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ON THE ISOTRIVIALITY OF FAMILIES OF ELLIPTIC SURFACES 0 ∗ KEIJI OGUISO AND ECKART VIEHWEG 0 0 2 A family f : X → B of projective complex manifolds is called birationally n a isotrivial, if there exists a finite cover B′ → B, a manifold F and a birational J map ϕ from F ×B′ to X× B′. The morphism f is isotrivial, if ϕ can be chosen B 0 to be biregular. 1 One can ask, tempted by the corresponding property for families of curves, ] whether f is birationally isotrivial whenever B is an elliptic curve or C∗ and the G Kodaira dimension of a general fibre non-negative. Assuming that all fibres of f A are minimal models, one could even hope that f is isotrivial. . h Both problems have an affirmative answer, if local Torelli theorems hold true t a for the fibres of f (or, as explained in 1.4, for some étale cover), and both have m been solved by Migliorini [13] and Kovács [10] for families of surfaces of general [ type (see also [21], [4] or [2]). In this note we want to extend their methods to 2 surfacesofKodairadimensiononeandthereby completetheproofofthefollowing v 0 theorem. 0 1 Theorem 0.1. All smoothprojective familiesof minimalsurfaces ofnon-negative 2 Kodaira dimension over complex elliptic curves or over C∗ are isotrivial. 1 9 The projectivity assumption is essential. Indeed there exist smooth, highly 9 / non-projective families of K3-surfaces over P1, called twistor spaces. h t Let Mh be the quasi-projective moduli scheme of polarized manifolds with a numerically effective canonical divisor and Hilbert polynomial h (see [20]). If Y m is a complex algebraic manifold, Φ : Y → M a morphism, étale over its image, : h v andifΦisinduced bya“universal” family, then0.1implies thatY isalgebraically i X hyperbolic for deg(h) = 2 (see also [11]). r IfY¯ isasmoothcompactificationwithS = Y¯−Y anormalcrossing divisor, one a might hope, that Ω1 (logS) (or some symmetric product) contains a subbundle Y¯ F, isomorphic to Ω1 (logS) over Y, with F numerically effective and det(F) Y¯ big. This positivity property holds true for moduli schemes of curves, and it has recently been verified by Zuo [22] if the fibres of the universal family over Y satisfy the local Torelli theorem. If B is an elliptic curve, or if the fibres X of f allow an étale cover which is an b elliptic surface without multiple fibres, the proof of the isotriviality is quite easy. In the first case, the proof is given at the beginning of section 4, in the second the necessary arguments are sketched in 4.2 and 4.3, as special cases of the proof of 0.1 for elliptic surfaces, given in section 7. We thank Egor Bedulev, Fabrizio Catanese, Daniel Huybrechts, Yujiro Kawa- mata and Qi Zhang for helpful remarks and comments. The first named author ∗ Supported by a fellowship of the Humboldt foundation. This work has been partly supported by the DFG Forschergruppe “Arithmetik und Geometrie”. 1 2 KEIJI OGUISO AND ECKART VIEHWEG would like to thank the members of the “Forschergruppe Arithmetik und Geome- try” at the University of Essen, in particular Hélène Esnault, for their hospitality and help. Notations 0.2. Indiscrepancy totheintroductionX andB willdenotecomplex projective manifolds of dimension three and one, and f : X → B will be a family of surfaces, i.e. a flat projective morphism with two dimensional connected fibres X = f−1(b). We fix an open dense subscheme B ⊂ B, such that b 0 f = f| : X = f−1(B ) −−→ B 0 X0 0 0 0 is smooth, and we write S = B −B and ∆ = f∗(S). 