Table Of Content

1 Deformation theory of singular symplectic n-folds 0 0 2 Yoshinori Namikawa n a J 9 Introduction 1 By a symplectic manifold (or a symplectic n-fold) we mean a compact ] G Kaehler manifold of even dimension n with a non-degenerate holomorphic 2- form ω, i.e. ωn/2 is a nowhere-vanishing n-form. This notion is generalized to A a variety with singularities. We call X a projective symplectic variety if X is a . h normalprojectivevarietywith rationalGorensteinsingularitiesandif the regu- at lar locus U of X admits a non-degenerate holomorphic 2-form ω. A symplectic m variety will play an important role together with a singular Calabi-Yau variety in the generalized Bogomolov decomposition conjecture. Now that essentially [ a few examples of symplectic manifolds are discovered, it seems an important 2 task to seek new symplectic manifolds by deforming symplectic varieties. In v this paper we shall study a projective symplectic variety from a view point of 3 1 deformation theory. If X has a resolution π : X˜ → X such that (X˜,π∗ω) is a 1 symplectic manifold, we say that X has a symplectic resolution. Our first re- 0 sults are concernedwith a birationalcontractionmapofa symplectic manifold. 1 0 0 Proposition (1.4). Let π : X˜ → X be a birational projective morphism h/ from a projective symplectic n-fold X˜ to a normal n-fold X. Let Si be the set t of points p∈X such that dimπ−1(p)=i. Then dimS ≤n−2i. In particular, a i m dimπ−1(p)≤n/2. : Proposition (1.6). Let π : X˜ → X be a birational projective morphism v from a projective symplectic n-fold X˜ to a normal n-fold X. Then X has only i X canonical singularities and its dissident locus Σ has codimension at least 4 in 0 r X. Moreover, if Σ\Σ0 is non-empty, then Σ\Σ0 is a disjoint union of smooth a varieties of dim n−2 with everywhere non-degenerate 2-forms. WhenX hasonlyanisolatedsingularityp∈X,everyirreduciblecomponent of π−1(p) is Lagrangian(Proposition (1.11)). In this situation it is conjectured that the exceptional locus is isomorphic to Pn/2 with normal bundle Ω1 . Pn/2 Similar resultsto (1.4)and(1.11)areobtainedindependently by Wierzba [Wi]. We shall exhibit four examples of birational contraction maps of symplectic 4-folds in (1.7). The second example shows that the Kaehler (projective) assumption of a symplectic manifold is not necessarily preserved under an ele- mentary transformation. The fourth example deals with a symplectic manifold 1 obtainedasaresolutionofcertainquotientofa Fanoschemeoflines onacubic 4-fold. As for a fiber space structure of a symplectic n-fold, see [Ma]. After we study the birational contraction map of a symplectic manifold in section 1, we shall prove our main theorem in section 2: Theorem (2.2). Let π : X˜ → X be a symplectic resolution of a projective symplectic variety X of dimension n. Then the Kuranishi spaces Def(X˜) and Def(X) are both smooth of the same dimension. There exists a natural map π : Def(X˜) → Def(X) and π is a finite covering 1. Moreover, X has a flat ∗ ∗ deformation to a smooth symplectic n-fold X . Any smoothing X of X is a t t symplectic n-fold obtained as a flat deformation of X˜. (2.2) was proved by Burns-Wahl [B-W] for K3 surfaces. Given a one- parameter flat deformation f : X → ∆ of such X as (2.2), by Theorem, we could have a simultaneous resolution ν : X˜ → X′ after a suitable finite base change X′ →∆′ of X by ∆′ →∆. The same situation as (2.2) naturally arises for Calabi-Yau 3-folds; but the resultsforthemareverypartialascomparedwithsymplecticcase(cf. Example (2.4)). Onthe other hand, it is natural to consider a symplectic variety which does nothaveasymplecticresolution;forexample,suchvarietiesappearinaworkof O’Grady[O]asthemodulispacesofrank2semi-stablesheavesonaK3surface with c = 0 and with even c ≥ 6. At the moment it is not clear when these 1 2 varietieshaveflatdeformationsto symplecticmanifolds. Butwecanprovethat such varieties have unobstructed deformations: Theorem (2.5). Let X be a projective symplectic variety. Let Σ ⊂ X be the singular locus. Assume that codim(Σ⊂X)≥4. Then Def(X) is smooth. Weshallgivearoughsketchofthe proofofTheorem(2.2)intheremainder. First note that X has only rational Gorenstein singularities. Then the existence of the map π follows from the fact that R1π O = 0 (cf. [Ko-Mo, ∗ ∗ X˜ (11.4)]). Let U be the complement of Σ in X and write U˜ for π−1(U). By (1.4) 0 and (1.6), we can prove, roughly speaking, that a deformation of X˜ (resp. X) is equivalent to that of U˜ (resp. U). (See Proposition (2.1).) From this fact it follows that π :Def(X˜)→Def(X) is finite. ∗ FinallywecomparethedimensionsoftangentspacesofDef(X)andDef(X˜) attheoriginandthenconcludethatDef(X)issmooth. SinceDef(X˜)issmooth by Bogomolov[Bo], we only have to prove that dimT1 =dimT1 is not larger X U than h1(X˜,Θ )=h1(U˜,Θ ). We need here a detailed description of the sheaf X˜ U˜ T1 :=Ext1(Ω1 ,O ) (Lemma (1.9), Corollary (1.10)). U U U The last statement will be proved in the following way. By the existence of a non-degenerate 2-form ω, there is an obstruction to extending a holomorphic 1Precisely, there are open subsets 0 ∈ V ⊂ Def(X˜) and 0 ∈ W ⊂ Def(X) such that π∗|V :V →W isapropersurjectivemapwithfinitefibers. 2 curve on X˜ sideways in a given one-parameter small deformation X˜ → ∆1. Therefore, if we take a general curve of Def(X˜) passing through the origin and take a corresponding small deformation of X˜, then no holomorphic curves survive (cf. [Fu, Theorem (4.8)]). Let t ∈ Def(X) be a generic point (that is, t is outside the union of a countable number of proper subvarieties of Def(X)). Since π : Def(X˜) → ∗ Def(X) is a finite covering,we may assume that X has a symplectic resolution t π : X˜ → X . By the argument above, X˜ contains no curves. By Chow t t t t lemma [Hi], there is a bimeromorphic projective map h : W → X such that t h is factored through π . Since h−1(p) is the union of projective varieties for t any point p ∈ X , π −1(p) is the union of Moishezon varieties. If π is not t t t an isomorphism, π −1(p) has positive dimension for some p ∈ X ; hence X˜ t t t contains curves, which is a contradiction. Thus π is an isomorphism and X is t t a (smooth) symplectic n-fold. The author thanks A. Fujiki for giving him invaluable informations on this topic. Thefirstversionofthis paperwaswrittenin1998. After thattheauthor wasinformedthat Wierzba [Wi]independently obtainedsimilar results to (1.4) and (1.11). 