aa r X i v : . [ m a t h . AG ] J a n On certain K-equivalent birational maps
Duo LiAbstract. We study K-equivalent birational maps which are resolved by asingle blowup. Examples of such maps include standard flops and twistedMukai flops. We give a criterion for such maps to be a standard flop or atwisted Mukai flop. As an application, we classify all such birational mapsup to dimension 5.
Contents1. Introduction 12. Proof of main results 43. Classifications when dim X ≤ E s.t. P ( E ) = P roj ( Sym ( E ∨ )) . By P n fibration α : X → Y, we mean that α − ( y ) is isomorphic to P n for every closed point y ∈ Y. In this article, we address the study of the birational map θ : X / / ❴❴❴ X + which is K-equivalent and can be resolved by a single blowup. We call θ abirational map of Simple Type . To be concrete, there is a closed smoothsubvariety P of X (resp. P + a closed smooth subvariety of X + ) and the blowup ϕ : Bl P ( X ) → X along P (resp. ϕ + : Bl + P ( X + ) → X + along P + ) satisfies that:(1) Bl P ( X ) ≃ Bl P + ( X + ) (we denote this variety by e X ). n certain K-equivalent birational maps Duo Li (2) θ ◦ ϕ = ϕ + , i.e., the diagram e X ϕ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ ϕ + ❇❇❇❇❇❇❇❇ X θ / / ❴❴❴❴❴❴❴ X + is commutative.(3) ϕ ∗ ( K X ) = ϕ + ∗ ( K X + ) in P ic ( e X ) . (4) θ is not an isomorphism.Actually, we know two concrete constructions of birational maps of simpletype as follows:1. standard flop (over a base C )(In article [8], standard flops are called ordi-nary flops instead):Suppose that there is a closed subvariety P of X satisfies that π : P = P C ( E ) → C is a projective bundle. The normal bundle of P satisfies that N P/X = π ∗ E + ⊗ O ( −
1) where E + is a vector bundle over C and O (1) is thetautological bundle for P C ( E ). Then we consider the blowup of X along P. Let P + = P C ( E + ) . We denote π + : P + → C the projective bundle morphism, thenthe exceptional divisor E ≃ P × C P + . For any fibre F of p + : E → P + , bya simple calculation, we have that O E ( E ) | F ≃ O ( −
1) (see Chapter 11, [5] orSection 1, [8]). By Fujiki-Nakano’s criterion(see [2] or Remark 11.10, [5]), thereis a blow-down ϕ + : e X → X + compatible with p + . Hence we obtain a birationalmap θ : X X + , we call θ a standard flop over C.
2. twisted Mukai flop (over a base C ):Suppose that there is a closed subvariety P of X satisfies that π : P = P C ( E ) → C is a projective bundle. The normal bundle of P satisfies that N P/X = Ω
P/C ⊗ π ∗ L for some line bundle L of C. Then we consider the blowupof X along P. Let P + = P C ( E ∨ ) . We denote π + : P + → C the projective bundlemorphism, then the exceptional divisor E ≃ P P (Ω P/C ) is a prime divisor of P × C P + , hence there is an induced projective bundle structure p + : E → P + . For any fibre F of p + , by a simple calculation, we have that O E ( E ) | F ≃ O ( − ϕ + : e X X + compatible with p + . Hence we obtain a birational map θ : X X + , we call θ a twisted Mukai flopover C, if L is a trivial line bundle, we call θ a Mukai flop for short.2 uo Li On certain K-equivalent birational maps Remark 1.1.
In the above constructions, X + is not necessarily projective.But it is reasonable to assume that there exists a flopping contraction β : X → X compatible with π : P → C. Under this assumption, X + can be provedprojective. For details, see Proposition 1.3 and Proposition 6.1, [8].There is a natural question: are all birational maps of simple type eitherstandard flops or twisted Mukai flops? Actually, we construct a new simpletype birational map in Example 3.8 and one of the main results of our articleis: Theorem 1.2.
A birational map of simple type, if dim X ≤ , is either astandard flop, a twisted Mukai flop or a flop as in Example 3.8. For properties about standard flops and twisted Mukai flops, we use [5] and[8] as our main references. We note that in articles [3] and [8], motivic and quan-tum invariance under a standard flop or a twisted Mukai flop was studied, wehope this article could offer useful information to further study in that direction.Now let us state the structure of this article: first, we prove that ϕ and ϕ + share a common exceptional divisor E and dim P = dim P + . Then we generaliseE. Sato’ classification results about varieties which admits two different projec-tive bundle structures to a relative version. The main result of this article isTheorem 2.6, we prove that a birational map of simple type is a standard flopor a twisted Mukai flop if and only if P and P + are projective bundles over acommon variety. As an application of our main result, we classify simple typebirational maps when dim X ≤ , here recent results of [6] and [17] play impor-tant roles in our classification. Acknowledgments.
The author is very grateful to Professor Baohua Fufor his support, encouragement and stimulating discussions over the last fewyears. The author is very grateful to Professor Chin-Lung Wang for his helpfulsuggestions and discussions. The author wishes to thank Yang Cao, Yi Gu,Wenhao Ou, Xuanyu Pan, Lei Zhang for useful discussions and thank ProfessorXiaokui Yang for his support and encouragement.3 n certain K-equivalent birational maps Duo Li
2. Proof of main resultsLet us start with the following observation, which shows that ϕ and ϕ + sharea common exceptional divisor E , hence E admits two projective bundle struc-tures. We will repeatedly use this fact in the rest of our article. Lemma 2.1.
The blowups ϕ and ϕ + share a common exceptional divisor E and dim P = dim P + Proof.
Let k = codim ( P, X ) and k + = codim ( P + , X + ) , then k, k + ≥ . Since ϕ and ϕ + are blowups, the corresponding exceptional divisors are E and E + , wehave the following two equations:(1) ϕ ∗ K X = K e X − ( k − E (2) ϕ + ∗ K X + = K e X − ( k + − E + . Since ϕ ∗ ( K X ) = ϕ + ∗ ( K X + ) , E is numerically equivalent to k + − k − E + , where k + − k − > . Since E is a projective bundle over P , for any point p ∈ P , the fiber F p of ϕ is isomorphic to P k − . For any line C in F p ≃ P k − , C · E = deg O e X ( E ) | C =deg O ( −
1) = − . As p varies, these negative lines in F p cover E. If E = E + , then there exists a line C in F p for some p ∈ P such that C isnot contained in E + . So C · E = C · ( k + − k − ) E + ≥ , contradicts to C · E < . Then E = E + and k + = k. (cid:3) By Propostion 1.14, [1], ϕ | E and ϕ + | E are non-isomorphic fibrations. Thereis a lower bound for the dimension of P. Lemma 2.2. P + 1 ≥ dim X. Proof.
The exceptional divisor E has two projective bundle structures over P and P + . We denote these two projective bundle morphisms in the followingdiagram: E p / / p + (cid:15) (cid:15) PP + where p = ϕ | E and p + = ϕ + | E . Since fibers of different extremal ray contractionscan meet only in points, ( p, p + ) : E −→ P × P + is a finite morphism to its image.So dim E ≤ P, which means that dim X ≤ P + 1 . (cid:3) uo Li On certain K-equivalent birational maps Now recall E.Sato’s classification about varieties which admit two projectivebundle structures (See Theorem A in [14]).
Theorem 2.3. If E has two projective bundle structures over projective spaces: E p / / p (cid:15) (cid:15) P n P n then E is isomorphic to either P n × P n ( in this case, dim E = 2 n ) or P P n (Ω P n ) (in this case, dim E = 2 n − ). Remark 2.4.
We now sketch the proof of E.Sato’s theorem in the case ofdim E = 2 n −
1. First, it can be proved that Φ = ( p , p ) : E −→ P n × P n isa closed immersion. Then Φ( E ) is a Cartier divisor in P n × P n , we denote thedefining equation of Φ( E ) by F ( X · · · , X n ; Y · · · , Y n ) . The key point is F isa homogeneous polynomial of bidegree (1 , . Moreover, it can be shown thatafter performing suitable linear transforms of ( X · · · , X n ) and ( Y · · · , Y n ) , F is of the form F = P ni =0 X i · Y i . If we view P P n (Ω P n ) as a closed subvariety of P n × ( P n ) ∗ by the Euler sequence:0 → Ω P n → O ( − n +1 → O → , then E has the same defining equation as that of P P n (Ω P n ) . Now we generalise E.Sato’s result to a relative version.
