Math. Z. 242, 687–707 (2002) Digital Object Identifier (DOI) 10.1007/s002090100374
#-Minimal models of uniruled 3-folds Massimiliano Mella Dipartimento di Matematica, Universit`a di Ferrara, 44100 Ferrara, Italia (e-mail:
[email protected]) Received: 3 March 2000 / in final form: 5 September 2001 / c Springer-Verlag 2002 Published online: 1 February 2002 –
Mathematics Subject Classification (1991):14J30, 14N05 1. Introduction The aim of minimal model theory is to choose, inside of a birational class of varieties, a “simple” element. This program has been fulfilled in dimension 2 by the Italian school of the beginning of the century and a decade ago in dimension 3. After its discovery, the theory of (−1)-curves has been used thoroughly to study algebraic surfaces. Unfortunately in the threefold case the program and the “simple” output objects are not easily handled. It is difficult to use them as a tool to understand the geometry of three dimensional varieties. Here, after [Re], we rephrase the standard minimal model program for uniruled varieties, using a polarizing divisor. As in [Re], it will be called #-minimal model. We will be able, under strong assumption on the variety studied, to govern the program and understand its output. Even if quite restrictive, the assumption needed are very geometric in nature. This allows to apply the #-program in various concrete situations. Indeed one purpose of this note is to give a generalized and unified treatment of various results on uniruled varieties, [Al] [CF] [Io]. We first state the #-program and observe some of its natural properties. This program is governed by a movable linear system H. The crucial observation is the following. If the generic element H ∈ H is a smooth surface of negative Kodaira dimension, then the #-program is well understood in a neighborhood of H. This allows to study both the steps of the program and the final output for specific families of 3-folds. For Fano varieties with bad
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anticanonical sections we improve previous results of Alexeev, [Al], Theorem 4.1. We describe the birational type of 3-folds T containing a big system of uniruled surfaces Corollary 5.5. Finally we are able to list the possible #models of small degree threefolds embedded in projective spaces. This is the higher dimensional counterpart of the classical result for surfaces of reduced degree ≤ 1, Theorem 5.8. In the appendix we collect some observations and examples about uniruled varieties embedded of small degree in projective spaces. This part is based on Castelnuovo bound and its generalization due to Harris, [Ha]. This note is a revised version of a preprint that circulated in 1996, inspired by a long stay at Warwick University and by suggestions of M. Reid. I also benefited a lot from many conversations with A. Corti and I would like to thank V. Alexeev for pointing me out the papers of Campana–Flenner, [CF], and Sano, [Sa]. The referee’s comment on the first version of this paper made it clearer and, I hope, easier to read. I would like to thank both Trento University for supporting me along the first part of this research and Centro Nazionale delle Ricerche (grant n. 203.01.66).
2. Notations and preliminaries All the varieties are defined over C and our notations are compatible with [KMM]. Let X be a variety we will denote with N E(X) ⊂ N1 (X) the closure of the cone of effective cycles inside the vector space of 1-cycles modulo numerical equivalence. By means of the intersection product to any Q-Cartier divisor D we associate the hyperplane D⊥ = {[Z] : D·Z = 0} ⊂ N1 (X). We are interested in the part of the cone where −KX is positive. If X is terminal this part is locally polyhedral and the rays delimiting it are called extremal rays. A surjective morphism f : X → Y with connected fibers between normal varieties is called a Mori fiber space if −KX is f -ample, rkP ic(X/Y ) = 1 and dimX > dimY . Given a projective morphism f : X → Y and A, B ∈ Div(X) ⊗ Q then A is f -numerically equivalent to B (A ≡f B) if A · C = B · C for any curve contracted by f . A is f -linearly equivalent to B ( A ∼f B ) if A − B ∼ f ∗ M , for some line bundle M ∈ P ic(Y ), we will suppress the subscript when no confusion is likely to arise. 2.1 A variety X is called uniruled if it admits a generically finite dominant rational map p : Y × P1 −−>X. By means of Miyaoka–Mori characterization of uniruled varieties, [MM], the minimal model program and Miyaoka characterization of minimal 3-folds, [Mi], a threefold is uniruled if and only if its Kodaira dimension is negative. A uniruled 3-fold is always birational to a Mori fiber space.
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Let H be a linear system of Weil divisors, not necessarily complete, on a variety X. H ∈ H a general member of this linear system. We will say that H is movable if h0 (X, nH) > 1 for some n > 0. Observe that the push forward of a movable H is again movable. In other words the elements of H cannot be contracted by a birational projective morphism. 2.2 The following is a particular case of [BS, Prop. 1.4]. Let X be a terminal 3-fold and f : X → W a Mori fiber space with generic fiber F Pr . Assume there exists an f -ample line bundle H ∈ P ic(X) with H|F ∼ O(1). Then X and W are smooth and (X, H) = (P(E), O(1)), with E = ϕ∗ H a rk(r + 1) vector bundle on W . 2.3 We want here to recall some basic properties of terminal Q-factorial 3folds. We will use them throughout the paper without explicit mentioning, as a reference see for instance [Me]. Let X be a Q-factorial terminal 3fold and x ∈ Sing(X) a singular point. Then X has isolated singularities and any irreducible surface S trough x is singular at x. Moreover if we let i(X) = inf {t ∈ N : tKX ∈ P ic(X)} then for any integral divisor A ⊂ X is i(X)A ∈ P ic(X). 3. #-Minimal models Let T be a terminal Q-factorial uniruled 3-fold. As observed in 2.1, there exists a birational modification Φ : T −−>T 1 −−> . . . −−>T k to a Mori space T k . Φ is obtained as a chain of birational modifications associated to extremal rays on T , T 1 , · · · . We are looking for a way to choose these extremal rays in a “natural” way. To do this we will borrow an idea of Reid, [Re, (2.3)]. Fix a movable linear system H. We will choose an extremal ray, say R+ [z], having the smallest value H · z. Definition 3.1 Let T be a terminal Q-factorial uniruled 3-fold and H a movable linear system with generic element H ∈ H on T . Assume that H is nef, by abuse of language we will say that H is nef, then ρ = ρH = ρ(T, H) =: sup {m ∈ Q|H +mKT is an effective Q-divisor} ≥ 0, is the threshold of the pair (T, H), see [Re, (2.1)]. Theorem 3.2 There exists a pair (T # , H# ), with generic element H # ∈ H# such that: i) there is a birational map ϕ : T −−>T # with ϕ∗ H = H# ; ii) π : T # → W is a Mori space; iii) ρ(T, H)KT # + H # ≡π OT # .