1. Birational contraction maps of symplectic n-folds A symplectic n-fold means a symplectic manifold of dimension n. We shall state three lemmas which will be used later. The first lemma is essentially a linear algebra. Lemma (1.1). Let V be a complex manifold with dimV =2r and let ω be an everywhere non-degenerate holomorphic 2-form on V (i.e. ∧rω is nowhere- vanishing.) Let E be a subvariety of V with dimE >r. Then ω| is a non-zero E 2-form on E. Lemma (1.2). Let f :V →W be a birational projective morphism from a complex manifold V to a normal variety W. Let p ∈ W and assume that the germ (W,p) of W at p has rational singularities. Assume that E := f−1(p) is a simple normal crossing divisor of V. Then H0(E,Ωî ) = 0 for i > 0, where E Ωî :=Ωi /(torsion). E E Proof. DenotebyF· (resp. W)theHodgefiltration(resp. weightfiltration) · of Hi(E) := Hi(E,C). Note that these two filtrations give a mixed Hodge structure on Hi(E). Since E is a proper algebraic scheme, GrW(Hi(E)) = 0 j for j >i. Assume that H0(E,Ωî ) 6= 0. Then Gri GrW(Hi(E)) 6= 0. By the Hodge E F i symmetry Gr0GrW(Hi(E))6=0, and hence Gr0(Hi(E))=Hi(E,O )6=0. F i F E Ontheotherhand,(Rif O ) =0fori>0because(W,p)hasonlyrational ∗ V p singularities. Take a sufficiently small open neighborhood V′ of f−1(p) in V. There exists a commutative diagram of Hodge spectral sequences 3 Hk(V′,Ωj ) >Hj+k(V′,C) V′     (1) y y Hk(E,Ωˆj ) >Hj+k(E,C) E Note that Hi(V′,C) ∼= H2(E,C). Denote by F· (resp. F·) the filtrations 1 2 on Hi(V′,C) (resp. Hi(E,C)) induced by the spectral sequences. There is a natural surjection Gr0 Hi(V′,C)→Gr0 Hi(E,C). As (Rif O ) =0 for i> F1 F2 ∗ V 0 0, Hi(V′,OV′) = 0. Therefore we have GrF01H2(V′,C) = GrF02Hi(E,C) = 0. Since the second spectral sequence degenerates at E terms, Hi(E,O ) = 0, 1 E which is a contradiction. Lemma(1.3). Let V be a symplectic n-fold and let H be a smooth 3- dimensionalsubvarietyofV containingasmoothrationalcurveC withN | ∼= H/V C O⊕n−3. Assume that N ∼= O(−1)⊕O(−1), O(−2)⊕O or O(−3)⊕O(1). C/H Then Hilb(V) is smooth of dimension (n-2) at [C]. Moreover, in this case, N ∼=O⊕(n−2)⊕O(−2) or O⊕(n−4)⊕O(−1)⊕O(1)⊕O(−2). C/V Proof. We shall prove that the Hilbert scheme (functor) has the T1-lifting property at [C]; then Hilb(V) is smooth at [C] by [Ra, Ka 1]. Let S be m the spectrum of the Artinian ring A = C[t]/(tm+1). Set V := V × S . m m S0 m Let C ⊂ V be an infinitesimal displacement of C to m-th order, and let m m C :=C × V . We have to prove that m−1 m Vm m−1 H0(N )→H0(N ) Cm/Vm Cm−1/Vm−1 is surjective. Let ω be a non-degenerate 2-form on V. Then ω lifts to an element ω ∈ H0(Ω2 ) in such a way that ∧n/2ω ∈ H0(Ωn ) is a m Vm/Sm m Vm/Sm nowhere vanishing section (that is, if we identify H0(Ωn ) with A , then Vm/Sm m ∧n/2ω corresponds to an invertible emement of A ). The 2-form ω induces m m m a pairing Θ | ×Θ | →O . Vm/Sm Cm Vm/Sm Cm Cm Since this pairing vanishes on Θ × Θ and since ω is non- Cm/Sm Cm/Sm m degenerate, one has a surjection α :N →Ω1 m Cm/Vm Cm/Sm by the exact sequence 0→Θ →Θ | →N →0. Cm/Sm Vm/Sm Cm Cm/Vm Letusfirstconsiderthecasewhenm=0. Byassumption,wehaveN | ∼= H/V C O⊕n−3; hence by the exact sequence 4 0→N →N →N | →0 C/H C/V H/V C weseethatN isisomorphictoO(−1)⊕O(−1)⊕O⊕(n−3),O(−3)⊕O(1)⊕ C/V O⊕(n−3),O(−2)⊕O(−1)⊕O(1)⊕O⊕(n−4) orO(−2)⊕O⊕(n−2). Bytheexistence ofthe surjectionα ,the firsttwocasesareexcluded. Inparticular,itis checked 0 that H1(Ker(α ))=0. Note that this implies that H1(Ker(α )) =0 for all m 0 m because there are exact sequences 0→Ker(α )→Ker(α )→Ker(α )→0. 