L’INSTITUTFOURIER DE ANNALES

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A L

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L’INSTITUT FOURIER

Angelo Felice LOPEZ, Roberto MUÑOZ & José Carlos SIERRA On the extendability of elliptic surfaces of rank two and higher Tome 59, n^o1 (2009), p. 311-346.

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ON THE EXTENDABILITY OF ELLIPTIC SURFACES OF RANK TWO AND HIGHER

by Angelo Felice LOPEZ,

Roberto MUÑOZ & José Carlos SIERRA (*)

†Dedicated to the memory of Giulia Semproni

Abstract. — We study threefolds X ⊂ P^r having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.

Résumé. — On étudie les variétés de dimension troisX ⊂P^r qui ont comme section hyperplane une surface lisse avec une fibration elliptique. On prouve d’abord un théorème général sur les plongements possibles de ces surfaces de nombre de Picard égal à deux. Dans un deuxième temps, on prouve des résultats plus précis pour les fibrations de Weierstrass de rang supérieur ou égal à deux. En particulier, on prouve qu’une fibration de Weierstrass de rang deux qui n’est pas une surface K3 n’est pas une section hyperplane d’une variété de dimension trois localement intersection complète. On donne, de plus, des conditions sous lesquelles, pour beaucoup des plongements de fibrations de Weierstrass de rang quelconque, toute variété de dimension trois comme ci-dessus est un cône.

Keywords:Elliptic surfaces, hyperplane sections, Mori fiber spaces.

Math. classification:14J30, 14J27, 11G05.

(*) The first author was partially supported by the MIUR national project “Geome- tria delle varietà algebriche" COFIN 2002-2004 and by the INdAM project “Geometria birazionale delle varietà algebriche". The author would like to thank the CIMAT (Gua- najuato, Mexico) for the wonderful hospitality during part of this research. The second author was partially supported by MCYT project BFM2003-03971. The third author was also partially supported by the program “Profesores de la UCM en el extranjero.

Convocatoria 2006".

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1. Introduction

The minimal model program has highlighted the importance, among the basic building blocks in the study of birational equivalence classes of algebraic varieties, of Mori fiber spaces, that is morphisms f : X → Y with connected fibers such thatXis normal projective withQ-factorial terminal singularities, Y is normal projective, dimY < dimX, −KX f-ample and ρ(X)−ρ(Y) = 1. In dimension3, where the minimal model program has been accomplished, a lot of work has been dedicated to the study of the case whenY is a point, that is whenX is a Fano variety ([22, 23, 42, 46]).

Perhaps the next interesting case is whenY is a curve and here also several papers have appeared (see [7] and references therein).

In the present article we also study the latter case, but from a different point of view, that we wish to outline here. Given a Mori fiber spacef : X → Y with Y a curve and general fiber F, in many cases we can take a projective embeddingX ⊂P^r withOX(1)∼=OX(−KX+hF), h >>0.

Now a general hyperplane sectionS=X∩H inherits an elliptic fibration and will often haveρ(S) =ρ(X) = 2(the latter happens, for example when X is smooth, h²(OX) = 0and pg(S)>0, by a theorem of Moishezon [40, Thm.7.5]).

Reversing this scenery it seems therefore interesting to take a projective embeddingS⊂P^N of an elliptic surfaceS, for example with Picard number two, and study which threefoldsX⊂P^N⁺¹ can haveS as hyperplane section. In the literature there is also a lot of work performed in this direction, mostly based on the following two techniques. The first one is adjunction theory (see [5]), that, to say in a few words in the specific cases we are describing, aims first at extending the fibration to the threefold and then to study the properties of the threefold using the extended fibration. A nice example of this is the result of Badescu ([3, Thm.7]), that classifies pairs (X, L)with X a normal projective variety, L an ample line bundle onX such that there existsS∈ |L|that is aP^s-bundle over a curve. The second technique is more recent and is based on a theorem of Zak [50, page 277]

and on the theory of Gaussian maps [49]. For this point of view we mention here the study of smooth Fano threefolds and Mukai varieties [9, 10], and, more recently, of Enriques-Fano threefolds and threefolds with hyperplane sections pluricanonical surfaces of general type [29].

To explain the results proved in this article, employing both techniques above, we start with a few definitions.

In the sequel all varieties are over the complex numbers.