0 We will call f a family of minimal surfaces, if the non-singular fibres X , for b b ∈ B , are minimal models of non-negative Kodaira dimension, but we will not 0 require f to be a relative minimal model in a neighborhood of f−1(S). The dualizing sheaves of B, X and of f will be denoted by ω , ω and ω = B X X/B ω ⊗f∗ω−1. X B If D is an effective normal crossing divisor on X, Ωi (logD) = Ωi (logD ) X X red denotes the sheaf of logarithmic differential forms. Starting from section three, the general fibre F of f is assumed to be a minimal elliptic surface of Kodaira dimension κ(F) = 1 and starting with section four, we will assume that B is an elliptic curve and S = ∅, or that (B,S) = (P1,{0,∞}). 1. Families of surfaces and isotriviality The positivity results for direct images of powers of dualizing sheaves, due to Fujita, Kawamata and the second named author (see [15], 7.2 and the references given there) can be presented in a nice form, if the base is a curve and if the smooth fibres are minimal. Definition 1.1. Let X be a projective manifold and U ⊂ X an open dense subset. An invertible sheaf L on X is called i) semi-ample with respect to U, if for some µ and all multiples µ of µ the 0 0 map ϕµ : H0(X,Lµ)⊗C OX −−→ Lµ is surjective over U. ii) ample with respect to U, if L is semi-ample with respect to U and if ϕ µ induces an embedding U → P(H0(X,Lµ)) for µ sufficiently large. Lemma 1.2. Let f : X → B be a family of minimal surfaces of non-negative Kodaira dimension, smooth over B = B −S. 0 a) Then f ων is numerically effective, for all ν ≥ 1. ∗ X/B b) If f is semi-stable, then the following conditions are equivalent: i) For some ν > 0 and for all multiples ν of ν f ων is ample. 0 0 ∗ X/B ii) There exists some η > 0 such that f ωη contains an ample subsheaf. ∗ X/B iii) ω is semi-ample with respect of X = X −f−1(S) and for a general X/B 0 fibre F of f one has κ(ω ) = κ(F)+1. X/B iv) f is not birationally isotrivial. Corollary 1.3. Let τ : Y → X be generically finite. If f ◦ τ : Y → B is birationally isotrivial, then the same holds true for f : X → B. ISOTRIVIAL FAMILIES OF ELLIPTIC SURFACES 3 Proof. We may assume both, f and f◦τ to be semi-stable. The natural inclusion ω → τ ω induces an inclusion f ων → (f◦τ) ων , for all ν > 0. Hence X/B ∗ Y/B ∗ X/B ∗ Y/B if f is not birationally isotrivial, the condition ii) in 1.2 b) is satisfied. For a smooth projective family f : X → B consider the polarized variation 0 0 0 of Hodge-structures R2f C . If B is an elliptic curve or C∗, then this variation 0∗ X0 0 of Hodge-structures is necessarily trivialized over some étale cover B′ → B . In 0 0 fact, theinduced morphism fromtheuniversal cover CofB to theperioddomain 0 of polarized Hodge-structures is constant (see for example [19], §3). Combined with 1.3 one obtains: Corollary 1.4. If there exists an étale covering τ : Y → X , such that the fibres 0 0 0 of f ◦τ satisfy the local Torelli theorem, and if B is an elliptic curve over C∗, 0 0 0 then f is birationally isotrivial. Remark 1.5. The assumptions of 1.4 hold true for all families of minimal surfaces of Kodaira dimension zero. The same argument can be used to prove the corresponding statement for families of curves of genus g ≥ 1. For families of minimal surfaces the birational isotriviality is equivalent to the isotriviality. As well-known, the trivialization even exists over an étale cover of B . 0 Lemma 1.6. A smooth projective family f : X → B of minimal surfaces (or 0 0 0 curves) of non-negative Kodaira dimension is birationally isotrivial, if and only if there exists a finite étale cover B′ → B and a surface (or curve) F with 0 0 X × B′ ≃ F ×B′. 0 B0 0 0 Proof. It is easy to find a finite cover B′′ → B and an isomorphism 0 0 ϕ : X × B′′ −−∼→ F ×B′′ 0 B0 0 0 of polarized manifolds. In fact, there exists a coarse moduli space M of polarized h manifolds, and Kollár and Seshadri constructed a finite cover of M which carries h a universal family (see [20], p. 298). Of course one may assume B′′ → B to be 0 0 Galois with group G. In different terms, one has a lifting of the Galois action on B′′ to F × B′′, giving X as a quotient. Let H be the ramification group of a 0 0 0 point b ∈ B′′. Then H acts trivially on the fibre F ×{b}. 0 On the other hand, the automorphism group of a polarized manifold of non- negative Kodaira dimension is finite, hence the action of H on F × B′′ must 0 locally be the pullback under pr of the action on B′′. Necessarily the same holds 2 0 true globally and X × (B′′/H) = (F ×B′′)/H = F ×(B′′/H). 0 B0 0 0 0 2. A vanishing theorem Asin[13], [10], [21], [4]or[2]wewillusevanishing theorems forthecohomology ofdifferentialformswithlogarithmicpoles. However, wehavetoallowpolesalong somedivisor Π, transversal totheelliptic fibration. Inorder tofindsuchadivisor, we will be forced to modify f and to allow some additional singular points in the 0 fibres. 4 KEIJI OGUISO AND ECKART VIEHWEG Assumption 2.1. Let X,W and B be normal proper algebraic varieties of dimension three, two and one respectively, and let g X −−→ W J (cid:10) f J^ (cid:10)(cid:29) h B be morphisms with connected fibres. Consider an effective divisor Υ and a prime divisor Π on X, and an invertible sheaf L on W. Let B = B −S be open and 0 dense in B, X = f−1(B ), W = h−1(B ) 0 0 0 0 and denote by f , g , Π , L and h the restrictions to X and W , respectively. 0 0 0 0 0 0 0 Assume: i) Π is a section, i.e. g| : Π → W is an isomorphism. 0 Π0 0 0 ii) X is non-singular and ∆ = f∗(S) as well as ∆ + Π are normal crossing divisors. iii) h : W → B is smooth. 0 0 0 iv) g : X → W is a flat family of curves. 0 0 0 v) f : X → B is smooth outside of a finite subset T of X . 0 0 0 0 vi) The sheaf L is ample with respect to W . 0 vii) h Lν ∼= f (g∗Lν ⊗O (−ν ·Υ)), for all ν > 0. In particular Υ is supported ∗ ∗ X in ∆. viii) degω (S) ≥ 0. B Definition 2.2. a) For ι : X −T → X define Ωi (log∆)∼ = ι Ωi (log∆). X/B ∗ X−T/B b) Ωi (log∆)′ = Im(Ωi (log∆) −−→ Ωi (log∆)∼). X/B X X/B c) We use the same notation for the sheaves of differential forms with logarithmic poles along Π: Ωi (log(∆+Π))∼ = ι Ωi (log(∆+Π)) and X/B ∗ X−T/B Ωi (log(∆+Π))′ = Im(Ωi (log(∆+Π)) → Ωi (log(∆+Π))∼) X/B X X/B Since Π does not meet the non-smooth locus T of f , the sheaf 0 Ω2 (log(∆+Π))′ X/B is invertible in a neighborhood of Π and (2.2.1) Ω2 (log(∆+Π))′ = Ω2 (log∆)′ ⊗O (Π). X/B X/B X By definition one has the exact sequences (2.2.2) 0 −−→ f∗ω (S) −−→ Ω1 (log(∆+Π)) −−→ Ω1 (log(∆+Π))′ −−→ 0 B X X/B (2.2.3) 0 −−→ f∗ω (S)⊗Ω1 (log(∆+Π))∼ −−→ Ω2 (log(∆+Π)) −−→ B X/B X −−→ Ω2 (log(∆+Π))′ −−→ 0. X/B The main result of this section is ISOTRIVIAL FAMILIES OF ELLIPTIC SURFACES 5 Proposition 2.3. Assuming 2.1 H0(X,Ω2 (log(∆+Π))′ ⊗g∗L−1 ⊗O (Υ−Π)⊗f∗ω (S)−2) = X/B X B H0(X,Ω2 (log∆)′ ⊗g∗L−1 ⊗O (Υ)⊗f∗ω (S)−2) = 0. X/B