0 m m−1 Next consider the following commutative diagram with exact columns and exact rows 0 0 0       y y y 0 −−−−→ H0(Ker(α )) −−−−→ H0(Ker(α )) −−ψ−m−→ H0(Ker(α )) 0 m m−1       y y y (2) 0 −−−−→ H0(N ) −−−−→ H0(N ) −−φ−m−→ H0(N ) C/V Cm/Vm Cm−1/Vm−1    τ0 τm τm−1 y y y 0 −−−−→ H0(Ω1) −−−−→ H0(Ω1 ) −−ϕ−m−→ H0(Ω1 ) C Cm/Sm Cm−1/Sm−1 As we remarked above, H1(Ker(α ))=0 for all m, hence τ , τ and τ m 0 m m−1 are surjective. By the same reason,ψ is also surjective. By the Hodge theory, m ϕ is surjective. It now follows from the diagram above that φ is surjec- m m tive. Thus the Hilbert scheme Hilb(V) is smooth at [C]. Its dimension equals h0(N )=n−2. C/V Proposition (1.4). Let π : X˜ → X be a birational projective morphism from a projective symplectic n-fold X˜ to a normal n-fold X. Let S be the set i of points p∈X such that dimπ−1(p)=i. Then dimS ≤n−2i. In particular, i dimπ−1(p)≤n/2. Proof. For a non-empty S , we take anirreducible componentR of π−1(S ) i i i in such a way that (1) by π, R dominates an irreducible component of S with dimS , and i i i (2) a general fiber of R →π(R ) has dimension i. i i Put l := n−dimR . By definition dimS = n−l−i. We shall prove that i i dimS ≥ n−2l; if this holds, then i ≤ l, and hence dimS ≤ n−2i. When i i l ≥ n/2, then clearly dimS ≥ n−2l. We assume that l < n/2. We shall i derive a contradiction assuming that dimS < n−2l and assuming that S is i i irreducible. WhenS isnotirreducible,itisenoughonlytoreplaceS byπ(R ). i i i Take a birational projective morphism ν : Y → X˜ in such a way that F :=(π◦ν)−1(S )becomesadivisorofasmoothn-foldY withnormalcrossings. i Set f =π◦ν. 5 A non-degenerate 2-form ω on X˜ is restricted to a non-zero 2-form on R i because dimR > n/2 (Lemma (1.1)). Therefore we have a non-zero element i ν∗ω| ∈H0(F,Ωˆ2). F F For a general point p ∈ S , the fiber F of the map F → S is a normal i p i crossing variety. Hence, if we take a suitable open set U of S and replace F i i by (π◦ν)−1(U ), then the sheaf Ωˆ2 has a filtration f∗Ω2 ⊂F ⊂Ωˆ2 with the i F Si F exact sequences 0→F →Ωˆ2 →Ωˆ2 →0 F F/Si 0→f∗Ω2 →F →f∗Ω1 ⊗Ωˆ1 →0 Si Si F/Si Letusprovethatthe 2-formν∗ω| is notthe pull-backofany2-formonS . F i Writen=2r. Assumethatω′ :=ν∗ω| isthepull-backofa2-formonS . Then F i ∧r−lω′ = 0 because dimS < 2r−2l. On the other hand, take a general point i q ∈R . Since R is a submanifold of X˜ of codimension l aroundq, it is checked i i by linear algebra that ∧r−l(ω| ) 6= 0 in a open neighborhood of q ∈ R . Let Ri i F′ be the union of irreducible components of F which dominate R by ν. Since i ν∗ω|F′ = (ν|F′)∗(ω|Ri), ∧r−l(ν∗ω|F′) 6= 0. Since (∧r−lω′)|F′ = ∧r−l(ν∗ω|F′), this implies that ∧r−lω′ 6=0, which is a contradiction. LetF bethefiberofF →S overp∈S . NotethatF isanormalcrossing p i i p variety for a general point p ∈ S . Then, by the exact sequence, one can see i that H0(F ,Ωˆ1 )6=0 or H0(F ,Ωˆ2 )6=0. p Fp p Fp Take an l+i dimensional complete intersection H = H ∩H ...