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Definition 3.3 According to [Re], (T # ,H#) will be called a #-Minimal Model of the pair (T, H). Proof of the Theorem. We will proceed in an inductive way. Let (T0 , H0 ) = (T, H). H0 is nef by hypothesis and T0 is uniruled. Therefore to (T0 , H0 ) are naturally associated: – the nef value t0 = sup{m ∈ Q|mKT0 + H0 is nef }, – a rational map ϕ0 : T0 −−>T1 , which is either an extremal contraction, or a flip, of an extremal ray in the face spanned by t0 KT0 + H0 , – a movable linear system H1 := ϕ0∗ H0 on T1 . One inductively defines ϕi : Ti −−>Ti+1 and (Ti+1 , Hi+1 ) as follows. First note that ti−1 KTi + Hi is nef. Let δ = sup{d ∈ Q|dKTi + (ti−1 KTi + Hi ) is nef } and define ti := δ + ti−1 . The second step is to prove that there exists an extremal ray [Ci ] ⊂ N E(Ti ) in the face spanned by ti KTi +Hi . By construction ti KTi +Hi = δKTi +(ti−1 KTi +Hi ). Furthermore ti−1 KTi + Hi is nef and (ti + ,)KTi + Hi is not nef. Then there exists a curve Ci , with KTi · Ci < 0, in the face spanned by ti KTi + Hi . Let us define ϕi : Ti −−>Ti+1 the birational modification associated to the extremal ray [Ci ] ⊂ N E(Ti ) and Hi+1 := ϕi∗ Hi . The inductive process is therefore composed by divisorial contractions and flips. Since T0 is uniruled after finitely many steps we reach a Mori fiber space π : Tk → W . Claim 3.4. (Tk , Hk ) satisfies iii). Proof of the claim. Let ϕi : Ti −−>Ti+1 a birational modification of the inductive procedure. By Kleiman’s criteria the cone of nef divisors is the closure of the ample cone. The cone of nef divisors is therefore contained in the closure of the cone of effective divisors. So that we always have ti ≤ ρHi . Let us observe that the threshold is preserved after any single birational modification. On one hand ρHi+1 ≥ ρHi because the push forward of an effective divisor is effective. On the other hand for any ϕi ϕ∗i KTi+1 = KTi − αE, for some α ≥ 0 (if ϕi is a flip then ϕ∗i is the trace morphism and α = 0). That is to say ϕ∗i (rKTi+1 + Hi+1 ) = rKTi + Hi − (r − ti )αE, thus ρHi ≥ ρHi+1 . Let ρ = ρHi and π := ϕk : Tk → W the Mori space ending the process. Since π is a fibration, then (tk + ,)KTk + Hk cannot be effective, therefore tk = ρ. Let (T # , H# ) = (Tk , Hk ) to conclude.
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Remark 3.5 Along the proof of Claim 3.4 we proved that the threshold is preserved by #-MMP modifications. Note that H# is relatively nef. Furthermore if the rational map defined by |mH| is birational then H# is relatively ample. In general the #-Minimal Model is not uniquely determined. This problem arise when two extremal rays, spanned by the same nef divisor have not disjoint exceptional loci. For example let T = E × F1 and H = π ∗ A, where E is a smooth curve of genus g > 0, A ∈ P ic(E) a movable divisor and π : T → E the natural projection. Then there are two extremal rays spanned by H itself, one of divisorial type and the other of fiber type. The order in which the rays are contracted determines the #-minimal model (either a P1 -bundle or a P2 -bundle). As mentioned in the introduction we are not able to handle the whole #program. We will restrict ourselves to study the following special situation. Let (T, H) a pair with ρH < 1. If there is a smooth surface S ∈ H it is possible to describe in detail the #-program in a neighborhood of the surface S, see also [CF, §2]. Proposition 3.6 Let ϕi : Ti −−>Ti+1 a birational modification in the #program relative to (T, H), with ρH < 1. Assume that S ∈ Hi is a smooth surface. Then ϕi (S) = S is a smooth surface and ϕi|S : S → S is either an isomorphism or the contraction of a disjoint union of (-1)-curves. Proof. S is smooth and Ti is terminal Q-factorial therefore S ∩ Sing(Ti ) = ∅. In particular Hi is a Cartier divisor. Claim 3.7. If ϕi is a flip then S is disjoint from the flipping curves Proof of the claim. Let C be a curve flipped by ϕi then KTi · C > −1, [Mo2], and C is not on the smooth locus of Ti . So that C ⊂ S and S · C is an integer. By definition (ρKTi + Hi ) · C ≤ 0 and by hypothesis ρ < 1. Therefore S · C = 0 and the claim is proved. Case 3.8 (ϕi contracts a divisor E onto a curve) The generic fiber F of ϕi is out of Sing(Ti ) and KTi · F = −1. Since (ρKTi + Hi ) · F ≤ 0, then Hi · F = 0 and S ∩ E is the disjoint union of (-1)-curves. Case 3.9 (ϕi is a divisorial contraction to a point) We can assume that Hi is ϕi ample, let E the exceptional divisor and B = ∪Bi = S ∩ E. Moreover Hi|E ample and therefore let E the exceptional divisor and B = ∪Bi = S ∩ E. Moreover Hi|E ample and therefore B is connected. Let ϕS := ϕi|S : S → S, then by adjunction formula KS · Bi < 0. Furthermore ϕS is birational therefore B is a (-1)-curve. Then E is smooth along B
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and E · B = −1. By construction −KT · B > 1, therefore −KE · B = −KT · B − E · B ≥ 3. ˜ → E a relatively minimal resolution of E and L = µ∗ (−E|E ). Let µ : E ˜ Then Then L · µ∗ B = 1. Let [µ∗ B] ≡ bi [Ci ] for some [Ci ] ∈ N E(E). i
−3 ≥ KE˜ · µ∗ B = and 1=
i
bi KE˜ · Ci
bi L · Ci .