X B Remark 2.4. If f is semistable, Ω2 (log∆)∼ = ω and Ω2 (log∆)′ is a X/B X/B X/B subsheaf, say ω′ , of ω . Then 2.3 says that X/B X/B H0(X,ω′ (Υ)⊗f∗ω (S)−2 ⊗g∗L−1) = 0. X/B B Proof of 2.3. The statement is compatible with blowing up W and X, as long as the centers are contained in h−1(S) and f−1(S), respectively. In fact, for τ : X′ → X and ∆′ = τ∗∆ Ω2X/B(log∆)′ ⊗OX(Υ) = τ∗(Ω2X′/B(log∆′)′ ⊗OX′(τ∗Υ)). Blowing up W (and hence X) we may assume that W is non-singular. For µ sufficiently large, Lµ(−h∗(S) ) is ample with respect to W . red 0 Hence, blowingupW andreplacing µbysomemultiple, wewillfindaneffective divisor Σ in W such that Lµ(−Σ) is globally generated and big, and such that Σ = h∗(S) . Moreover, if η : W → P(H0(W,Lµ(−Σ)) denotes the induced red red morphism, we can also assume that there exists an effective relatively anti-ample exceptional divisor E. Replacing Lµ(−Σ) by Lµ·ν(−ν ·Σ−E), we may assume finally that Lµ(−Σ) is ample. The assumption 2.1, vii), implies that g∗Σ ≥ µ·Υ. Since Π is a section, for some ρ > 0 the map 0 g∗g O (ρ·Π) −−→ O (ρ·Π) ∗ X X is surjective over X . After blowing up X, one finds an effective divisor Γ , 0 1 supported in ∆, with g∗g O (ρ·Π)−−→→O (ρ·Π−Γ ). ∗ X X 1 Let Σ ,... ,Σ be the irreducible components of Σ. For all ν, sufficiently large, 1 r and for all Σ′ = r ǫ Σ ≥ 0, with ǫ ∈ {0,1}, Pi=1 i i i g∗(Lµ·ν)⊗O (ρ·Π−g∗(νΣ+Σ′)−Γ ) X 1 is big and generated by its global sections. Choosing ν larger than ρ and larger than themultiplicities of the components ofg∗(Σ ) one finds ǫ ,... ,ǫ such that red 1 r N = ν ·µ does not divide the multiplicities of the components of Γ = g∗(ν ·Σ+Σ′)+Γ . 1 By construction Γ = ∆ , Γ ≥ N ·Υ, and g∗(LN)⊗O (ρ·Π−Γ) is globally red red X generated and big. Let us write Γ (N −ρ)·Π+Γ L′ = g∗(L)⊗O Π− = g∗(L)⊗O Π− . X(cid:16) h i(cid:17) X(cid:16) h i(cid:17) N N Claim 2.5. For all m ≥ 0 and for i+j < 3, Hi(X,Ωj (log(∆+Π))⊗L′−1 ⊗f∗ω (S)−m) = 0. X B 6 KEIJI OGUISO AND ECKART VIEHWEG Before proving 2.5, let us deduce 2.3. Using 2.5 and the long exact cohomology sequence induced by (2.2.3) ⊗L′−1 ⊗f∗ω (S)−2 one obtains an embedding of B Γ H0 := H0(X,Ω2 (log(∆+Π))′ ⊗g∗L−1 ⊗O −Π+ ⊗f∗ω (S)−2) X/B X(cid:16) hNi(cid:17) B = H0(X,Ω2 (log(∆+Π))′ ⊗L′−1 ⊗f∗ω (S)−2) X/B B into H1 := H1(X,Ω1 (log(∆+Π))∼ ⊗L′−1 ⊗f∗ω (S)−1). X/B B Since Ω1 (log(∆ + Π))′ → Ω1 (log(∆ + Π))∼ is surjective outside of a finite X/B X/B set of points, H1 is a quotient of H′1 := H1(X,Ω1 (log(∆+Π))′ ⊗L′−1 ⊗f∗ω (S)−1). X/B B Applying 2.5, for j = 1, i = 1, to (2.2.2), one finds an injective map H′1 −−→ H2(X,L′−1) and, 2.5, for j = 0, i = 2, implies that bothgroups are zero. Hence all the groups, H′1, H1 and H0, are zero. Since Γ ≥ Υ one obtains 2.3 from H0 = 0. hNi Proof of 2.5. By the choice of L′ one has g∗(LN)⊗O (ρ·Π−Γ) = L′N ⊗O (−(N −ρ)·Π−Γ′) X X for Γ′ = Γ−N· Γ . Since N does not divide the multiplicities of the components hNi of Γ, one finds Γ′ = ∆ . The sheaf g∗(LN)⊗O (ρ·Π−Γ) contains the inverse red red X image of an ample invertible sheaf on W. All this remains true, if we replace L′ by L′ ⊗f∗ω (S)m, and L by L⊗h∗ω (S)m. So we may assume m to be zero. B B If δ : X → PM is the morphism given by the global sections of the ν-th power of g∗(LN) ⊗ O (ρ· Π − Γ), for ν sufficiently large, then δ| = δ| can at X X−Γred X0 most contract components of the fibres of g . In particular the maximal fibre 0 dimension of δ| is one. X0 The divisor H of a general section of g∗(LN·ν)⊗O (ρ·Π−Γ)ν is smooth, and X Π+Γ+H a normal crossing divisor. By [5], 6.2 a) and 4.11 b), Hi(X,Ωj (log(Π+Γ+H))⊗L′−1) = 0 X for i+j 6= 3, and Hi(H,Ωj (log(Π+Γ)| )⊗L′−1) = 0, H H for i+j 6= 2. Considering the long exact sequence for 0 −−→ Ωj (log(Π+Γ))⊗L′−1 −−→ Ωj (log(Π+Γ+H))⊗L′−1 X X −−→ Ωj−1(log((Π+Γ)| )⊗L′−1 −−→ 0 H H one obtains 2.5. ISOTRIVIAL FAMILIES OF ELLIPTIC SURFACES 7 3. Families of elliptic surfaces Let us return to the family f : X → B of minimal elliptic surfaces of Kodaira dimension one, with f : X → B smooth. By [12] or [16], for all ν ≥ 0 and 0 0 0 b ∈ B with X = f−1(b), 0 b (3.0.1) f ων ⊗C(b) = H0(X ,ων ). ∗ X/B b Xb Fibrewise, for ν sufficiently large and divisible, H0(X ,ων ) defines the Iitaka b Xb map X → W to a non-singular curve W , and by (3.0.1) b b b f∗f ων −−→ ων ∗ X/B X/B defines the relative Iitaka map X −−→→g W ⊂ P(f ων ), ∗ X/B whose restriction g to X is a morphism, and W = g(X ) is smooth over B . 0 0 0 0 0 Blowing up X, as always with centers in f−1(S), we can factor f as g X −−→ W J (cid:10) f J^ (cid:10)(cid:29) h B where X, W and B are non-singular projective manifolds and where g : X → 0 0 W is a flat projective family of curves. Using the corresponding property for 0 X → W , one finds b b dimHi(g−1(w),ων ) = 1, g−1(w) for i = 0,1 and w ∈ W . Hence g ων is invertible for all ν ≥ 0. Moreover, 0 0∗ X0/W0 g∗g ω = ω (−Γ˜(0)) 0 0∗ X0/W0 X0/W0 for some divisor Γ˜(0). We will need several properties of elliptic threefolds, i.e. threefolds with an elliptic fibration. The results needed, due to Kawamata, Fujita, Nakayama, Mi- randa, Dolgachev-Gross and Gross are recalled in [8], together with more precise references. For elliptic threefolds occurring as the total space of a family of elliptic surfaces, [7] is an excellent source. The properties and definitions needed from the theory of elliptic surfaces, in particular Kodaira’s classification of the singular fibres, can be found in [1]. By [8], Lemma 1.2, blowing up W with centers in W − W one finds a flat 0 relative minimal model g : X → W, extending g : X → W . Before stating m m 0 0 0 this result in 3.2, we will use it to define the multiple locus and the discriminant divisor. In fact to this aim it would be sufficient to know the existence of g over m a subscheme W with codim(W −W ) ≥ 2. 1 1 Let ∆(g ) be the smallest subvariety such that m g−1(W −∆(g )) −−→ W −∆(g ) m m m is smooth. Notations 3.1. An irreducible one-dimensional component of ∆(g ) belongs to m one of the following, according to the fibre E = g−1(w) over the general point w m of the component: 8 KEIJI OGUISO AND ECKART VIEHWEG a) E is a multiple fibre. We denote those components by Σ ,... ,Σ and call 1 r Σ = r Σ the multiple locus. To Σ we attach the multiplicity m of the Pi=1 i i i general fibre, and Γ = g−1(Σ ) , hence m ·Γ = g−1(Σ ). i m i red i i m i b) Let j : W → P1 denote the rational map, induced by the j-invariant. Let D ,... ,D be the components of the discriminant locus whose image is ∞. 1 s To D we attach the multiplicity b of D in j−1(∞). In particular, if the i i i general fibre over D is a Newton polygon, then b is nothing but the length i i of the polygon, (i.e. type I ). We write J = s b D . bi ∞ Pi=1 i i c) If E is not a multiple fibre, nor a Newton polygon, we denote the corresponding components by D ,... ,D and we attach a number b to D s+1 ℓ i i according to Kodaira’s classification (see [1], for example): type I∗ II III IV II∗ III∗ IV∗ n b 6 2 3 4 10 9 8 i d) D = ℓ D will be called the discriminant locus, and Si=1 i ℓ ℓ b D = J + b D X i i ∞ X i i i=1 i=s+1 the discriminant divisor. Remark that Σ and J can have common components, corresponding to I . ∞ m n The component of the discriminant locus with general fibre of type I∗ will occur n in J = s b D with multiplicity n and in ℓ b D with multiplicity 6. ∞ Pi=1 i i Pi=s+1 i i Lemma 3.2. Blowing up W with centers in W−W , there exists a flat morphism 0 g : X → W, with g−1(W ) = X and g | = g , such that m m m 0 0 m X0 0 a) W −W is a normal crossing divisor. 0 b) X has at most Q-factorial terminal singularities. m c) g ω is an invertible sheaf δ. m∗ Xm/W d) δ12 ≃ O ( ℓ b D ) = O (J + ℓ b D ), and, for all ν ≥ 0 W Pi=1 i i W ∞ Pi=s+1 i i r ν(m −1) ω[ν] = g∗ δν ⊗O i Γ . Xm/W m Xm(cid:16)X m i(cid:17) i i=1 e) The j-invariant defines a rational map j : W → P1, regular in a neighborhood of h−1(S), and j∗(∞) = J . ∞ By [8], lemma 1.2, 3.2 holds true if the discriminant locus is a normal crossing divisor and if one allows further blow ups. Hence one obtains 3.2 over the complement in W of finitely many points of W . Since X is non-singular, since 0 0 g : X → W is flat and since g ων is invertible, b), c) and d) extend to W . 0 0 0 0∗ X0/W0 0 Let us recall the following property of the multiple locus Σ, first observed by Iitaka. Lemma 3.3. Keeping the notations introduced above, Σ(0) = Σ∩W is étale over 0 B and the fibres of (g−1Σ(0)) → Σ(0) are reduced. 0 0 red Proof. LetuswriteagainΓ(0) = g∗Σ(0). Thenω = g∗(g ω )⊗O (Γ(0)−Γ(0)). 0 X0 0 0∗ X0 X0 red If Σ(0) → B is not étale, there exists some b ∈ B such that Σ(0)| contains a 0 0 Wb multiple point. This remains true, if we replace B by any finite cover B′ → B . 0 0 0 ISOTRIVIAL FAMILIES OF ELLIPTIC SURFACES 9 In particular, in order to prove the first part of 3.3, we may assume that Σ(0) = s Σ , for Σ the image of a section of W → B . The same can be assumed Pi=1 i i 0 0 for the second part. In fact, if Σ(0) → B is étale but some fibre of Γ(0) → Σ(0) 0 red non reduced, then the same remains true after replacing B by an étale covering. 0 Consider for some r ≥ 1 a point v ∈ W which lies exactly on r of the compo- b nents Σ of Σ(0), say i v ∈ Σ ∩...∩Σ ∩W . 1 r b Let E denote the reduced fibre of g or g over v, let Γ = (g∗Σ ) and let m be b i i red i the multiplicity of Γ in g∗Σ . Finally let M be the multiplicity of E as a fibre of i i g : X → W and Γ .W the intersection cycle, a positive multiple of E. b b b i b For all µ ≥ 1 the natural map g∗g ωµ → ωµ induces an isomorphism 0 0∗ X0 X0 s g∗g ωµ −−∼=→ ωµ − m µ·(mi −1) Γ 0 0∗ X0 X0(cid:16) X iD m E i(cid:17) i i=1 where a = a−[a] denotes the fractional part of a real number a. Since a similar (cid:10) (cid:11) equation holds true for g , one obtains b r µ·(m −1) µ(M −1) i (3.3.1) m ·(Γ .W ) = M ·E. X iD m E i b D M E i i=1 Choosing for µ the lowest common multiple l = lcm(m ,... ,m ) the left hand 1 r side of (3.3.1) is zero, hence M divides l. Choosing µ = M, one finds that each m divides M, hence M = l = lcm(m ,... ,m ). For µ = M −1 = r ·m −1 one i 1 r i i has (M −1)(m −1) 1 (M −1)2 1 i = r ·m −(r +1)+ and = M −2+ . i i i m m M M i i Therefore (3.3.1) implies that r Γ .W = E. This is only possible for r = 1 Pi=1 i b and if Γ .W is reduced. 1 b Remark 3.4. Let Σ be an irreducible component of the multiple locus Σ and 1 let Γ = g−1(Σ ) . The fibres of Γ ∩X → Σ ∩W are either smooth elliptic 1 1 red 1 0 1 0 curves or Newton polygons. Assume the latter, i.e. that Σ is contained in the 1 discriminant locus. Then Γ is non-normal. However, since the fibres of Γ over 1 1 points in Σ ∩W have at most ordinary double points as singularities, the non- 1 0 normal locus must be étale over Σ ∩W , hence over B . Altogether, replacing 1 0 0 B by an étale covering, we can assume that Σ(0) = Σ∩W consists of sections 0 0 and that the same holds true for the non-normal locus of the reduced multiple divisors. In order to apply the vanishing stated in 2.3, we would like to restrict ourselves to semistable families X → B. However, in doing so, one would have to allow W to be singular, and 3.2 would not apply. The following technical construction will serve as a replacement. Lemma 3.5. Let f : X → B be a family of elliptic surfaces of Kodaira dimension one, with f : X → B = B − S smooth and relatively minimal. Assume that 0 0 0 S consists of at least two points, if B = P1. Then there exists a finite covering 10 KEIJI OGUISO AND ECKART VIEHWEG τ : B′ → B, with B′ = τ−1(B ) étale over B and a diagram of projective 0 0 0 morphisms X′ −−η−′→ Xs −−σ−′→ X g′ gs g    Wy′ −−−η→ Wys −−−σ→ Wy h′ hs h    By′ −−=−→ By′ −−−τ→ By with (as always, the index refers to the restrictions to B and B′): 0 0 0 i) η and η′ are isomorphisms. σ and σ′ are fibre products. 0 0 0 0 ii) X′, W′ and Xs are non-singular, Ws is normal with at most rational Goren- stein singularities. iii) fs = hs ◦gs : Xs → B′ is semistable, hence the fibres of hs : Ws → B′ are reduced, and for f′ = h◦g the fibres ∆′ = f′−1(B′−B′) and h′−1(B′ −B′) 0 0 are normal crossing divisors. iv) Let Σ′ be the multiple locus for g′ in W′. Then Σ′∩W is the disjoint union 0 of sections, as well as the non-normal locus of g′∗(Σ′) ∩X0. red v) δ′ = g∗′ωX′/W′ is invertible, and j : Ws → P1 is regular in a neighborhood of (hs)−1(B′ −B′). 0 vi) Let D′ denote the discriminant locus. Then h′−1(B′ − B′) + D′ + Σ′ is a 0 normal crossing divisor in a neighborhood of h′−1(B′ −B′). 0 vii) δ′12 = OW′(Pℓi=1biDi′) = OW′(J∞′ + Pℓi=s+1biDi′), where Pℓi=1biDi′ is the discriminantdivisor, defined in 3.1 (in particular, components corresponding to I∗, occur twice). b viii) Let Σ′,... ,Σ′ be the components of the multiple locus which dominate B′. 1 r Then for all ν > 0 one has r ν ·(m −1) f∗′ωXν′/B′ = h′∗(cid:16)ωWν ′/B′ ⊗OW′(cid:16)Xh mi iΣ′i(cid:17)⊗δ′ν(cid:17). i i=1 ix) Let D′ ,... ,D′ be those components of ℓ D′, which dominate B′. s+1 ℓ′ Pi=s+1 i Then for all multiples ν of 12 r ℓ′ ν ·(m −1) ν ν ·b f∗′ωXν′/B′ = h′∗(cid:16)ωWν ′/B′ ⊗OW′(cid:16)Xh mi iΣ′i + 12J∞′ + X 12iDi′(cid:17)(cid:17). i i=1 i=s+1 x) gsων = (η ◦g′) Ω2 (log∆′)ν and both sheaves are reflexive. ∗ Xs/B′ ∗ X′/B′ Proof. We may assume, that ∆+D+Σ is a normal crossing divisor and that the j-invariant defines a morphism in a neighborhood of h−1(S). We choose B′ to be ramified over S of order divisible by the multiplicities of the components of h−1(S), and such that X × B′ has a stable reduction B fs : Xs → B′. 3.3 and 3.4 allow to assume that iv) holds true. Choosing forWs the normalizationofW× B′, the fibres of hs arereduced and B Ws has at most rational Gorenstein singularities. Obviously fs factors through Ws. W′ is a desingularization of Ws, such that vi) holds true, and such that the flat relative minimal model, described in 3.2, exists over W′. If we take for X′