∩H 1 2 n−l−i of very ample divisors of X passing through a general point p ∈ S . (When i l + i = n, we put H = X.) Then H has only rational singularities. Put H˜ := f−1(H) and put g := f| . Note that g−1(p) = F is a divisor of H˜ H˜ p with normal crossings. By Lemma (1.2) H0(F ,Ωî ) = 0 for i > 0, which is a p Fp contradiction. Corollary (1.5). Let π :X˜ →X be a birational projective morphism from a projective symplectic n-fold X˜ to a normal n-fold X. Then any π-exceptional divisor is mapped onto an (n−2)-dimensional subvariety of X by π. Proof. Take a π-exceptionaldivisor E. For some i≥1 we cantake the E as an R in the proof of Proposition (1.4). Then dimπ(E)≥n−2. i Let X be a normal variety of dim n with canonical singularities. Let Σ be the singularlocus of X. By [Re] there is a closedsubset Σ ⊂Σ such that each 0 pointofΣ\Σ hasananalyticopenneighborhoodinX isomorphicto(rational 0 double point) ×(Cn−2,0). The locus Σ is calledthe dissidentlocus. Generally 0 we have dimΣ ≤ n−3. But, when X has a symplectic resolution, we have a 0 stronger result. Proposition (1.6). Let π : X˜ → X be a birational projective morphism from a projective symplectic n-fold X˜ to a normal n-fold X. Then X has only 6 canonical singularities and its dissident locus Σ has codimension at least 4 in 0 X. Moreover, if Σ\Σ is non-empty, then Σ\Σ is a disjoint union of smooth 0 0 varieties of dim n−2 with everywhere non-degenerate 2-forms. Proof: (1.6.1) Σ has no (n-3)-dimensional irreducible components. WeshallderiveacontradictionbyassumingthatΣhasan(n−3)-dimensional irreduciblecomponent. LetH :=H ∩H ∩...∩H beacompleteintersection 1 2 n−3 ofveryampledivisorsofX. TheH intersectsthe(n−3)-dimensionalcomponent infinitepoints. Letp∈H beoneofsuchpoints. LetH′ :=π−1(H). Sincethere are no exceptional divisors of π lying on the (n−3)-dimensional component of Σ, π| : H′ → H gives a small resolution of H around p. Pick an irreducible H curveC fromπ|−H1′(p). TheC isisomorphictoP1,anditsnormalbundleNC/H′ in H′ is isomorphic to one of three vector bundles O(−1)⊕O(−1), O(−2)⊕O orO(−3)⊕O(1). NotethattheHilbertschemeHilb(X˜)hasatmostdimension (n−3)becauseC canonlymoveinX˜ alongthe(n−3)-dimensionalcomponent of Σ. However this contradicts Lemma (1.3). (1.6.2). By(1.6.1)weonlyhavetoobservetheirreduciblecomponentsofΣ with dimension n−2. So we replace Σ by an irreducible component of Σ with dim n−2. We shall derive a contradiction by assuming that dimΣ =n−3. 0 Let H := H ∩H ∩...∩H be a complete intersection of very ample 1 2 n−3 divisorsof X. Then H˜ :=π−1(H) is a crepantresolutionof H. Set Λ:=Σ∩H and Λ := Σ ∩H. Note that Λ consists of finite points. Write τ : H˜ → H 0 0 0 for the restriction π| of π to H˜. Every fiber of τ has at most dimension one H˜ because, if some fibers are 2-dimensional, then there is a prime divisor of X˜ lying on Σ , which contradicts Corollary (1.5). 