i
L is nef and KE˜ is µ-nef. Moreover L · C = 0 if and only if µ(C) = pt. ˜ P2 , see for Therefore there is one i such that KE˜ · Ci ≤ −3, and E instance [CKM]. Furthermore L is ample and KE · B = KE˜ · B = −3. That is KTi · B = −2 and (Hi|E )2 = Hi · B = 1. By relative spannedness, [AW], Hi|E is spanned and thus h0 (E, Hi|E ) ≥ 3. This is enough to prove that ∆(E, Hi|E ) = 0 and (E, E|E ) (P2 , O(−1)). Using the Proposition 3.6 we can control #-minimal model and its output. Corollary 3.10 Let T be a terminal Q-factorial uniruled 3-fold and H a movable and nef linear system. Set (T # , H# ) a #-minimal model of (T, H). Assume that ρH < 1 and H is base point free. Then H # ∈ P ic(T # ), H# has at most base points and H # is smooth. Proof. By Bertini Theorem H is smooth therefore we can apply Proposition 3.6 in an inductive way up to reach a model (T # , H# ). We will need the following relative version of Corollary 3.10. Definition 3.11 Let T be a 3-fold and H a movable linear system, with dimH ≥ 1. Assume that H = M + F , where M is a movable linear system without fixed component and F is the fixed component. A pair (T1 , H1 ) will be called a log minimal resolution of the pair (T, H) if there is a morphism µ : T1 → T , with the following properties: – T1 is terminal Q-factorial – µ−1 ∗ M = H1 , where H1 is a Cartier divisor, dimBsl(H1 ) ≤ 0 – a general element H1 ∈ H1 is a minimal resolution of a general element M ∈ M. Corollary 3.12 For any pair (T, H) of an irreducible Q-factorial 3-fold T and a movable linear system H, with dimH ≥ 1, there exists a log minimal resolution.
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Remark 3.13 If T is an irreducible uniruled 3-fold and H a movable linear system of Weil divisors, we will call #-Minimal Model of the pair (T, H), a #-Minimal Model of a log minimal resolution. Note that this is well defined only up to birational equivalence. Using Corollary 3.12 we can describe a pair (T, H), with dimH ≥ 1, in terms of a birational Mori fiber space. A #-Minimal Model of a log minimal resolution. It is useful to have a way to ensure that the resulting model is effectively different from the starting one. The following Lemma is in this direction and will be used in the next section to study Q-Fano 3-folds. Lemma 3.14 Let (T0 , H0 ) be a pair consisting of a uniruled 3-fold and a movable linear system with dimH0 ≥ 1. Let µ : (T1 , H1 ) → (T0 , H0 ) a log minimal resolution and (T2 , H2 ) a #-minimal model. Let Hi ∈ Hi generic elements. Assume that H0 is Cartier and −KH0 , −KH2 are ample. Then 2 ≤ K 2 . Furthermore if H is normal and singular then K 2 < K 2 . KH 0 H2 H0 H2 0 Proof. −KH2 is ample. Then ρH1 < 1 and by construction there are two well defined morphisms ν : H1 → H2 and µ : H1 → H0 . Moreover µ is induced by a log minimal resolution µ : T1 → T0 . If we define the ai ’s by ∗ KH1 = (KT1 + H1 )|H1 = µ (KT0 + H0 ) − ai E i |H1 = µ∗ KH0 − ai Ei ; then ai ≥ 0. By Corollary 3.10 ν is a morphism between two smooth surfaces, thus KH1 = ν ∗ KH2 + bi Ei , with bi ≥ 0, where Ei are either µ or ν exceptional. This yields to 2 2 ∗ KH + = K + 2ν K · (a + b )E i i i H H 2 0 2 (ai + bi )Ei · (ai + bi )Ei . By construction (ai + bi )Ei · (aj + bj )Ej = (µ∗ KH0 − ν ∗ KH2 ) · (aj + bj )Ej If ν∗ Ej = 0 then ( (ai + bi )Ei ) · (a + b )E = ( (ai + bi )Ei ) · j j j ∗ µ KH0 ≤ 0; while if µ∗ Ej = 0 then ( (ai + bi )Ei ) · (aj + bj )Ej = −ν ∗ KH2 · (aj + bj )Ej . So in any case 2 2 ∗ · µ · ν ∗ KH2 , = K + (a + b )E K + (a + b )E KH i i i i i i H H2 0 0
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where in the right hand side the last two terms are non positive. To prove the last assertion, let us start with a simple but useful observation. There are not birational contractions, that is morphisms with connected fibers, from a del Pezzo surface to a normal singular surface. In particular H1 is not a del Pezzo surface. That is there exists the class of a cycle [C1 ] ∈ N E(H1 ) such that KH1 · C1 ≥ 0. On the other hand H2 is Fano thus ν is not an isomorphism. There exists at least a (-1)-curve, say E0 , contracted therefore µ∗ (E0 ) = 0. by ν. But µ is a log minimal resolution 2 > K2 . Thus ( (ai + bi )Ei ) · µ∗ KH0 < 0, and KH H0 2 Let us derive another technical result, which will be useful in the appendix. Lemma 3.15 Let ϕi : Ti −−>Ti+1 a step of the #-program of a pair (X, H). Assume that Hi is irreducible and let B ⊂ Ti any curve not contained in the exceptional locus of ϕi . Then Hi · B ≤ Hi+1 · ϕi∗ (B). Proof. If ϕi is a morphism projection formula Hi+1 · ϕi∗ B = ϕ∗i Hi+1 · B = (Hi + αE) · B, allows to conclude. Assume now that ϕi is the flip of the curve C. Let us consider the following diagram Z DD DD q }} DD }} } DD } } " } ~ ϕi _ _ _ _ _ _ _ / Ti+1 Ti A AA zz AA zz AA z f A |zzz g p
W
where f and g are the contractions of [C] and [C + ], respectively, and p, q is a resolution of ϕi . Since Hi is f -nef, there exists an M ∈ P ic(W ) such that m(Hi + f ∗ M ) is spanned for m 0. Let HZ := p∗ (m(Hi + f ∗ M )) −1 ∗ ∗ ∗ then H Z = q∗ (m(Hi+1 + g M )). Furthermore q (m(Hi+1 + g M )) = HZ + ai Ei , with ai ≥ 0. Therefore again by projection formula Hi · B ≤ Hi+1 · ϕi∗ B.
4. General elephants of Q-Fano 3-folds In this section we will apply the #-program to study Q-Fano 3-folds with “bad” anticanonical class. Let T be a Q-Fano 3-fold and assume that the general element of |−KT |, has worse than Du Val singularities. According to
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the general elephant conjecture varieties of this kind shouldn’t exist. Alexeev proved that such a T , if one further assume that dimφ|−KT | (T ) = 3, is birational to a Q-Fano 3-fold on which a general anticanonical element has only Du Val singularities, [Al, Th 4.3]. Furthermore he proved that any Q-Fano 3-fold admits a Gorenstein model, at the expense of introducing canonical singularities, [Al, Th 4.8]. As observed by Corti, [Co2, 1.19], one should expect to have a Gorenstein terminal model for a Q-Fano 3-fold with “big” anticanonical system. Using #-Minimal Model Program we will show that a Q-Fano 3-fold T with “big” anticanonical system and whose general anticanonical element has worse than Du Val singularities, is birational to a smooth Fano 3-fold. In such a way we meet Corti’s expectations for this particular class, conjecturally empty, of Q-Fano 3-folds. Theorem 4.1 Let T be a Q-Fano such that dimφ|−KT | (T ) = 3 and the general element in | − KT | has worse than Du Val singularities. Then T is birational to a smooth Fano 3-fold T # of Fano index ≥ 2. Proof. Let H0 = | − KT | and µ : (T1 , H1 ) → (T, H0 ) a log minimal resolution of (T, H0 ). A general element in H0 has worse than Du Val singularities. Then the canonical class of H1 , KH1 = K + H1 , is negative on infinitely many curves dense in T1 . In particular ρH1 < 1. Let ν : T1 −−>T2 the result of the #-program applied to (T1 , H1 ). π : T2 → W the fiber type contraction associated to an extremal ray, say [Z] ∈ N E(T2 ). By Corollary 3.10, H2 ∈ P ic(T2 ) and a general element in H2 is smooth. Furthermore since dimφH0 (T ) = 3 then H2 is relatively ample and 1 KT2 · Z = − H · Z < −1. ρ There are the following possibilities: – dimW = 1. The general fiber F of π is either P2 or Q2 and T2 is birational to W × P2 , see for instance the appendix in [Co1]. T has rational singularities and h1 (T, OT ) = 0 then W P1 . In particular T2 is rational. – dimW = 2. π : T2 → W is a conic bundle with a section. So that T is birational to P1 × W . Again h1 (T, OT ) = 0, and via Castelnuovo rationality criteria, W and henceforth T are rational. – dimW = 0. Then ρ(T2 ) = 1 and H2 is ample. The general element H2 is a smooth del Pezzo surface and H2 has at most base points. By assumptions dim(H2 ) ≥ h0 (T, −KX ) ≥ 4 so that h0 (H2 , H2|H2 ) ≥ 3. By general properties of del Pezzo surfaces |H2 | is spanned by global sections. If T2 is smooth or dimW > 0, we have finished. Assume T2 is a singular Q-Fano. Let x ∈ Sing(T2 ) and L2 = |H2 ⊗ Ix |. |H2 | is ample and spanned
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therefore L2 has only base points. In particular the general element is a singular normal irreducible surface. Let (T3 , L3 ) a log minimal resolution of (T2 , L2 ), and (T4 , L4 ) a #-model. With π4 : T4 → W2 the Mori fiber space associated. Observe that also in this case ρL3 < 1. We have now to analyze the morphism π4 as before. Assume that L4 is π4 -trivial. Since dim(L2 ) ≥ 2 and the generic element is irreducible then L2 is not composite with a pencil and dimW2 = 2. Let ϕ2 : T2 −−>V2 the map defined by L2 with dimV2 = 2. Let Φ = Φ|H2 | : T2 9→ Pn , the morphism defined by sections of |H2 | and l a curve contracted by ϕ2 . Any hyperplane passing trough Φ(x) and any other point of Φ(l) must contain Φ(l). That is Φ(l) is a line trough Φ(x) ∈ Pn . Let us come back to the morphism π4 . The generic fiber f0 of π4 , is contained in L4 . The #-program is an isomorphism in a neighborhood of f0 . So that the strict transform of H2 is a birational section and we conclude as above that the 3-fold is rational. Assume that L4 is π4 -ample. If dimW2 > 0 we conclude that T4 is rational. If dimW2 = 0 then L4 is ample and a generic element L4 ∈ |L4 | is 2 ≥ 1. a smooth del Pezzo surface. Furthermore by Lemma 3.14 KL2 4 > KH 2 0 In particular |L4 | is spanned and since it is also ample then h (T4 , L4 ) ≥ 4. We can therefore iterate our argument and, by Lemma 3.14, it must stop after at most 7 steps, since KL2 k ≤ 9. 5. 3-folds with a big uniruled system Definition 5.1 Let T be a terminal Q-factorial 3-fold and H a movable linear system. We will say that (T, H) is a pair with a big uniruled system if H ∈ H is nef and big and H is a smooth surface of negative Kodaira dimension. Lemma 5.2 Let (T, H) be a pair with a big uniruled system. Then T is uniruled and ρ(T, H) < 1. Proof. Let f ⊂ H be a generic rational curve such that KH · f < 0. H is nef and big therefore KT · f < 0 and by [MM] T is uniruled. For m 0 let m(KT + H) = αH + B, with f ⊂ B. H is nef and m(KT + H) · f < 0 then either B is not effective or α < 0. Thus in any case ρH < 1. Our first aim is to list the #-Minimal Models of pairs with a big uniruled system. Theorem 5.3 Let (T, H) be a pair with a big uniruled system. Then (T # ,H# ) is one of the following: i)
a Q-Fano 3-fold of index 1/ρ > 1, with KT # ∼ −1/ρH # and Φ|H # | birational, the complete classification is given in [CF] and [Sa]:
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(P(1, 1, 2, 3), O(6)) (X6 ⊂ P(1, 1, 2, 3, a), X6 ∩ {x4 = 0}), with 3 ≤ a ≤ 5 (X6 ⊂ P(1, 1, 2, 2, 3), X6 ∩ {x3 = 0}) (X6 ⊂ P(1, 1, 1, 2, 3), X6 ∩ {x0 = 0}) (P(1, 1, 1, 2), O(4)) (X4 ⊂ P(1, 1, 1, 1, 2), X4 ∩ {x0 = 0}) (X4 ⊂ P(1, 1, 1, 2, a), X4 ∩ {x4 = 0}), with 2 ≤ a ≤ 3 (P3 , O(a)), with a ≤ 3, (Q3 , O(b)), with b ≤ 2 (X3 ⊂ P(1, 1, 1, 1, 2), X3 ∩ {x4 = 0}), (X3 ⊂ P4 , O(1)) (X2,2 ⊂ P5 , O(1)) a linear section of the Grassmann variety parametrising lines in P4 , embedded in P9 by Plucker coordinates (P(1, 1, 1, 2), O(2)), the cone over the Veronese surface # ) (P2 , O(2)) ii) a bundle over a smooth curve with generic fiber (F, H|F
# and with at most finitely many fibers (G, H|G ) (S4 , O(1)), where S4 is the cone over the normal quartic curve and the vertex sits over an hyper-quotient singularity of type 1/2(1, −1, 1) with f = xy − z 2 + tk , for k ≥ 1, [YPG], iii) a quadric bundle with at most cA1 singularities of type f = x2 + y 2 + # ∼ O(1), z 2 + tk , for k ≥ 2, and H|F iv) (P(E), O(1)) where E is a rk 3 vector bundle over a smooth curve, v) (P(E), O(1)) where E is a rk 2 vector bundle over a surface of negative Kodaira dimension.