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Definition 1.1. — A subvarietyY ⊂P^N is calledextendableif there exists a subvariety X ⊂P^N+1 and a hyperplane H = P^N ⊂ P^N⁺¹ such thatY =X∩H,X is different from a cone overY anddimX = dimY+ 1.

Such anX is called anextensionofY.

If Y is extendable to a varietyX as above with locally complete inter- section singularities, we will say thatY isl.c.i. extendable. Similarly we can definesmoothly extendable ifX is smooth, ornormally extend- ableifXis normal,l.c.i.-terminal extendableifX has terminal locally complete intersection singularities, l.c.i.r.s. extendable if X is locally complete intersection with rational singularities.

We will study extensions of elliptic surfaces, as in the ensuing

Definition 1.2. — LetSbe a smooth irreducible projective surface and letBbe a smooth irreducible curve. We will denote byd(B)the minimum degree of a very ample line bundle onB. Anelliptic fibrationπ:S →B is a surjective morphism whose general fiber is a smooth connected curve of genus one. If a smooth surfaceS has an elliptic fibration we will call S anelliptic surface.

A simple but important point for us is that, in many cases (see Proposi- tion 4.7), extensions of elliptic surfaces are in factMori fiber spaces.

Our first result, which, together with Theorem 4.8 below, can be con- sidered a more precise version of [41, 3.5.2], studies which embeddings can occur for some extendable elliptic surfaces S ⊂ P^N with Picard number two.

Theorem 1.3. — Let S ⊂P^N be a smooth surface having an elliptic fibrationπ:S →Bwith general fiberf. Suppose thatN¹(S)∼=Z[C]⊕Z[f] for some divisorCand that the hyperplane bundle ofS isHS ≡aC+bf, so that we can also suppose, without loss of generality, thatC.f >1. Let X ⊂P^N⁺¹ be any l.c.i. extension of S and suppose furthermore that one of the following holds:

(i) X has rational singularities andg(B)>0;

(ii) X hasQ-factorial terminal singularities,B∼=P¹ andκ(S) = 1;

(iii) κ(S) = 1andH¹(S, KS+HS−f1−. . .−f_d(B)) = 0for every set of smooth distinct fibersf_i’s.

Then

(a, C.f)∈ {(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(3,3)}.

Moreover if, in addition,Xis locally factorial, then(a, C.f)6∈ {(1,7),(1,8), (1,9)}.

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The vanishing condition in (iii) above is satisfied in many cases, see for example Remark 4.3. Moreover we observe that there are examples of smoothly extendable smooth elliptic surfaces withκ(S) = 1, N¹(S)∼= Z[C]⊕Z[f],HS ≡aC+bf and(a, C.f)∈ {(1,3),(1,4),(1,5),(1,6),(2,4), (3,3)}(see examples 6.1-6.6).

Our results can be made a lot more precise if we assume a little bit more on the fibration: In caseS has a Weierstrass fibration (see Definition 3.1) andρ(S) = 2, thenSis often non l.c.i. extendable (but it can be extendable in higher rank, see Example 6.7).

Corollary 1.4. — LetS ⊂P^N be a smooth surface having a Weier- strass fibration π : S → B with general fiber f and section C. Set n =

−C², g=g(B)and suppose thatρ(S) = 2,n>1and(g, n)6= (0,1).

(i) If(g, n)6= (0,2)then S is not l.c.i. extendable.

(ii) If(g, n) = (0,2) then any possible l.c.i. extensionX ⊂P^N⁺¹ of S is an anticanonically embedded Fano threefold withρ(X) = 1 and h¹(OX) =h²(OX) = 0.

In the case (ii), which turns out to be exactly the K3-Weierstrass case (see Proposition 3.6(iii)), we can be a little bit more precise.

Corollary 1.5. — LetS ⊂P^N be a smooth surface having a Weier- strass fibration π: S →P¹ with general fiber f and section C such that C² =−2. Suppose that ρ(S) = 2. Let HS ∼aC+bf be the hyperplane bundle ofS and letg(S)be the sectional genus ofS. We have:

(i) S is not l.c.i.-terminal extendable.

(ii) If(a, b, g(S))6∈ {(3,7,13),(3,8,16),(3,10,22),(3,11,25),(3,13,31), (3,14,34),(4,9,21),(4,11,29),(4,13,37),(5,11,31),(5,12,36)}, then S is not l.c.i. extendable.

(iii) If (a, b, g(S)) 6∈ {(3,7,13),(3,8,16),(3,9,19),(3,10,22),(3,11,25), (3,12,28),(3,13,31),(3,14,34),(3,15,37), (4,9,21), (4,10,25),(4, 11,29),(4,12,33),(4,13,37),(5,11,31),(5,12,36)}, then S is not normally extendable.

We end this introduction with a non extendability result (regardless of the singularities of the extension) for Weierstrass fibrations with more spe- cial assumptions on the embedding line bundle,but with no assumption on the rank on the Picard group.

Theorem 1.6. — LetS⊂P^N be a smooth surface having a Weierstrass fibrationπ:S→B with general fiberf and section C. Setn=−C² and

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g=g(B). Suppose that the hyperplane bundle ofSis of typeH_S ≡aC+bf and thatn>1.

Then S is not extendable if any of the following conditions is satisfied:

(i) g= 0 anda>6 with(a, b, n)6= (6,7,1), or (ii) g>1 anda≡0 (mod3),a>6, or

(iii) g >1, S is linearly normal and either a >7, b >an+ 5g−1 or a= 5,b>6n+ 7g−3.

Acknowledgments. The authors wish to thank Mike Roth for several helpful discussions.

2. Background material

We recall in this section, for the reader’s convenience, some results of adjunction theory that will be used in the sequel.

Throughout this section we will denote by X an irreducible normal n- dimensional projective variety with terminal singularities,n>2, byA an ample line bundle onX and by r the index of singularities of X, that is the smallest positive integerr such thatrKX is a Cartier divisor.

Definition 2.1. — [5, Def.1.5.3] Thenefvalue of (X, A)is τ(X, A) = min{t∈R:KX+tAis nef}.

By Kawamata’s rationality theorem [24, Thm.4.1.1], ifKXis not nef, the nefvalue is a rational number and we can writerτ = ^u_v for someu, v∈N. By the Kawamata-Shokurov base-point free theorem [30, Thm.3.3], the linear system|m(vrKX +uA)| is base-point free for m >> 0 and using the Stein factorization of the morphism defined by this linear system one gets a morphism φ(X, A) : X → Y with connected fibers onto a normal projective varietyY. This morphism depends only on the pair(X, A)and is called thenefvalue morphism of(X, A).

We mention here several useful results of general adjunction theory re- lated to the nefvalue.

Proposition 2.2. — [5, Prop.7.2.2]Letτ be nefvalue of(X, A). Then either

(i) τ=n+ 1and(X, A)∼= (Pⁿ,O_Pn(1)); or

(ii) τ=nand(X, A)∼= (Q,OQ(1)),Q⊂Pⁿ⁺¹ a quadric; or

(iii) τ=nand(X, A)is a(Pⁿ⁻¹,O_Pn−1(1))-bundle over a smooth curve underφ(X, A); or

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(iv) τ6nandK_X+nAis nef and big.

In the range n−1< τ < nwe have

Theorem 2.3. — [5, Thms.7.2.3 and 7.2.4]Assume thatX isQ-facto- rial and suppose that KX +nA is nef and big. Then KX+nA is ample and ifτ is the nefvalue of(X, A)we haveτ 6n−1 unlessτ =n−¹₂ and (X, A)is a generalized cone over(P²,O_P2(2)).

Whenτ6n−1 we have

Theorem 2.4. — [5, Thm.7.3.2] Suppose thatτ(X, A)6n−1. Then KX+ (n−1)Ais ample unless τ(X, A) =n−1 and either

(i) rKX ∼ −r(n−1)A; or

(ii) (X, A)is a quadric fibration over a smooth curve underφ(X, A); or (iii) (X, A)is a scroll over a normal surface underφ(X, A); or

(iv) φ=φ(X, A) :X →Y is birational. Moreover ifX is factorial then φ(X, A)is the simultaneous contraction to distinct smooth points of divisorsE_i ∼=Pⁿ⁻¹ such thatE_i ⊂Reg(X),O_E_i(E_i)∼=O_Pn−1(−1) andA_|E_i ∼=O_Pⁿ⁻¹(1). AlsoL:= (φ∗(A))^∗∗ and KY + (n−1)Lare ample andKX+ (n−1)A∼=φ^∗(KY + (n−1)L).

WhenKX+ (n−1)Ais nef and big we can define the first reduction of (X, A). By the Kawamata-Shokurov base-point free theorem [30, Thm.3.3]

we have a birational morphism π : X → X⁰ with connected fibers and normal image associated to the linear system|mr(KX+(n−1)A)|form >>

0. SetA⁰ := (π_∗(A))^∗∗. The pair(X⁰, A⁰)is called the first reduction of (X, A). Then we have

Theorem 2.5. — [5, Thm.7.3.4] Assume that X is factorial and that n>3and let(X⁰, A⁰)be the first reduction of(X, A). Suppose thatn−2<

τ(X⁰, A⁰)< n−1. Then either

(i) n= 4, τ(X⁰, A⁰) = ⁵₂ and(X⁰, A⁰)∼= (P⁴,O_P4(2)); or

(ii) n= 3, τ(X⁰, A⁰) = ³₂ and(X⁰, A⁰)∼= (Q,O_Q(2)),Q⊂P⁴ a quadric;

or

(iii) n= 3, τ(X⁰, A⁰) = ⁴₃ and(X⁰, A⁰)∼= (P³,O_P3(3)); or

(iv) n = 3, τ(X⁰, A⁰) = ³₂, φ(X, A) has a smooth curve as image and (F, A⁰_|F)∼= (P²,O_P2(2))for a general fiber F of φ(X, A).

3. Weierstrass fibrations

We collect in this section some notation and facts about Weierstrass fibrations that will be used in the sequel.

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Let π:S →B be a minimal elliptic surface, that is there are no (−1)- curves in the fibers ofπ. Suppose moreover thatπhas a sections:B→S.

On each reducible fiber contract any irreducible component not meeting s(B). Hence we obtain a new (singular) elliptic surfaceπ⁰ :S⁰→B with a section and whose fibers are all both reduced and irreducible. In this context a global Weierstrass equation can be given and the following concept appears (see [39]).

Definition 3.1. — Let S be a surface and let B be a smooth curve.

AWeierstrass fibration π:S →B is a flat and proper map such that every geometric fiber has arithmetic genus one (so that it is either a smooth genus one curve, or a rational curve with a node, or a rational curve with a cusp), with general fiber smooth and such that there is given a section of πnot passing through the singular point of any fiber.

We will say that a Weierstrass fibration π : S → B is smooth if S is smooth.

Remark 3.2. — By the above discussion the notions of Weierstrass fibrationandelliptic surface with section can be freely interchanged whenS is smooth andρ(S) = 2.

We recall from [39] some well-known facts about Weierstrass fibrations (see also [38]).

Definition 3.3. — Let π : S → B be a Weierstrass fibration with sectionC ⊂S. We define thefundamental line bundle L of π as the dual line bundle ofπ_∗N_C/S onB. We will setn= degL.

Remark 3.4. — AsL= (R¹π_∗OS)^∗, the fundamental line bundleLdoes not depend on the given sectionC. Moreovern=−C²>0andL ∼=OB if and only ifS is a productB×F, withF an elliptic curve [39, (II.3.6) and (III.1.4)].

Lemma 3.5. — [39, (II.3.5), (II.3.7) and (II.4.3)] Letπ : S → B be a Weierstrass fibration with section C and fundamental line bundle L. We have:

(i) π_∗O_S ∼=π_∗O_S(C)∼=O_B,R¹π_∗O_S(uC) = 0for every u>1.

(ii) π∗OS(mC)∼=OB⊕ L⁻²⊕ L⁻³⊕. . .⊕ L^−mfor every m>2.

The invariants and Kodaira type of a smooth Weierstrass fibration are given as follows.

Proposition 3.6. — [39, (III.1.1), (III.4), (IV.1.1) and (VII.1.3)] Let π:S →B be a smooth Weierstrass fibration with sectionC, general fiber

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f and fundamental line bundleL with n= degL >1 and g =g(B). We have:

(i) KS ≡(n+2g−2)f,q(S) =h¹(OS) =g,pg(S) =h⁰(KS) =n−1+g andh^1,1(S) = 10n+ 2g.

(ii) κ(S) =−∞if and only ifS is a rational surface if and only ifg= 0 andn= 1.

(iii) κ(S) = 0if and only if S is a K3 surface if and only ifg = 0 and n= 2.

(iv) κ(S) = 1if and only if(g, n)6∈ {(0,1),(0,2)}.

(v) LetA∈PicS. ThenA≡αC+βfif and only ifA∼=OS(αC)⊗π^∗M, for someM ∈Pic^βB.

By the above Proposition, the rank of the Picard group of a smooth Weierstrass fibration satisfies

26ρ(S)6h^1,1(S) = 10n+ 2g.

On the other hand, for surfaces with pg > 0 the Picard number ρ is in general strictly less than h^1,1 and by Hodge theory one expects at least pg independent conditions for a given cycle to be algebraic. The typical picture one conjectures is that the generic surface in its moduli space has low Picard number (as it is for general surfaces inP³).

For Weierstrass fibrations overP¹this prediction turns out to be true. In the latter case Miranda [38] constructed a moduli space for such fibrations and Cox [11, MainThm.] (see also [26, Cor.1.2]) proved that a general (in the countable Zariski topology) Weierstrass fibration π : S → P¹ with n>2, satisfiesρ(S) = 2.

Remark 3.7. — It is likely that there exist smooth Weierstrass fibrations π : S → B with ρ(S) = 2, where B is a smooth curve of genus g, for everyg >1. According to a suggestion of R. Kloosterman they should be constructed as follows. LetEbe the elliptic curve with J-invariant zero and, for anyn>1, letP1, . . . , P6n ∈B and letC be a cyclic covering of degree 6ramified atP₁, . . . , P_6n. IfCdoes not have a nonconstant morphism toE thenρ(S) = 2. To see this note that there is a line bundleLof degreenon Bsuch thatL⁶∼=OB(P1+. . .+P6n). We have therefore a nonzero section s∈H⁰(L⁶)giving rise to the Weierstrass data (L,0, s)onB and, by [39, II.5], to a smooth Weierstrass fibrationπ:S →B. By “Tate’s algorithm"

[39, IV.3.1] all fibers ofπare irreducible (the singular ones being cuspidal).

By Shioda-Tate’s formula [39, Cor.VII.2.4], we have thatρ(S) = 2 if the Mordell-Weil group of sections ofπhas rank zero, that is if all sections are of finite order. But a section of infinite order gives, as in [26, section 6],

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a nonconstant morphism from C to the elliptic curve E with J-invariant zero.

The following remark will be useful.

Remark 3.8. — Letπ:S→B be a smooth Weierstrass fibration with sectionCand general fiberf. Ifρ(S) = 2thenN¹(S)∼=Z[C]⊕Z[f]. To see this note that, sinceN¹(S)is torsion free, we haveN¹(S)∼=Z[A]⊕Z[A⁰]for someA, A⁰∈PicS. On the other handZ[C]⊕Z[f]has rank two, therefore there are integersa >1, a⁰ >1, u, v, u⁰, v⁰ such that aA ≡uC+vf and a⁰A⁰ ≡u⁰C+v⁰f. NowaA.f =u, whenceadivides both uandv and we getA≡u1C+v1f for some integersu1, v1. SimilarlyA⁰≡u⁰₁C+v⁰₁f.

To study the extendability of Weierstrass fibrations we will need the following simple results.

Lemma 3.9. — Letπ:S →B be a smooth Weierstrass fibration with sectionC, general fiberf and fundamental line bundleLwithn= degL>

1and g=g(B). Let D≡αC+βf. For g>1 andP ∈Pic⁰B we will set D_P =D⊗π^∗P. We have:

(i) H¹(D) = 0 if either α = 1 and β > 2g−1 or α > 2 and β >

αn+ 2g−1.

(ii) Ifg>1andP ∈Pic⁰Bis general, thenH¹(DP) = 0if eitherα= 1 andβ>g−1orα>2andβ >αn+g−1.

(iii) IfH ≡aC+bf, g >1 and P ∈ Pic⁰B is general, thenH¹(H− 2D_P) = 0if eithera−2α= 1andb−2β >g−1ora−2α>2and b−2β>αn+g−1.

(iv) |D| is base-point free ifα>2andβ >αn+ 2g.

(v) Dis very ample ifα>3 andβ>αn+ 2g+ 1.

(vi) IfD is very ample thenα>3andβ >αn+ 1.

Proof. — To see (i) note that, by Proposition 3.6(v), we have D ