the intersections( E + b E ) · l , ( E + b E ) · l are either both zero or both positive, and this gives the secondstatement. We show that there exist two disjoint smooth prime divisors F, b F ⊂ X , differentfrom E , E , b E , b E , such that: • F and b F are P -bundles, with fibers l ⊂ F and b l ⊂ b F respectively, such that F · l = b F · b l = − • the intersections ( E + b E ) · l , ( E + b E ) · b l , ( E + b E ) · l , ( E + b E ) · b l are all positive. Notice that X i is again a Mori dream space.
35e have codim N ( E , X ) = 3 (see 3.3.4). Consider a special Mori program for − E (which exists by Proposition 2.4), and let G , . . . , G s ⊂ X be the P -bundles determinedby the Mori program. Recall from Lemma 2.7 that G , . . . , G s are pairwise disjoint smoothprime divisors, with 2 ≤ s ≤
3, such that every G i is a P -bundle with G i · r i = −
1, where r i ⊂ G i is a fiber; moreover E · r i >
0. In particular G i = E and G i ∩ E = ∅ , thus G i = E and G i = b E . Finally, if G i = b E , by 3.3.8 we have ( E + b E ) · r i > E + b E ) · r i > { G , . . . , G s } contains at least two divisors distinct from b E , say G and G . Then we set F := G and b F := G , and we are done.Otherwise, we have s = 2 and G = b E . Then Lemma 2.8 applies, and by 2.8(4) thereexists a smooth prime divisor b G , having a P -bundle structure with fiber b r , such that: b G · b r = − , G ∩ b G = ∅ , b G = E , and b E · b r = 1 . In particular b G = b E and b G ∩ b E = ∅ , therefore b G = E and b G = b E . By 3.3.8 we have( E + b E ) · b r > E + b E ) · b r >
0, thus we set F := G and b F := b G . As soon as F (respectively b F ) intersects one of the divisors E i , then F · f i > E i · l > b F · f i > E i · b l > b E i . In particular we have F · f > b F · f >
0, where f is a general fiber of ψ .Suppose for instance that F ∩ E = ∅ . If E · l = 0, then E contains some curve l , butthis is impossible because ( E + b E ) · l > E ∩ ( E ∪ b E ) = ∅ ; thus E · l > F · f = 0, then F contains an irreducible curve f which is a fiber of the P -bundlestructure on E . Let π : F → G be the P -bundle structure on F , and π ∗ : N ( F ) → N ( G )the push-forward. Notice that π ( f ) is a curve, because f and l are not numericallyequivalent in X , and hence neither in F .Consider the surface S := π − ( π ( f )). Then π ∗ ( N ( S, F )) = R π ∗ ([ f ] F ), hence N ( S, F ) =ker π ∗ ⊕ R [ f ] F = R [ l ] F ⊕ R [ f ] F , and N ( S, X ) = R [ l ] ⊕ R [ f ].Since b E · f >
0, we have S ∩ b E = ∅ , and there exists an irreducible curve C ⊂ S ∩ b E . Thus [ C ] ∈ N ( S, X ), so that C ≡ λl + µf with λ, µ ∈ R . On the other hand C ∩ ( E ∪ b E ) = ∅ (because C ⊂ b E ) and0 = ( E + b E ) · C = λ ( E + b E ) · l, which by 3.3.9 yields λ = 0, µ = 0 and [ f ] = (1 /µ )[ C ] ∈ N ( b E , X ), a contradiction with3.3.4.Therefore F · f >
0. We have f ≡ f + b f (see 2.8(4)), and F · b f ≥ F = b E (see 3.3.9), hence F · f > For every i ∈ { , . . . , k } let F i , b F i ⊂ X i be the transforms of F, b F . Let us showthat for any i ∈ { , . . . , k − } r { i , i } , the divisors F i and b F i are disjoint from Locus( Q i ).By contradiction, suppose for instance that this is not true for F , and let j ∈ { , . . . , k − } r { i , i } be the smallest index such that F j intersects Locus( Q j ). Recall from 3.3.7 that σ i is a flip for every i ∈ { , . . . , k − } r { i , i } ; in particular, Q j is a small extremal ray,and σ j is a flip. 36fter 2.8(3), σ is regular on the divisors E , E , b E , b E , and Locus( Q j ) is disjoint fromtheir images in X j .Recall from 2.6(4) the definition of A i ⊂ X i , for i ∈ { , . . . , k } : A ⊂ X is theindeterminacy locus of σ − , and for i ≥
2, if σ i − is a divisorial contraction (respectively, if σ i − is a flip), A i is the union of σ i − ( A i − ) (respectively, the transform of A i − ) and theindeterminacy locus of σ − i − .If j >
0, by the minimality of j , F j does not intersect the loci of the previous flips,hence it can intersect A j only along the images of E and E . Therefore(3.3.12) Locus( Q j ) ∩ F j ∩ A j = ∅ . Let α j : X j → Y j be the contraction of Q j . Suppose first that α j is finite on F j . ThenLocus( Q j ) = Exc( α j ) F j , and since F j ∩ Locus( Q j ) = ∅ , we have F j · Q j >
0. Henceevery non trivial fiber of α j must have dimension 1, otherwise α j would not be finite on F j .If j = 0, then α is a small contraction of a smooth Fano variety with one-dimensionalfibers, which is impossible, see [AW97, Theorem 4.1].Suppose that j >
0. If C ⊂ X j is an irreducible curve in a fiber of α j , then C mustintersect F j , hence C A j by (3.3.12); in particular C Sing( X j ) (recall that Sing( X j ) ⊆ A j by 2.6(4)). Then [Ish91, Lemma 1] yields − K X j · C ≤
1, and [Cas09, Lemma 3.8] impliesthat C ∩ A j = ∅ . We conclude that Locus( Q j ) ⊆ X j r A j , and this is again impossible by[AW97, Theorem 4.1], because − K X j · Q j > α j ) | X j r A j : X j r A j → Y j r α j ( A j ) is asmall contraction of a smooth variety with one-dimensional fibers.Suppose now that α j is not finite on F j . Then there exists an irreducible curve C ⊂ F j with [ C ] ∈ Q j ; in particular C is disjoint from the images of E , E , b E , b E in X j . Considerthe transform e C ⊂ F ⊂ X of C , so that e C is disjoint from E , E , b E , b E .Recall that F intersects both E ∪ b E and E ∪ b E by 3.3.8. We assume that F intersects E and E , the other cases being analogous. Then E · l > e C ≡ λl + µC , where C ⊂ F ∩ E is a curve, λ, µ ∈ R , and µ ≥
0. In particular C ∩ E = ∅ , therefore0 = E · e C = λE · l . On the other hand E · l > λ = 0and e C ≡ µC . Recall that the map X X j is regular on F by the minimality of j ,and call C ′ the image of C in X j . We deduce that C ≡ µC ′ in X j , so that [ C ′ ] ∈ Q j .But C ′ is contained in the image of E , which is disjoint from Locus( Q j ), and we have acontradiction. We show that k = 2 in (3.3.2), so that i = 0 and i = 1.By contradiction, suppose that k >
2. Recall from 3.3.7 that σ i is a flip for every i ∈ { , . . . , k − } r { i , i } , so equivalently we are assuming that the Mori program (3.3.2)contains some flip.We define an integer m ∈ { k − , k − , k − } and a morphism η : X m +1 → X k as follows: • if σ k − is a flip, set m := k − η := Id X k ;37 if σ k − is a contraction and σ k − is a flip, set m := k − η := σ k − : X k − → X k ; • if both σ k − and σ k − are contractions, set m := k − η := σ k − ◦ σ k − : X k − → X k .It follows from these definitions that Q m is a small extremal ray of X m and σ m : X m X m +1 is a flip; let Q ′ m +1 be the corresponding small extremal ray of X m +1 . Set moreover e ϕ := ϕ ◦ η : X m +1 → Y . X σ + + ❤ ❣ ❢ ❞ ❝ ❜ ❵ ❴ ❫ ❭ ❬ ❩ ❳ ❲ / / ❴❴❴ ψ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲ X m σ m / / ❴❴❴ X m +1 η / / e ϕ ●●●●●●●●● X kϕ (cid:15) (cid:15) Y We keep the same notations as in the proof of Lemma 2.8; in particular we set T i := σ ( E i ) ⊂ X k for i = 1 ,
2. Notice that when m = k − m = k − η is justthe smooth blow-up of T ⊂ X k (respectively, of T ∪ T ⊂ X k ).Recall that for j = 1 , Q i j
6⊂ N ( D i j , X i j ), in particular σ i j is finite on D i j ;this implies that η is finite on D m +1 .We show that every fiber of e ϕ has dimension 1. Indeed this is true for ϕ by 2.8(1).Moreover η is an isomorphism over X k r ( T ∪ T ), therefore e ϕ has one-dimensional fibersover Y r ϕ ( T ∪ T ). On the other hand, we know by 2.8(3) that there exist open subsets U ⊆ X and V ⊆ Y such that ϕ ( T ∪ T ) ⊂ V , both ψ : U → V and ϕ | ϕ − ( V ) : ϕ − ( V ) → V are conic bundles, and σ | U : U → ϕ − ( V ) is just the blow-up of T and T . This impliesthat e ϕ | e ϕ − ( V ) : e ϕ − ( V ) → V is a conic bundle, in particular it has one-dimensional fibersover ϕ ( T ∪ T ) ⊂ V .Recall from 2.8(1) that ϕ is finite on D k = η ( D m +1 ), and since η is finite on D m +1 , wededuce that e ϕ must be finite on D m +1 . Notice also that D m +1 ⊃ A m +1 ⊇ Locus( Q ′ m +1 )(see 2.6(4)). As in the proof of Lemma 2.8, using [Cas09, Lemma 3.8] we see that everyfiber of e ϕ which intersects Locus( Q ′ m +1 ) is an integral rational curve.Let C ⊂ X m +1 be an irreducible curve with [ C ] ∈ Q ′ m +1 , and set S := e ϕ − ( e ϕ ( C )), sothat S is an irreducible surface.Since, by 3.3.10, F and b F have positive intersection with a general fiber of ψ in X , F m +1 and b F m +1 have positive intersection with every fiber of e ϕ in X m +1 . In particular, F m +1 and b F m +1 intersect S .On the other hand by 3.3.11 the divisors F m and b F m in X m are disjoint from Locus( Q m ),therefore F m +1 and b F m +1 are disjoint from Locus( Q ′ m +1 ). We deduce that:(3.3.14) F m +1 ∩ C = b F m +1 ∩ C = ∅ and dim( F m +1 ∩ S ) = dim( b F m +1 ∩ S ) = 1 . For i = 1 , G i the image of E i in X m +1 , so that T i = η ( G i ) and ϕ ( T i ) = e ϕ ( G i ).Notice that A k r ( T ∪ T ) = η ( A m +1 r ( G ∪ G )).Recall that the open subset V ⊆ Y was defined in (2.10) as V := Y r ϕ ( A k r ( T ∪ T )) = Y r e ϕ ( A m +1 r ( G ∪ G )) .
38y 2.7(1) and 2.8(2) we have Locus( Q ′ m +1 ) ∩ ( G ∪ G ) = ∅ . In particular C ⊆ Locus( Q ′ m +1 ) ⊆ A m +1 r ( G ∪ G ), thus e ϕ ( C ) ⊆ Y r V. On the other hand we also have e ϕ ( G ∪ G ) = ϕ ( T ∪ T ) ⊂ V , therefore we deducethat e ϕ ( G ∪ G ) ∩ e ϕ ( C ) = ∅ and hence( G ∪ G ) ∩ S = ∅ . Finally by 3.3.9 we have F ∩ b F = ∅ in X , and by 3.3.11 the divisors F and b F are disjointfrom the locus of every flip in the Mori program (3.3.2). This implies that F m +1 ∩ b F m +1 ⊆ G ∪ G , therefore: F m +1 ∩ b F m +1 ∩ S = ∅ . Together with (3.3.14), this yields that C , F m +1 ∩ S , and b F m +1 ∩ S are pairwise disjointcurves in S .Let C ′ be an irreducible component of b F m +1 ∩ S . Since e ϕ | S : S → e ϕ ( C ) is a fibration inintegral rational curves, we have C ′ ≡ λC + µf where λ, µ ∈ R and f ⊂ S is a fiber. Then0 = F m +1 · C ′ = µF m +1 · f while F m +1 · f >