Before giving the proof let us make a few comments about the output. Remark 5.4 The singular varieties at the point ii) are canonically birational, to a smooth projective bundle. In particular there exist singular terminal Mori fiber spaces with generic fiber (P2 , O(2)) and containing a smooth ruled surface. This corrects an error in [Me, Prop 3.7] and [CF, Prop 3.4], where such varieties have not been detected. To have an example it is enough to take a smooth projective bundle over a curve, say X = P2 × P1 , blow up a conic C which sits in a fiber G and contract the strict transform of G. In this way we produce a Veronese cone singularity. The smooth surface is a generic element of the strict transform of divisors in |OX (2, 1) ⊗ IC |. Going on blowing up hyperplane sections and contracting down we get hyper-quotient index two singularities. Indeed along the proof we will prove that reversing this process we obtain a canonical desingularization of the singular points that can appear in such a situation. The same is true also for singular quadric bundles at point iii). The varieties at point v) are birational to a projective bundle over either P2 or a ruled surface, but not in a canonical way. Indeed this is achieved following the #-program on the base surface by means of links on the 3-fold T # .
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Theorem 5.3 and Remark 5.4 allow to extend the result of [CF] to the following class of 3-folds. Corollary 5.5 Let T be an irreducible threefold and H a linear system with generic element H. Assume that one of the following is satisfied: – (T, H) is a pair with a big uniruled system – dimΦH (T ) = 3 and the Kodaira dimension of a resolution of H is negative. Then T is birational to one of the following: – – – –
H × P1 , a terminal sestic in either P(1, 1, 1, 2, 3) or P(1, 1, 2, 2, 3), a terminal quartic in P(1, 1, 1, 1, 2), a terminal cubic in P4 .
Proof of the Corollary. Let (T1 , H1 ) a log resolution of (T, H). Under both assumptions (T1 H1 ) is a pair with a big uniruled system. Then a #-model of (T1 , H1 ) is in the list of Theorem 5.3. We can now argue exactly as in [CF, Rem. 1.6]. Proof of the Theorem. By Sect. 3 we have a Mori fiber space π : T # → W with ρKT # + H # ≡π 0 and H # is smooth. Case 5.6 (Fano case) Assume that ρKT # ≡ H # and rkP ic(T # ) = 1, then H # is ample and generated by global sections outside a finite set of smooth # points. H # is smooth and KH # = (KT # + H # )|H # = (1 − 1/ρ)H|H #
therefore H # is a del Pezzo surface. The classification of these 3-folds is known [CF] and [Sa].
Case 5.7 (Fiber type case) Let F be a generic smooth fiber. Then (ρKT # + # = −KF . H # )|F ∼ OF thus 1/ρH|F If dimW = 1 then either F P2 and ρ = 1/3, 2/3 or F Q2 and ρ = 1/2. Furthermore all fibers are irreducible and reduced and the morphism is flat. Furthermore the delta genus of any fiber is defined and it is semi-continuous. This simple observation allows to give a biregular description of these varieties in the following way. 5.7.1 (F P2 and ρ = 1/3) Then T # P(ϕ∗ H # ) by (2.2). 5.7.2 (F P2 and ρ = 2/3) We will prove that the only possible fibers are P2 or S4 . The latter with the vertex sitting on a terminal hyper-quotient singularity of index 2 and type 1/2(1, −1, 1) f = xy − z 2 + tk , with k ≥ 1. Incidentally observe that −(KT # + H # )|F ∼ OF (1). Thus T # has
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a birational section and it is birational to a projective bundle. By (2.2) all the fibers on the Gorenstein locus are P2 . Assume that there is a singular non Gorenstein point x and a singular fiber G x. By Fujita classification and flatness, the only possible singular fiber is the cone over the normal quartic curve. In particular |H # | is relatively very ample. Using this remark we will be able to describe a birational map from T # to a projective bundle over W factored by elementary links in the Sarkisov category, [Co1]. Let ϕ : T # 9→ PN an embedding given by the sections of L = |H # + ∗ π M |, for some very ample M on W . Let ν : Y → T # the blow up of ϕ(x) with exceptional divisor E = ei Ei . Let L1 and L2 two generic element of |L ⊗ Ix |. Then ν ∗ Li = LiY + E and ν ∗ G = GY + γi Ei , for integers γi ≥ ei . Since L is very ample then L1Y · L2Y · GY = 0 and L1Y · L2Y · Ei is the degree of Ei as subvariety of PN −1 . Therefore by projection formula (5.1) 4 = L1 · L2 · G = γi deg(Ei ). GY F4 and G is embedded in P5 by ϕ. Let E0 an irreducible component of the exceptional divisor that contains the rational normal curve of degree 4. Then degE0 > 2, γi = ei = 1 and GY |E ∼ LY |E is ample. So that E is irreducible and reduced of degree 4. It is now immediate to observe that E is either the Veronese surface or the cone over a rational normal quartic curve. Let f be a fiber of the ruling on GY then GY · f = −1. Moreover by adjunction formula KY · f = −1. This is enough to prove that GY ⊂ Y is contractible to a 3-fold T1 . Moreover T1 is smooth along the exceptional locus of the contraction. Iterating this process we get a resolution of the singularities of T # . Keep in mind that ϕ(T # ) ⊂ PN and we are making smooth blow ups of points in the ambient variety. After finitely many steps we have the following picture
T#
} ν }}} } } ~} }
...
Y @ @
@@ @@ @@
T1
}} }} } }} ~} }
. . .A
AA AA AA AA
Tk
}} }} } } }~ }
YA
AA AA η AAA
T
in particular T is a smooth projective bundle over W and Tk has a quotient singularity of type 1/2(1, −1, 1). We have to describe an analytic neighborhood of the point x ∈ T # . By results of Pinkham, [Pi] and Koll`ar–Shepherd-Barron, [KSB, pag 313316], the versal deformation of the singularity in S4 the versal deformation
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of the singularity in S4 has only one smooth component of dimension 1, with generic fiber a Veronese surface. Locally this family is the quotient of C4 /(xy − z 2 + t) by a cyclic group of order 2 acting on (x, y, z, t) with weights (1, −1, 1, 0). The local equation of T # can be obtained by this one by a base change t → tk . 5.7.3 (F Q2 and ρ = 1/2) Then by Fujita list the only possible singular fiber G is a quadric cone. In particular |H # | is relatively very ample. Repeating the same argument of the quartic cone we exhibit a canonical birational map to a smooth quadric bundle over W , factored by elementary links in the Sarkisov Category. Then argue by [KM, Prop 4.10.1] and [Cu]. To derive that the possible singularities of T # are cA1 points of the type k[x, y, z, t]/(x2 + y 2 + z 2 + tk ). Another approach is to use the versal deformation of the cone singularity. That is given by x2 + y 2 + z 2 + t. If dimW = 2 the only possibility is F P1 and ρ = 1/2. # 5.7.4 (F P1 and ρ = 1/2) . Then H|F = O(1) and by (2.2) W is
smooth and T # P(ϕ∗ H # ). Let S in H # a smooth section then ϕ|S : S → W is a birational morphism, therefore W is either ruled or the blow up of a ruled surface. To have a better description of these conic bundles consider the #minimal model of (W, G = (ϕ|S )∗ (H|S )), [Re]. Let (W # , G# ) the #Minimal Model of (W, G) and H the push forward of ϕ∗ H on W # . Then W # is either P2 or a ruled surface and (T # , H # ) is birational to (P(H), O(1)) T # _ _ _/ P(H) ϕ
W
ψ
/ W #.
It is possible to factor the birational map (T # , H # )−−>(P(H), O(1)) by elementary links in the Sarkisov category. The links are described in the following way. Let B ⊂ W a (-1)-curve then D = ϕ∗ (B) Fa , for some a ≥ 0. If a = 0 then the link is simply the blow down of D. If a > 0 then let C0 the exceptional section of the rational scroll D. Then NC0 /T ∼ O(−1) ⊕ O(−a). It is possible to antiflip it (or flop if a = 1) to get the following ν / T T _ _ _/ T˜ ϕ
W
ψ
/ W #.
where ν is the weighted blow up with weights (1, 1, a).
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There exists a natural geometric interpretation of the conditions imposed in Theorem 5.3. Theorem 5.8 Let Td ⊂ Pn be a degree d non degenerate 3-fold. Suppose that d < 2n − 4, then any #-Minimal Model (T # , H# ) of (Td , O(1)) is in the list of Theorem 5.3. Remark 5.9 This result is an answer to a question of Mumford, [Mu, 2.15 pg 66], about reduced degree varieties. The reduced degree of a variety Td ⊂ Pn of dimension k and degree d is by definition rd :=
d . n+1−k
We could rephrase Theorem 5.8 saying that 3-folds of reduced degree < 2 have a #-minimal model in the list of Theorem 5.3. If we forget about the polarization and consider only the 3-fold T # then the birational type of this class of 3-folds is described in Corollary 5.5. Observe that the 3-folds listed admit an embedding satisfying the numerical criteria. Theorem 5.8 can be interpreted as the three dimensional generalization of the following classical result for surfaces, [GH, Proposition at page 525]. Let S ⊂ Pn be a non degenerate irreducible surface of degree d ≤ n − 1 then S is either a rational scroll or the Veronese surface. In the threefold case we loose the biregular character of the classification and smoothness. Furthermore the varieties listed can have arbitrarily high irregularity. Consider for example P2 × C, with C a curve of genus g ≥ 2 embedded by (O(1), L). Where L is a non special very ample divisor of degree g+3. Even the irregularity of the base of conic bundles is unbounded. As the following example shows. Let T = F1 ×C, where F1 is the blow up of P2 and C is a genus g curve. Consider the linear system H = |(C0 +2f, L)|, with degL > 10g − 4. Then 1 KT + H 2 is nef and defines a conic bundle structure of T onto P1 × C. It is a simple calculation then to observe that the embedding defined by H satisfies the degree assumption. This is not surprising indeed. The surface classification is essentially based on the fact that the unique uniruled curve of degree d ≤ n is the rational normal curve. While uniruled surfaces of degree d < 2n − 2 have arbitrary irregularity. Proof of the Theorem. Let ν : X → T a resolution of singularities and H = ν ∗ O(1). By Lemma A.2 in the appendix (KX + H) · H 2 < 0. So that by adjunction formula, H ∈ H is a smooth surface of negative Kodaira dimension and (X, H) is a pair with a big uniruled system.
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Consider the proof of Proposition 3.6. If the nef value of the pair (Ti , Hi ) is positive the birational modification associated is the blow down of (P2 , O(−1)). If we restrict to study ample linear systems H it is possible to generalize all #-Minimal Model Theory to arbitrary dimensions. Corollary 5.10 Let Xd ⊂ Pn a non degenerate k-fold with k > 3 and only Q-factorial terminal singularities. Assume that d < 2(n − k) − 2 then a #-minimal model (X # , H # ) of (X, O(1)), in adjunction theory language (X # , H # ) is the first reduction, is one of the following: a Q-Fano n-fold of Fano index 1/ρ > k − 2, with KT # ∼ −1/ρH # and Φ|H # | birational, the complete classification is given in [Fu] if X # is Gorenstein and in [CF] and [Sa] in the non-Gorenstein case. # ii) a projective bundle over a smooth curve with fibers (F, H|F ) (Pk−1 , i)
# O(1)), or a quadric bundle with at most cA1 singularities, with H|F ∼ O(1); iii) (P(E), O(1)) where E is a rk(k − 1) ample vector bundle either on P2 or on a ruled surface.