0 WeshallshowthatΛisasmoothcurveandthatExc(τ)islocallyisomorphic to the product of Λ and a tree of P1’s. If so,then H must have rationaldouble points of the same type along Λ and this is a contradiction. A contradiction will be deduced in severalsteps. (i)Takea pointp ∈Λ . We only haveto arguelocally aroundp . Since H 0 0 0 hasrationalsingularitiesandsinceτ−1(p )is1-dimensional,τ−1(p )isatreeof 0 0 P1’s. Let C ,...,C be the irreducible components of τ−1(p ). Let us compute 1 m 0 the normal bundle N . Take a sufficiently small open neighborhood U of Ci/H˜ ∪C ⊂H˜. Since H has only rationalsingularities,we have H1(U,O )=0. Let i U I be the defining ideal of C in U. Then, by the exact sequence H1(U,O )→ i i U H1(C ,O /I2)→H2(U,I2)=0weknowthatH1(C ,O /I2)=0. Byanother i U i i i U i exact sequence H0(C ,O /I2) → H0(C ,O /I )(= C) → H1(C ,I /I2) → i U i i U i i i i H1(C ,O /I2) = 0, we know that H1(C ,I /I2) = 0 because the first map is i U i i i i surjective. Therefore, NCi/H˜ ∼=O(−1)⊕O(−1), O(−2)⊕O or O(−3)⊕O(1). ByLemma(1.3),theHilbertschemeHilb(X˜)issmoothofdimension(n−2) at [C ]. This fact tells us two things. i 7 (i-a): Each C moves inside H˜; in fact, if C is rigid in H˜, then Hilb(X˜) i i possibly has only (n-3) parameter at [C ] corresponding to a displacement of i C ⊂X˜ along Σ , which is a contradiction. i 0 (i-b): We have NCi/H˜ ∼= O⊕O(−2), in particular, NCi/X˜ ∼= O⊕(n−2) ⊕ O(−2). This fact can be proved by using Grothendieck’s Hilbert scheme (cf. [Ko 1, Chap. I]): Let Hilb(X˜/X) be the relative Hilbert scheme for π : X˜ → X. Since C is contained in a fiber of π, Hilb(X˜) coincides with Hilb(X˜/X) at i [C ]. Therefore Hilb(X˜/X) is smooth of dimension (n-2) at [C ]. Moreover, i i the irreducible component of Hilb(X˜/X) containing [C ] dominates an (n−2)- i dimensional irreducible component of Σ by the map Hilb(X˜/X) → X. By the universal property of the relative Hilbert scheme, we have Hilb(H˜/H) ∼= Hilb(X˜/X)× H, and hence Hilb(H˜/H) is smooth of dimension 1 at [C ] by X i Bertini theorem. Since Hilb(H˜) coincides with Hilb(H˜/H) at [C ], this implies i that Hilb(H˜) is smooth of dimension 1 at [Ci]. Therefore we have NCi/H˜ ∼= O⊕O(−2). (ii) We shall prove that Λ is irreducible around p ∈ Λ . By (i-a) there 0 0 are no flopping curves in Exc(τ), hence τ is a unique crepant resolution of H. Therefore, we can construct τ locally around p in the following manner. Let 0 Λ ,...,Λ betheirreduciblecomponentsofΛatp . BlowupH atfirstalongthe 1 n 0 defining ideal I of the reduced subscheme Λ and take its normalization. We 1 1 shallprovethatτ isfactorizedbythiscompositionofblow-upandnormalization. We shall argue along the line of [Re 2, §2.12-15]. First note that H is a cDV point by [Re 1, Theorem (2.2)]. Let us view H as a total space of a flat family of surface rational double points over a disc ∆1. The τ then can be viewed as a simultaneous (partial) resolution of this flat family. Let F ,...,F be the 1 l irreduciblecomponentsofExc(τ)whichdominateΛ . Thereisauniquepositive 1 divisor F =Σa F such thatF meets eachgeneralfiber H˜ (t∈∆1) in the sum i i t of the Artin’s fundamental cycles for the rational double points H ∩Λ . Since t 1 there are no rigid τ-exceptional curves, any τ-exceptional curve C moves along some Λ . If C moves along Λ with i > 1, then (−F.C) = 0. If C moves i i along Λ , then (−F.C) ≥ 0 by the definition of F. Therefore, −F is τ-nef 1 divisor. At a general point of Λ , τ O (−F) coincides with the defining ideal 1 ∗ H˜ sheaf I of the reduced subscheme Λ . Since every fiber of τ has dimension 1 1 ≤1, τ∗OH˜(−F)∼=I1 (cf. [Re 2, (2.14)]. Since −F is a τ-nef, τ-big divisor, the naturalmapτ∗τ O (−F)→O (−F)issurjective. Thustheidealτ−1I ⊂O ∗ H˜ H˜ 1 H˜ is invertible. Let H′ be the blowing up of H along I . Then τ is factorized as 1 1 H˜ →H′ →H. 1 Take an irreducible component of the singular locus of the resulting 3-fold which dominates Λ . Blow up the 3-fold along the defining ideal of this irre- 1 ducible component with reduced structure, and then take the normalization. Repeating such procedure resolves singularities along generalpoints of Λ . De- 1 note by τ :H →H the resulting 3-fold. Next take an irreducible component 1 1 8 of Sing(H ) which dominates Λ . Blow up H alongthe defining idealof it and 1 2 1 take the normalization. By repeating them, τ is finally decomposed as H˜ =H →τn H τn→−1 ...→τ1 H n n−1 We shall derive a contradiction by assuming n ≥ 2. By (i-b) there is a smooth surface E ⊂ H˜ which has a P1-bundle structure containing C as a i i fiber. These surfaces E are mapped onto the same irreducible component of Λ i by τ; indeed, if C ∩C 6= ∅ and τ(E ) 6= τ(E ), then E ∩E = {one point}, i j i j i j which is a contradiction because both E and E are Cartier divisors of H˜. i j Moreover, τ(E ) 6= Λ . Indeed, suppose to the contrary. Then C ,...,C are i n 1 m all contracted to a point by τ . At the same time, all exceptional divisors of τ n lying on Λ ,...,Λ are contracted to curves. By the construction of τ ’s, this 1 n−1 i is a contradiction. On the other hand, the decomposition of τ explained above depends on the ordering of the irreducible components of Λ. Thus we have τ(E )6=Λ for any i k k ≥1, which is obviously a contradiction. (iii) Let E ⊂H˜ be a smooth divisor mentioned above. It has a P1-bundle i structure containing C as a fiber. We shall prove i (iii-a): Exc(τ) is a divisor with simple normal crossings; (iii-b): Exc(τ)=S E ; 1≤i≤m i (iii-c): If C ∩C =∅, then E ∩E =∅. If C ∩C 6=∅, then E ∩E is a i j i j i j i j section of at least one of the P1-bundles E and E ; i j Firstweshallprove(iii-b). SinceΛisirreducibleby(ii),H˜ =H andτ =τ 1 1 in the notationof (ii). Theτ is decomposedinto blowing ups alongirreducible 1 reduced centers (followed by normalizations): H˜ →...→σ3 H(2) →σ2 H(1) →σ1 H Bytheconstruction,Exc(σ )hasafibrationoveranirreduciblecurvewhose k general fiber is isomorphic to P1 or a reducible line pair. When a general fiber of the fibration is irreducible, the special fiber must be irreducible. Indeed, if the special fiber contains more than one irreducible component, then the proper transform of some of them to H˜ becomes a rigid rational curve, which contradicts (i-a). In this case Exc(σ ) is irreducible. k Whenageneralfiberofthefibrationisreducible,thespecialfibermusthave one or two irreducible components because, if it has more than two irreducible components, then the proper transform of some of them to H˜ becomes a rigid rational curve. Ifthespecialfiberhasexactlytwoirreduciblecomponents,thenExc(σ )has k exactly two irreducible components. We shall prove that if the special fiber is irreducible, then