0, hence µ = 0 and [ C ′ ] ∈ Q ′ m +1 . Therefore C ′ ⊆ Locus( Q ′ m +1 ) ∩ F m +1 , a contradiction because Locus( Q ′ m +1 ) ∩ F m +1 = ∅ . Since k = 2, X is smooth and σ : X → X is just the blow-up of two disjointsmooth subvarieties T , T ⊂ X , of codimension 2. In fact, we have A = T ∪ T (see2.6(4)), and by (2.10) the description in 2.8(3) and 2.8(4) holds with V = Y and U = X .In particular, Y is smooth, ϕ : X → Y and ψ : X → Y are conic bundles, ρ X − ρ Y = 3, andthe divisors Z = ψ ( E ) and Z = ψ ( E ) are disjoint in Y . Moreover we have ψ ( F ) = Y by 3.3.10.The situation is very similar to the case where ϕ is finite on ( E ) s in 3.2.24, with thedifference that the E i ’s do not need to be products. In the same way we use Lemma 3.2.25to show that [ ψ ( l )] ∈ NE( Y ) belongs to an extremal ray of Y , whose contraction is a smoothconic bundle ζ : Y → Y ′ , finite on Z and Z ; in particular Y ′ is smooth of dimension n − ψ ′ := ζ ◦ ψ : X → Y ′ is equidimensional and hence flat, and ρ X − ρ Y ′ = 4.Moreover the general fiber of ψ ′ is a Del Pezzo surface S containing curves f , b f , f , b f , l ,hence N ( S, X ) = ker( ψ ′ ) ∗ and ψ ′ is quasi-elementary. X ψ ′ (cid:15) (cid:15) ψ ❇❇❇❇❇❇❇❇ σ / / X ϕ (cid:15) (cid:15) Y ′ Y ζ o o We show that the conic bundle ϕ : X → Y is smooth.By contradiction, suppose that this is not the case, and let ∆ ϕ ⊂ Y be the discriminantdivisor of ϕ . Recall that this is an effective, reduced divisor in Y such that ϕ − ( y ) issingular if and only if y ∈ ∆ ϕ .Consider also the discriminant divisor ∆ ψ ⊂ Y of the conic bundle ψ : X → Y . Since ϕ issmooth over Z and Z , the divisors ∆ ϕ , Z , Z are pairwise disjoint, and ∆ ψ = ∆ ϕ ∪ Z ∪ Z .39he fibers of ψ over Z ∪ Z are singular but reduced, hence ψ − ( y ) is non-reduced ifand only if ϕ − ( y ) is. Let W ⊂ ∆ ϕ be the set of points y such that ψ − ( y ) (equivalently, ϕ − ( y )) is non-reduced. Then W is a closed subset of Y , and W ⊆ Sing(∆ ϕ ) (see forinstance [Sar82, Proposition 1.8(5.c)]). Moreover by [Wi´s91, Proposition 4.3] we know that − K Y · C > C ⊂ Y not contained in W .For i = 1 , N ( Z i , Y ) ≤
1, because ζ ( Z i ) = Y ′ and hence ζ ∗ ( N ( Z i , Y )) = N ( Y ′ ). This yields Z ⊥ = Z ⊥ = ∆ ⊥ ϕ = N ( Z , Y ) = N ( Z , Y ) (see Remark 3.1.2). Thethree divisors ∆ ϕ , Z , Z are numerically proportional, nef, and cut a facet of NE( Y ), whosecontraction β : Y → P sends ∆ ϕ , Z , Z to points (see [Cas08, Lemma 2.6]). Even if a pri-ori we do not know whether every curve contracted by β has positive anticanonical degree,the general fiber of β does not meet W , therefore it is a Fano manifold. Moreover NE( β )is generated by finitely many classes of rational curves (see [Cas08, Lemma 2.6]). Thus thesame proof as [Cas09, Lemma 4.9] yields that Y ∼ = P × Y ′ , and ∆ ϕ = { pts } × Y ′ .In particular ∆ ϕ is smooth, hence W = ∅ and Y is Fano. Because Y ∼ = P × Y ′ , Y ′ isFano too, so that each connected component of ∆ ϕ is simply connected. However this isimpossible, because by a standard construction the conic bundle ϕ defines a double coverof every irreducible component of ∆ ϕ , obtained by considering the components of the fibersin the appropriate Hilbert scheme of lines, see [Bea77, § § ϕ is an elementary contraction, this double cover is non-trivial; on the other hand it is also´etale, because every fiber of ϕ is reduced, and we have a contradiction. Since ϕ : X → Y is smooth, every fiber of the conic bundle ψ : X → Y is reduced.Then [Wi´s91, Proposition 4.3] shows that Y and Y ′ are Fano. Finally c Y ′ ≤ (cid:4) Remark 3.3.18.
Let X be a Fano manifold, ϕ : X → Y a surjective morphism, and D ⊂ X a prime divisor. We have N ( ϕ ( D ) , Y ) = ϕ ∗ ( N ( D, X )), hence codim N ( D, X ) ≥ codim N ( ϕ ( D ) , Y ). In particular, if Y is a Fano manifold, then c Y ≤ c X . In this final section we prove the results stated in the introduction.
Proof of Theorem 1.1.
We have c X ≥ codim N ( D, X ) ≥
3. If c X = 3, Theorem 3.3 yields( ii ). If instead c X ≥
4, applying iteratively Theorem 3.3, we can write X = S ×· · ·× S r × Z ,where S i are Del Pezzo surfaces, r ≥
1, and Z is a Fano manifold with c Z ≤ D dominates Z under the projection, up to reordering S , . . . , S r we can assume that D dominates S × · · · × S r × Z . Then codim N ( D, X ) ≤ ρ S − i ).Suppose instead that D = S × · · · × S r × D Z , where D Z ⊂ Z is a prime divisor. Then3 ≥ c Z ≥ codim N ( D Z , Z ) = codim N ( D, X ) ≥ , and the inequalities above are equalities. Therefore again by Theorem 3.3 we have a flat,quasi-elementary contraction Z → W , where W is a Fano manifold with dim W = dim Z − ρ Z − ρ W = 4. Then the induced contraction X → S × · · · × S r × W satisfies ( ii ). (cid:4) roof of Corollary 1.3. We have c X ≥ codim N ( D, X ) ≥
3. Suppose that X is not aproduct of a Del Pezzo surface with another variety. Then Theorem 3.3 shows that c X = 3and there is a quasi-elementary contraction X → T where T is a Fano manifold, dim T = n −
2, and ρ X − ρ T = 4. If n = 4, [Cas08, Theorem 1.1] implies that ρ T ≤
2, hence ρ X ≤ n = 5 follows similarly. (cid:4) Lemma 4.1.
Let X be a Fano manifold, D ⊂ X a prime divisor, and ϕ : X → Y acontraction. Then codim N ( ϕ ( D ) , Y ) ≤ .Suppose moreover that codim N ( ϕ ( D ) , Y ) ≥ . Then X ∼ = S × T and Y ∼ = W × Z , where S is a Del Pezzo surface, W is a blow-down of S , and codim ϕ ( D ) ≤ . More precisely,one of the following holds: ( i ) ϕ ( D ) is a divisor in Y , and dominates Z under the projection; ( ii ) ϕ ( D ) = { p } × Z and D = C × T , where C ⊂ S is a curve contracted to p ∈ W .Proof. We have codim N ( ϕ ( D ) , Y ) ≤ codim N ( D, X ) ≤ N ( ϕ ( D ) , Y ) ≥
4. Then, again by Theorem 1.1, X ∼ = S × T where S is a Del Pezzo surface, and D dominates T under the projection. Therefore Y ∼ = W × Z , ϕ is induced by two contractions S → W and f : T → Z , and ϕ ( D ) dominates Z under theprojection.In particular dim W ≤ N ( ϕ ( D ) , Y ) ≥ ρ Z , hence ρ W ≥ codim N ( ϕ ( D ) , Y ) ≥
4. This implies that dim W = 2, thus W is a blow-down of S , and ϕ ( D ) has codimension1 or 2 in Y .If ϕ ( D ) is a divisor, we have ( i ). Suppose that codim ϕ ( D ) = 2, and consider thefactorization of ϕ as S × T ψ → W × T ξ → W × Z . Then ξ = (Id W , f ) induces an isomorphism W × { t } → W × { f ( t ) } for every t ∈ T . If t is general, we have dim ϕ ( D ) ∩ ( W × { f ( t ) } ) = 0and ψ ( D ) ∩ ( W × { t } ) ∼ = ϕ ( D ) ∩ ( W × { f ( t ) } ). This implies that ψ ( D ) has codimension 2in W × T , hence D is an exceptional divisor of ψ , which gives the statement. (cid:4) Corollary 1.4 follows from Theorem 1.1, while Corollary 1.5 is a straightforward appli-cation of Lemma 4.1 (just take the Stein factorization of ϕ ). Proof of Corollary 1.6.
By taking the Stein factorization, we can assume that ϕ is a con-traction; we also assume that ρ Y ≥
6. By [Cas08, Lemma 2.6] we know that Y has someelementary contraction ψ : Y → Z , and dim Z ≥ ρ Z ≥ D ⊂ X depending on ψ , as follows. If dim Z = 2, let D ⊂ X be any prime divisor such that dim ψ ( ϕ ( D )) = 1. If ψ is birational and divisorial, let D ⊂ X be a prime divisor such that ϕ ( D ) ⊆ Exc( ψ ). If ψ is birational and small, its liftingin X (see [Cas08, § n − , n − sm ; let D be its exceptional divisor: then again we have ϕ ( D ) ⊆ Exc( ψ ).In any case dim N ( ϕ ( D ) , Y ) ≤
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