Proof. Let us argue as in Theorem 5.8. By Lemma A.2 the threshold is< 1/(k − 2). By adjunction theory on terminal varieties, [Me], after finitely many blow downs of (Pk−1 , O(−1)), we find a Mori space structure. If X # is Fano and Gorenstein then rkP ic(X # ) = 1. Moreover the index is an integer ≥ dimX − 1, so that Fujita, [Fu], classification applies. All other cases are well described in [CF], [Sa] and [Me]. Remark 5.11 An equivalent statement was proved by Ionescu for smooth varieties in [Io]. As in the threefold case let me stress that the irregularity is unbounded. Xd is terminal Q-factorial. Under the minimal model conjecture, Corollary 5.10 furnishes the classification of #-models of irreducible non degenerate k-folds embedded in Pn of degree d < 2(n − k) − 2. Appendix In this appendix we sum up some observations about small degree varieties. Let us consider Tdk ⊂ Pn , a non degenerate irreducible k-fold. It is quite natural to predict that for “small” degree d, with respect to n, T must be uniruled. Our first aim is to find the region of the (n, d) plane banned to non uniruled varieties. Example A.1 Let E be an elliptic curve and Fk+1 ⊂ Pk an hypersurface of degree k + 1. Consider T = E × Fk+1 together with the natural projections p1 : T → E and p2 : T → Fk+1 . Let OT (Le ) = p∗1 OE (e) ⊗ p∗2 OFk+1 (1), then OT (Le ) is very ample for any e ≥ 3; let ϕe : T 9→ Pn , the embedding
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associated to |OT (Le )|, then n = h0 (E, OE (e)) · h0 (F4 , OF4 (1)) − 1 = (k + 1)e − 1 while the degree d = OT (Le )k = k(k + 1)e. In this case κ(T ) = 0 and d = k(n + 1). The main message hidden in this example is the following. It is not possible to take arbitrarily high Veronese embeddings of a fixed immersion to construct varieties sitting in the region in which we are interested in. We need to evaluate the negativity of the canonical class on hyperplane sections. Lemma A.2 Let Tdk ⊂ Pn an irreducible non degenerate k-fold. Let a = min{t ∈ N|d < t(n − k) + 2}, and degenerate k-fold. Let a = min{t ∈ N|d < t(n − k) + 2}, and assume that a ≤ k. Let ν : X → T a resolution of singularities and Hi ∈ |ν ∗ O(1)|, for i = 1, . . . , k −1 generic hyperplane sections. Let C = ∩k−1 1 Hi , then (KX + (k − a)H) · C < 0. Proof. Let S := ∩k−2 1 Hi a smooth surface containing C. From genus formula on the surface S (A.1)
1 g(C) = 1 + [d(k − 1 − (k − a)) + ((k − a)H + KX ) · C]. 2
We want to bound the genus from above. To do this we will will use Castelnuovo inequality, [GH, pag 251]. Let h = h0 (C, H) = n − k + 2, then the genus of C is bounded by the following sum g(C) ≤ (d − h + 1) + (d − 2h + 3) + (d − 3h + 5) + . . . , where we sum only positive terms. Therefore in our hypothesis we have either a(a − 1) g(C) ≤ (a − 1)d − (n − k) − a + 1, 2 or a(n − k) − (2a − 1) < d < a(n − k) + 2 and g(C) ≤ ad −
a(a + 1) (n − k) − a. 2
So that combining with equation (A.1) we obtain the desired inequalities ((k − a)H + KX )) · C ≤ (a − 1)(d − a(n − k) − 2),
(A.2) respectively
((k − a)H + KX )) · C ≤ (a + 1)(d − a(n − k) − 2).
Theorem A.3 Let Tdk ⊂ Pn . Assume that d < k(n − k) + 2, then T is uniruled.
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Proof. Let ν : X → T a resolution of singularities. Fix Hi ∈ |ν ∗ O(1)|, for i = 1, . . . , k − 1, and C = ∩Hi . By Miyaoka–Mori criterion, [MM], it will be enough to prove that KX · C < 0. This is the content of Lemma A.2 for a ≤ k. The following example settles completely the question. There exist kfolds X of degree d with κ(X) = 0 and d = k(n − k) + 2. Example A.4 ([Ha]) Let us consider a (k+1)-fold V , scroll over a rational normal, curve embedded of degree n − k in Pn and a generic element T ∈ |(k + 1)O(1) ⊗ IFn−k−2 |, where F is a fiber of the scroll. Then for the “generic” V , [Ha], T is spanned, thus for the generic element KT ∼ OT and degT = k(n − k) + 2. This example is particularly interesting. It is the example of a non uniruled variety for which the reduced degree is less then the dimension. In the curve and surface case this is impossible but starting from 3-folds on there are plenty of non uniruled variety, even of general type, which have this property. Let us now come back to 3-folds. the next example gives a family of uniruled 3-folds for which d = 3n − 8 and n is arbitrarily large. Keep in mind that by Lemma A.2 this family is just on the edge of the uniruled sector. Example A.5 Let T4 ⊂ P4 , Z = H1 ∩ H2 , with Hi ∈ |OT (1)|, a smooth hyperplane curve and Ma = |OT (a) ⊗ IZa−1 |. Let us consider π : X = BlZ (T ) → T and Ha ∼ π ∗ OT (a) − (a − 1)E the strict transform of Ma . Then rkP ic(X) = 2 and X admits a fibration f : X → P1 in K3 surfaces (the fibers are the strict transform of surfaces in |OT (1) − Z|). Therefore the only extremal ray is the one of the blowing up. Ha ∼ f ∗ OP1 (a − 1) + π ∗ OT (1), thus it is very ample, let ϕa : X → Ya ⊂ Pn the morphism defined by the sections of |Ha |. Let d = Ha3 the degree of Ya ⊂ Pn then Claim A.6 d = 3n − 8. Proof of the claim. By construction E 3 = −8 and E 2 · π ∗ OT (1) = −4. The degree of Ya is d = Ha3 = 4a3 + 8(a − 1)3 − 12a(a − 1)2 = 12a − 8. To calculate n let Ha = f ∗ O(a) + E, tensoring with Ha the structure sequence of f ∗ O(a) we get 0 → OX (E) → OX (Ha ) → Of ∗ O(a) (E) → 0, E − KX is nef and big therefore 0 → H 0 (X, E) → H 0 (X, Ha ) → ⊕a H 0 (F, OF (1)) → 0. Finally this yields n = h0 (X, Ha ) − 1 = ah0 (F, OF (1)) = 4a.