Theorem 2.5.
Suppose that E admits two projective bundle structures: E p / / p (cid:15) (cid:15) P P If the following conditions are satisfied: (1) P and P are projective bundles over a common variety C, i.e. P = P C ( E ) and P = P C ( E ) for some locally free sheaves E i on C . (2) Let π i : P i −→ C denote the projective bundle morphism, we assumethat π ◦ p = π ◦ p . (3) dim P = dim P . then E is isomorphic to either P × C P or P P (Ω P /C ) . In the first case, dim E =2 dim P − dim C ; in the second case, dim E = 2 dim P − dim C − . n certain K-equivalent birational maps Duo Li Proof.
We assume that the rank of E i is k + 1 and let α = π ◦ p . Then thefollowing diagram : E α ❆❆❆❆❆❆❆❆ p / / p (cid:15) (cid:15) P π (cid:15) (cid:15) P π / / C commutes. Since every fiber of α has two projective bundle structures overprojective spaces, by Theorem 2.3, dim E = 2 dim P − dim C or dim E =2 dim P − dim C − . In the case of dim E = 2 dim P − dim C, we aim to show that E ≃ P × C P . Note that we have the following commutative diagram: E Φ=( p ,p ) / / α (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ P × C P z z ✉✉✉✉✉✉✉✉✉✉ C For any c ∈ C, let E c be the fiber of α. Then by Theorem 2.3, the restriction ofΦ to E c : Φ c : E c −→ π − ( c ) × π − ( c ) ≃ P k × P k is an isomorphism. So Φ is an isomorphism.In the case of dim E = 2 dim P − dim C − , we aim to show that E ≃ P P (Ω P /C ) . We note that Φ = ( p , p ) : E −→ P × C P is a closed immersion. First, Φis finite onto its image. Otherwise, there will be a curve contracted by both p i . Since for any curve C of P k , deformations of C cover P k , then there is a fibreof p contracted by p . So by the rigidity lemma, p = p as fibrations, which isa contradiction to our assumptions. Also, for an arbitrary c ∈ C, by Theorem2.3, Φ c : E c −→ π − ( c ) × π − ( c ) ≃ P k × P k is a closed immersion. Moreover, E c is a prime divisor defined by a homogeneous polynomial of bidegree (1 , . ThenΦ is a finite morphism to its image of degree one, so Φ is a closed immersion.Let X = P × C P , q i : X → P i denote the projection morphism and f : X → C be the base morphism. We view E as a prime divisor of X, then O X ( E ) ≃ q ∗ O (1) ⊗ q ∗ O (1) ⊗ f ∗ L where O (1) denotes the tautologicalline bundle of a given projective bundle and L is a line bundle of C. Note that6 uo Li On certain K-equivalent birational maps f ∗ ( O X ( E )) ≃ E ∨ ⊗ E ∨ ⊗ L . Since E is an effective divisor, there is a non-zeroglobal section of H ( C, f ∗ ( O X ( E ))) = H ( X, O X ( E )). Then this global sec-tion induces a morphism of sheaves O → E ∨ ⊗ E ∨ ⊗ L , hence a linear form β : E ⊗ E → L . Actually, the linear form β is non-degenerate. For any affine open subset SpecA of C, Φ( E ) is a Cartier divisor of P × A P . Then Φ( E ) is defined bya single equation F ( X · · · , X k ; Y · · · , Y k ) = 0 whose coefficients are elementsof A. For any local sections of E and E : e = ( x · · · , x k ), e = ( y · · · , y k ),we have that β ( e , e ) = F ( x · · · , x k ; y · · · , y k ) . Since F is a homogeneouspolynomial of bidegree (1 , β as :(2.1) β ( e , e ) = (cid:0) x · · · x k (cid:1) M y ... y k where M is a matrix in M n ( A ) . By Theorem 2.3 and Remark 2.4, det( M ) isinvertible in A m for any maximal ideal m of A, so M is an invertible matrix,which means that the linear form β ( − , − )is non-degenerate. So E ≃ E ∨ ⊗ L . Since P ( E ⊗ L − ) ≃ P ( E ) , in what follows we assume that E ≃ E ∨ . By the relative Euler sequence 0 → Ω P /C → π ∗ ( E ∨ )( − → O → , P P (Ω P /C ) is a closed subvariety of P × C P . As closed subvarieties of P × C P , on any affine piece of C, E and P P (Ω P /C ) are defined by the same equation P X i · Y i = 0 , so E ≃ P P (Ω P /C ) . (cid:3) Now we assume that P and P + are projective bundles over a common variety C , i.e. P ≃ P C ( E ) and P + ≃ P C ( E + ) where E , E + are locally free sheaves ofrank n + 1 on C , π : P = P C ( E ) −→ C (resp. π + : P + = P C ( E + ) −→ C ) is theprojective bundle morphism. We let p = ϕ | E and p + = ϕ + | E . We summarise all the morphisms of our problem in the following diagram: E (cid:15) (cid:15) p + ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ p (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ e X ϕ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ ϕ + ❇❇❇❇❇❇❇❇ P / / π ' ' PPPPPPPPPPPPPPP X ❴❴❴ θ / / ❴❴❴ X + P + o o π + v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ C . n certain K-equivalent birational maps Duo Li Theorem 2.6.
Suppose that θ : ( X, P ) ( X + , P + ) is a birational map ofsimple type, where ( P, π ) and ( P + , π + ) are projective bundles over a commonvariety C and π ◦ p = π + ◦ p + . Then θ is a standard flop or a twisted Mukaiflop over base C. Proof.
Since E has two projective bundle structures over P and P + , by Theo-rem 2.5, E ≃ P × C P + or E ≃ P P (Ω P/C ) . Assume first that E ≃ P × C P + . Since E ≃ P P ( π ∗ E + ) and E ≃ P P ( N P/X ) , we can assume that N P/X = π ∗ E + ⊗ L where L is an invertible sheaf of P. Suppose that L = π ∗ L ′ ⊗ O ( − d ) , where O (1) is the tautological line bundle of π : P → C and L ′ is a line bundle of C. Since P C ( E + ) = P C ( E + ⊗ L ′ ) , we canassume that N P/X = π ∗ E + ⊗ O ( − d ) . We aim to show that d = 1 . Since ϕ is a blowup, ω E ≃ ϕ ∗ ω X | E ⊗O E (( n +1) E ) where n +1 = codim ( P, X ) . Let h = p ◦ π. We now calculate ϕ ∗ ω X | E ≃ p ∗ ω X | P as follows:(2.2) ϕ ∗ ω X | E ≃ p ∗ ω P ⊗ (det N P/X ) − ≃ p ∗ ω P ⊗ h ∗ (det E + ) − ⊗ p ∗ O (( n + 1) d ) . So ω E ≃ p ∗ ω P ⊗ h ∗ (det E + ) − ⊗ p ∗ O (( n + 1) d ) ⊗ O E (( n + 1) E ) . For any fiber F ≃ P n of p + , h ( F ) = c for some point c ∈ C. The morphism p maps F isomorphically to π − ( c ) = P n , which is illustrated as follows: F p | F (cid:15) (cid:15) / / E h ❅❅❅❅❅❅❅❅❅ p (cid:15) (cid:15) p + / / P + π + (cid:15) (cid:15) π − ( c ) / / P π / / C Since O E ( E ) | F ≃ O ( − , then we have that { p ∗ ω P ⊗ h ∗ (det E + ) − ⊗ p ∗ O (( n + 1) d ) ⊗ O E (( n + 1) E ) }| F ≃ O (( n + 1)( d − . We know that ω E | F ≃ O ( − n − , so d = 1 which means that N P/X ≃ π ∗ E + ⊗ O ( −
1) and θ is a standard flop over C. Assume now that E ≃ P P (Ω P/C ) . By Theorem 2.5, we know that
E ≃ E + ∨ and E ֒ → P × C P + is a closed immersion. Since E ≃ P P ( N P/X ) , we can as-sume that N P/X = Ω p/C ⊗ L where L is an invertible sheaf of P. Suppose that L = π ∗ L ′ ⊗ O ( − d ) , where O (1) is the tautological line bundle of π : P → C and L ′ is a line bundle of C. We aim to show that d = 0 . uo Li On certain K-equivalent birational maps Since ϕ is a blowup, ω E ≃ ϕ ∗ ω X | E ⊗O E (( n +1) E ) where n +1 = codim ( P, X ) . Let h = p ◦ π, we now calculate ϕ ∗ ω X | E ≃ p ∗ ω X | P as follows:(2.3) ϕ ∗ ω X | E ≃ p ∗ ω P ⊗ (det N P/X ) − ≃ h ∗ ω C ⊗ p ∗ L ⊗ ( − n − . So ω E ≃ h ∗ ω C ⊗ p ∗ L ⊗ ( − n − ⊗ O E (( n + 1) E ) . For any fiber F ≃ P n of p + ,h ( F ) = c for some point c ∈ C. The morphism p maps F into π − ( c ) = P n +1 asa hyperplane, which is illustrated as follows: F p | F (cid:15) (cid:15) / / E h ❅❅❅❅❅❅❅❅❅ p (cid:15) (cid:15) p + / / P + π + (cid:15) (cid:15) π − ( c ) / / P π / / C Since O E ( E ) | F ≃ O ( − , then we have that { h ∗ ω C ⊗ p ∗ L ⊗ ( − n − ⊗ O E (( n + 1) E ) }| F ≃ O (( n + 1)( d − . We know that ω E | F ≃ O ( − n − , so d = 0 which means that N P/X ≃ Ω P/C ⊗ π ∗ L ′ and θ is a twisted Mukai flop over C. (cid:3) Note that by Lemma 2.2, we know that there is a lower bound for the di-mension of P. As an application of Theorem 2.6, we will classify the birationalmorphism θ when dim P reaches the lower bound. Here, what is different fromTheorem 2.6, we don’t assume that P and P + are projective bundles in advance,actually we obtain a result as follows: Theorem 2.7. (1) If dim X = 2 dim P + 1 , then θ is a standard flop. (2) If dim X = 2 dim P, then θ is a Mukai flop or a standard flop over acurve.Proof. (1) If dim X = 2 dim P + 1 , then dim E = 2 dim P. So the morphism( p, p + ) : E −→ P × P + is surjective. For any fiber F of p + , p | F : F −→ P is surjective. By Lazarsfeld’s theorem [7], P is a projectivespace, similarly, P + is also a projective space. By Theorem 2.6, ϕ is astandard flop.(2) If dim X = 2 dim P, then dim E = 2 dim P − . By Theorem 2 in [13], E ≃ P P (Ω P ) where P and P + are projective spaces or E ≃ P × C P + where C is a smooth curve and P (resp. P + ) is a projective bundle over C. By Theorem 2.6, ϕ is a Mukai flop or a standard flop. (cid:3) n certain K-equivalent birational maps Duo Li
3. Classifications when dim X ≤ X ≤ . As a first step, we have thefollowing easy corollary from Theorem 2.7.
Corollary 3.1. (1) If dim X = 3 , then by Lemma 2.2, we have P +1 ≥ , so dim P = 1 . Then θ is a standard flop by Theorem 2.7. (2) If dim X = 4 , then by Lemma 2.2, we have P + 1 ≥ , so dim P =2 . Then θ is a Mukai flop or a standard flop by Theorem 2.7. (3) If dim X = 5 , then by Lemma 2.2, we have P +1 ≥ , so dim P = 2 or . If dim P = 2 , then θ is a Mukai flop or a standard flop by Theorem2.7. In the rest of this section, we keep the assumption that dim X = 5 . As we seein Corollary 3.1, the remaining unknown case is dim P = 3 . In this situation,the exceptional divisor E admits two P bundle structures as follows: E p / / p + (cid:15) (cid:15) PP + . For an arbitrary fibre F of p + , it is a natural question to ask whether p ( F ) isextremal in the cone of curves N E ( P ) . Actually, we have the following lemma.
Lemma 3.2. If E admits two P bundle structures as above, then p ( F ) isextremal. Furthermore, the contraction morphism π : P → C is smooth.Proof. See Theorem 2.2, [6]. (cid:3)
If the picard number ρ ( P ) ≥ , i.e. dim C ≥ , we aim to show that θ is astandard flop or a twisted Mukai flop. First, there exists π + : P + → C makingthe following diagram commutative: E p / / p + (cid:15) (cid:15) P π (cid:15) (cid:15) P + π + / / C .
The main obstruction to apply our result Theorem 2.6 is that we don’t know,a priori, whether π or π + is a projective bundle. Actually, there is a criterionfor projective bundles as follows: 10 uo Li On certain K-equivalent birational maps Lemma 3.3.
Suppose that f : M → S is a smooth P k fibration. Let us considerthe exact sequence → G m → GL k +1 → P GL k → , then we have an exact sequence of ´etale cohomologies: H et ( S, GL k +1 ) d → H et ( S, P GL k ) → H et ( S, G m ) , if d is surjective, then f is a projective bundle. In particular, when dim S = 1 ,H et ( S, G m ) vanishes and d is surjective.Proof. See Theorem 0.1 in [9] and Lemma 1.2 in [10]. (cid:3)
Remark 3.4.
Note that when dim Z ≥ , the H et ( S, G m ) is not necessarilyvanished, things become much more complicated. For example, we consider anarbitrary smooth P fibration f : M → S which is not a projective bundle. Let R = M × S M, the projection p : R → M is always a projective bundle, as thediagonal morphism δ : M → R is a section of p (for details, see Chapter 3, Ex-ercise 4.24 and Chapter 4, [11]). This example shows that the projective bundlestructure of p can’t decent to f , it is the main difficulty in the classification ofsimple type birational maps when dim X = 5 . Lemma 3.5. If dim C = 1 , then θ is a twisted Mukai flop over C. If dim C = 2 , then θ is a standard flop over C. Proof.
First, we fix some notations. For any c ∈ C, we let P c be the fibre of π and P + c be the fibre of π + . Let h = π ◦ p and E c = h − ( c ) . Since π is anelementary contraction, the picard numbers ρ ( P c ) = ρ ( P + c ) = 1 . We observe that the exceptional divisor E admits two P bundle structuresover P and P + , so there are induced P bundle structures of E c over P c and P + c . If dim C = 1 , then dim E c = dim P c + dim P + c − . By Theorem 2 in [13] and ρ ( P ) = 1 , P z and P + z are projective spaces. Since C is a curve, H et ( C, G m ) = 0 , then P and P + are projective bundles over C. Then by Theorem 2.6, θ is atwisted Mukai flop over C. If dim C = 2 , then for any c ∈ C, dim P c = dim P + c = 1 . Then dim E c =dim P c + dim P + c , by the same argument as in Theorem 2.7, we know that P c and P + c are projective lines. So π and π + are smooth P fibrations. However,as we show in Remark 3.4, the projective bundle structure of p can’t decent to π. Fortunately, there is a criterion of projective bundles for elementary contrac-tions, see Lemma 3.6 below. Let l = − P c · K P . Since K P = π ∗ K C ⊗ K P/C , l = 2 . n certain K-equivalent birational maps Duo Li Then π is an elementary contraction of maximal length. By Lemma 3.6, π isa projective bundle. By Theorem 2.6, we know that θ is a standard flop over C. (cid:3) Lemma 3.6.
Suppose that π : P → C is an elementary contraction of an ex-tremal ray Γ . Let l (Γ) = max { K P · A | A is a rational curve whose numericalclass [ A ] is in Γ } . If π is an equidimensional fiber contraction of relative di-mension d and l (Γ) = − d − , then π is called an elementary contraction ofmaximal length and π is a projective bundle.Proof. See Theorem 1.3, [4]. (cid:3)
Now we assume that ρ ( P ) = 1 i.e. dim C = 0 . We recall the following resultabout smooth varieties which admits two P bundle structures: Lemma 3.7.
Let X be a complex projective manifold with Picard number ρ ( X ) =1 and E rank vector bundle on X. Assume that Z = P ( E ) → X admits anothersmooth morphism Z → Y of relative dimension and n = dim X ≥ . Then, (1) X and Y are Fano manifolds with ρ = 1 and there exists a rank vectorbundle E ′ on Y such that Z → Y is given by P ( E ′ ) . (2) if E and E ′ are normalised by twisting with line bundles (i.e., c = 0 or − ), then (( X, E ) , ( Y, E ′ )) is one of the following, up to exchanging thepairs ( X, E ) and ( Y, E ′ ) : (a) (( P , T P ) , ( P , T P )) , where T P is the tangent bundle of the projec-tive plane P , (b) (( P , N ) , ( Q , S )) , where N is a null-correlation bundle on P and S is the restriction to the − dimensional quadric Q of theuniversal quotient bundle of the Grassmannian G (1 , P ) , (c) (( Q , C ) , ( K ( G ) , L )) , where C is a Cayley bundle on Q , K ( G ) is the − dimensional Fano homogeneous contact manifold of type G which is a linear section of the Grassmannian G (1 , P ) and L the restriction of the universal quotient bundle on G (1 , P ) . Proof.
See Theorem 1.1, [17]. (cid:3)
Now apply the above Lemma 3.7 to our problem, note that E admits two P bundle structures and dim E = 4 , we can assume that P = P and the normalbundle N P/X = N ⊗ O ( d ) for some integer d ∈ Z . By the exact sequence forthe null-correlation bundle0 → N → T P ( − → O (1) → uo Li On certain K-equivalent birational maps E is a closed subvariety of P × Q and p is compatible with the projection P × Q → P (resp. p + is compatible with the projection P × Q → Q , fordetails, see Proposition 2.6, [16] and Chapter 1, Section 4.2, [12] ). Example 3.8.
Keep the notations as above, we can construct a simple typebirational map as follows:Let P = P and the normal bundle N P/X = N ⊗ O ( d ) . We denote by ϕ : e X → X the blowup of X along P, hence the exceptional divisor E has a P bundle structure over P. By Lemma 3.7, E has another P bundle structureover a quadric Q , we denote this projective bundle morphism by p + : E → Q . Then ω E ≃ ϕ ∗ ω X | E ⊗ O E (2 E ) ≃ p ∗ ω P ⊗ (det N P/X ) − ⊗ O E (2 E ) . Next we calculate the restriction of O E ( E ) on an arbitrary fiber F of p + . By Proposition 2.6, [16], F is a projective line in P and c ( N ) = 0 . So p ∗ ω P ⊗ (det N P/X ) − ⊗ O E (2 E ) | F = O ( − − d ) ⊗ O E (2 E ) | F . Since ω E | F ≃ ω E/Q | F = O ( − . So O E ( E ) | F ≃ O ( d + 1) . By Fujiki-Nakano’s criterion(see [2]or Remark 11.10, [5]), there is a blow-down ϕ + : e X → X + compatible with p + if and only if d = − . Also, X + constructed here is not necessarily projective. As in Remark 1.1,suppose that there exists a flopping contraction β : X → X contracting P, then X + can be proved projective by the same argument as Proposition 1.3, [8].For the sake of completeness, we here give a detailed proof as follows: First, − K e X · F = deg ( ω E ⊗ O E ( − E )) | F = − < . So F is a K e X negative curve.Next, we aim to show that [ F ] spans an extremal ray in N E ( e X ) , i.e., it hasa supporting divisor(nef and big). Let H be an ample line bundle on X and L = β ∗ H, where H is an ample line bundle on X. Then we consider the divisor L ′ k = kϕ ∗ L − ( ϕ ∗ H + λE )where k >> λ = H · ϕ ( F ) . We next show that this divisor is nef and bigfor large k and vanishes precisely on the ray spanned by [ F ] . For an irreduciblecurve A in e X, we assume firstly that A ⊆ E. Since ρ ( E ) = 2 , A is numericallyequivalent to aF + bG where G is a fiber of p and a, b are positive numbers.Then for any positive k, L ′ k · A ≥ L ′ k · A = 0 if and only if b = 0 . Now,suppose that A * E, then by the projection formula, L ′ k · A = k L · ϕ ( A ) − H · ϕ ( A ) − λE · A = kH · β ( ϕ ( A )) − H · ϕ ( A ) − λE · A. n certain K-equivalent birational maps Duo Li The intersection number with H, we denote it by < H, · >, defines a linearform on N ( X ) R . Since H is ample, there is a positive lower bound of < H, · > for any compact subset of N E ( X ) \ { } . So for large enough k, L ′ k · A > . In conclusion, for k >> , L ′ k is nef and big and vanishes precisely on the rayspanned by [ F ] . Then F is a K e X negative extremal curve, so X + is projective.We’d like to thank Will Donovan for pointing out that our Example 3.8 isalready known by Roland Aburf, see [15].In conclusion, we have proved the following theorem. Theorem 3.9.
A birational map of simple type, if dim X ≤ , is either astandard flop, a twisted Mukai flop or a flop as in Example 3.8. References [1] O.Debarre:
Higher-Dimensional Algebraic Geometry. Universitext. Springer-Verlag, NewYork, 2001. [2] A.Fujiki, S.Nakano:
Supplement to “On the inverse of monoidal transformation”.Publ.RIMS Kyoto Univ.7(1071), 637-644. [3] B.Fu, C.-L. Wang:
Motivic and quantum invariance under stratified Mukai flops. (Englishsummary) J. Differential Geom. 80 (2008), no. 2, 261-280. [4] A.H¨ o ring, C.Novelli: Mori contraction of maximal length. Publ. Res. Inst. Math. Sci. 49(2013), no. 1, 215-228. [5] D.Huybrechts:
Fourier-Mukai transforms in algebraic geometry. Oxford MathematicalMonographs. The Clarendon Press, Oxford University Press, Oxford, 2006. viii+307 pp. [6] A.Kanemitsu:
Fano 5-folds with nef tangent bundles.arxiv.org/abs/1503.04579. [7] R.Lazarsfeld:
Some applications of the theory of positive vector bundles. Ann. of Math.110 (1979) 593-606. [8] Y.P. Lee, H.-W. Lin, C.-L. Wang:
Flops, motives, and invariance of quantum rings.(English summary) Ann. of Math. (2) 172 (2010), no. 1, 243-290. [9] M.Maruyama:
On classification of ruled surfaces. Lectures in Mathematics, Departmentof Mathematics, Kyoto University, 3 Kinokuniya Book-Store Co., Ltd., Tokyo 1970 iv+75pp. [10] M.Maruyama:
On a family of algebraic vector bundles. Number theory, algebraic geome-try and commutative algebra, in honor of Yasuo Akizuki, pp. 95-146. Kinokuniya, Tokyo,1973. [11] James S. Milne: ´Etale cohomology. Princeton Mathematical Series, 33. Princeton Uni-versity Press, Princeton, N.J., 1980. xiii+323 pp. [12] C. Okonek, M. H. Schneider and H. Spindler:
Vector bundles on complex projectivespaces. Corrected reprint of the 1988 edition. With an appendix by S. I. Gelfand. ModernBirkh¨auser Classics. Birkh¨auser/Springer Basel AG, Basel, 2011. viii+239 pp. [13] G.Occhetta, Jaros law A. Wi´sniewski:
On Euler- Jaczewski sequence and Remmert- Vande Ven problem for toric varieties. Mathematische Zeitschrift, 2001, 241(1):35-44. [14] E.SATO:
Varieties which have two projective space bundle structures. J.Math.KyotoUniv. 25-3 (1985)445-457. uo Li On certain K-equivalent birational maps [15] Ed Segal: A new 5-fold flop and derived equivalence. Bull. Lond. Math. Soc. 48 (2016),no. 3, 533-538. [16] M.Szurek; Jaros law A. Wi´sniewski:
Fano bundles over P and Q . Pacific J. Math. 141(1990), no. 1, 197-208. [17] K.Watanabe: P -bundles admitting another smooth morphism of relative dimension one.J. Algebra 414 (2014), 105-119. Address of Duo Li: Yau Mathematical Sciences Center, Tsinghua University, Beijing,100084, P. R. China.
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