Exc(σ ) is also k irreducible. Suppose to the contrary. Denote by C the special fiber. Then 9 Exc(σ ) has exactly two irreducible components F and F′. Eachof them has a k fibration over an irreducible curve, and the special fiber moves (as a 1-cycle on H(k)) in both F and F′. Let F˜ (resp. F˜′) be the proper transform of F (resp. F′)byσ˜ :H˜ →H(k). TheF˜ (resp. F˜′)hasafibrationoveranirreduciblecurve containing the proper transformC˜ of C in a specialfiber. The special fiber has only one irreducible componentC˜ because if it contains more, then C˜ is a rigid rational curve 2 and this contradicts (i-a). Thus C˜ moves (as a 1-cycle) in both F˜ and F˜′. On the other hand, since C˜ coincideswithoneofC ’s,C˜shouldmoveasfibersinonlyonesmoothP1-bundle i by (i-b). This is a contradiction. As a consequence, we know that Exc(τ) has exactly m irreducible components. Since E ’s are contained in Exc(τ), (iii-b) holds. i We shall next prove (iii-c) and (iii-a). The first statement of (iii-c) is clear. Assume that C ∩C 6= ∅. Denote by p : E → ∆1 (resp. p : E → ∆1) the i j i i j j P1-bundlestructureofE (resp. E )whosecentralfiberover0∈∆1 isC (resp. i j i C ). The intersection E ∩E is multi-sections of p and p of degree n and n j i j i j i j respectively. Suppose that n >1 and n >1. i j Let C be the set of all irreducible curves on H˜ which are fibers of p or p . i j For l, l′ ∈C, we say l and l′ are equivalent if there is a sequence of the elements of C: l :=l, l , ..., l , l :=l′ such that l ∩l 6=∅ for any k. This is an 0 1 k0−1 k0 k k+1 equivalence relation of C. Take a general fiber l∗ of p and consider the set C(l∗) of all curves which i are equivalentto l∗. Note that C(l∗)is a finite setconsisting ofsmooth rational curves. Pick up an element l ∈ C(l∗) which is a fiber of p . Then there are i at least n fibers of p which intersect l. Similarly, for any element m ∈ C(l∗) i j which is a fiber of p , there are at least n fibers of p which intersect m. This j j i implies that C(l∗) is not a tree of P1’s. On the other hand, C(l∗) is contained in a fiber of τ : H˜ → H, which is a contradiction. Therefore, n = 1 or n = 1. One can assume that n = 1. In i j i this case, E ∩E is a section of p , and E intersects E with multiplicity one i j i j i along E ∩E because, if not, then it contradicts the fact that eachfiber of τ is i j a tree of P1’s. Since there are no triple points in E ∪E ∪...∪E , (iii-a) and 1 2 m (iii-c) hold. (iv)We shallprovethatE ∩E issectionsofbothP1-bundles E andE in i j i j (iii-c). Ifthis is proved,then Exc(τ) is locally the productof a one-dimensional disk ∆1 and a tree of P1’s. 2 By Theorem (2.2) from [Re 1], we know that H(k−1) has only cDV sinularities. Put p := σk(C). The germ (H(k−1),p) is then isomorphic to a hypersurface singularity x2+ f(y,z,w)= 0 with deg(f) ≥ 3. There is an involution ι of (H(k−1),p) defined by x →−x, y→y,z→z andw→w. SinceH˜ isauniquecrepantresolutionofH(k−1),theιliftstoan involution˜ι of(H˜,σ˜−1(C)). By˜ι, F˜ andF˜′ areinterchanged. Therefore,ifthespecial fiber for F˜ is reducible, then the special fiber for F˜′ is also reducible. Any σ˜-exceptional divisor doesnotcontain C˜. SinceC˜ does notmoveinF˜ orF˜′,C˜ mustberigidinH˜. 10