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Let us further note that there is only one extremal ray on Ta . Then the #-minimal program of the pair (Ya , O(1)) consists just in the contraction of E. The #-minimal model of (Ya , O(1)) is (T4 , O(a)). In such a way we have constructed varieties with Fano #-minimal model, embedded inside our area with n arbitrarily large. Remark A.7 It is important to stress the following. There are only finitely many deformation types of Q-Fano 3-folds, [Ka2]. Hence only finitely many complete linear systems of bounded degree on a Q-Fano 3-fold. The above example shows that, choosing a non complete linear system, one can get Q-Fano #-models with arbitrarily high codimension. Definition A.8 ([Ko]) Let T be a uniruled variety and H an ample line bundle. We say that T is uniruled of H-degree at most d if there is a covering family of rational curves {Cλ } such that Cλ · H ≤ d. Let us consider a 3-fold Td ⊂ Pn , with d < 3(n − 3) + 2. By Lemma A.2 T is uniruled. Our aim is to bound its O(1)-degree of uniruling. Example A.9 Let T = P1 × F4 , where F4 ⊂ P3 is a quartic surface. Consider the line bundle La of bidegree (a, 1) on the product T . Then L3 = 12a and h0 (T, OT (L)) = 4(a + 1). La is very ample for any a ≥ 1 and embeds T ⊂ Pn as a 3-fold of degree d = 3n − 9. Furthermore the La -degree of uniruling is a. Example A.9 shows that there is no hope to bound the uniruled degree of 3-folds embedded in our area. On the other hand the behavior of uniruled degree is not completely uncontrolled. Theorem A.10 Let Td ⊂ Pn and assume that d < 3n − 7. If d < 2n − 4 then Td is uniruled of O(1)-degree at most 5. Assume that 2n − 4 ≤ d < 3(n − 3) + 2. Let d , δ= 3(n − 3) + 2 Then T is uniruled of O(1)-degree at most 3δ . (1 − δ) Proof. Let (T # , H # ) a #-model of (T, O(1)) and ρ the threshold. By a result of Kawamata, [Ka1], a dense subset of T # can be covered by rational curves B such that −KT # · B ≤ 6. Therefore H # · B ≤ 6ρ. Let BT the strict transform of B on T . Then by Lemma 3.15 we get H · BT ≤ H # · B ≤ 6ρ.
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It is therefore enough to get a bound on the threshold depending on d. If d < 2n − 4 we already know that ρ < 1 thus T is covered by rational curves of degree at most 5. Assume that 2n − 4 ≤ d < 3n − 8. Then by equation (A.2) with a = k = 3 in Lemma A.2 (A.3)
KT · H 2 ≤ 2(δ − 1)(3(n − 3) + 2).
By definition of threshold if (rKT + H) · H 2 < 0 then r > ρ, therefore substituting Equation (A.3) we obtain the required upper bound for the threshold δ(3(n − 3) + 2) ρ≤ , 2(1 − δ)(3(n − 3) + 2) which allows to conclude. Remark A.11 The bound in Lemma A.10 is not sharp. In fact we are estimating the threshold on KT ·H 2 and we are using Kawamata result. Nonetheless it is interesting to observe that the degree of the ruling is bounded by a function depending only on δ, that is on the ratio d/n. Similar results are true for irreducible varieties of arbitrary dimension, under the Minimal Model conjecture. References [Al] [AW]
Alexeev,V: General elephants of Q-Fano 3-folds, Comp.Math 91, 91–116 (1994) Andreatta,M. Wisniewski,J: A note on non vanishing and its applications, Duke Math. J. 72 (1993) 739–755 [BS] Beltrametti,M. Sommese,A.J: On the adjunction theoretic classification of polarized varieties, J. reine angew. Math. 427 (1992), 157–192 [CF] Campana,F. Flenner, H: Projective threefolds containing a smooth rational surface with ample normal bundle, J. reine angew. Math 440 (1993), 77–98 [CKM] Clemens, H. Koll´ar, J. Mori, S: Higher-dimensional complex geometry, Asterisque 166 (1988). [Co1] Corti, A. Factoring birational maps of threefolds after Sarkisov, J. Alg. Geo. 4 (1995), 223–254 [Co2] Corti, A. Del Pezzo surfaces over Dedekind schemes, Ann. of Math. 144 (1996), 641–683 [Cu] Cutkowski, S: Elementary contractions of Gorenstein threefolds, Math. Ann. 280 (1988), 521–525 [Fu] Fujita, T: Classification theories of polarized varieties, London Math. Soc. Lecture Note Series 155 Cambridge University press (1990) [GH] Griffiths, P.: Harris, J. Priciples of algebraic geometry, John Wiley & sons (1978) [Ha] Harris, J: A bound on the geometric genus of projective varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), 35–68 [Io] Ionescu, P: On varieties whose degree is small with respect to codimension. Math. Ann. 271 (1985), 339–348
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V.A. Iskovskikh: Fano 3-folds I, II, Math. USSR Izv. 11, 485–527 (1977), 12 469–506 (1978) [Ka1] Kawamata, Y: On the length of an extremal rational curve, Inv. Math. 105, 609–611 (1991) [Ka2] Kawamata, Y: Boundedness of Q-Fano threefolds Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), Contemp. Math. 131, Part 3, A. M. S. 439–445 (1992) [KMM] Kawamata, Y. Matsuda, K. Matsuki, K.: Introduction to the Minimal Model Program in Algebraic Geometry, Sendai, Adv. Studies in Pure Math. 10, KinokuniyaNorth-Holland, 283–360 (1987) [Ko] Koll´ar, J: Rational Curves on Algebraic Varieties, Ergebnisse der Math. 32, (1996), Springer [KM] Koll´ar, J. Mori, S: Classification of three-dimensional flips. J. of the A.M.S. 51992(), 533–703 [KSB] Koll´ar, J. Shepherd-Barron, N: Threefolds and deformations of surface singularities. Inv. Math. 91 299–338 (1988) [Me] Mella, M: Adjunction Theory on terminal varieties. Proceeding ”Complex Analysis and Geometry” editors V. Ancona, E. Ballico, A. Silva: Pitman Research Notes in Mathematics 366 153–16 (1997) [Mi] Miyaoka, Y: On the Kodaira dimension of minimal threefolds. Math. Ann. 281, 325–332 (1988) [MM] Miyaoka, Y., Mori, S: A numerical criterion for uniruledness. Ann. of Math. 124, 65–89 (1986) [Mo1] Mori, S: Threefold whose canonical bundle are not numerically effective. Ann. Math. 124, 133–176 (1982) [Mo2] Mori, S: Flip Theorem and the existence of minimal models for 3-folds. Journal AMS 1, 117–253 (1988) [Mu] Mumford, D: Stability of projective varieties. Ens. Math. 23, 39–110 (1977) [Pi] Pinkham, H: Deformation of algebraic varieties with Gm -action. Asterisque 20 (1974) [Re] Reid, M: Surfaces of small degree. Math. Ann. 275, 71–80 (1986) [Sa] Sano, T: Classification of Q-Fano d-folds of index greater than d − 2. Nagoya Math. J. 142 133–143 (1996) [U2] Koll´ar et al: Flip and abundance for algebraic threefolds. Asterisque 211 1992 [YPG] Reid, M: Young Person’s Guide to canonical singularities. Algebraic Geometry Bowdoin Part I, Proc. of Symp. of Pure Math. 46, 345–414 (1987)
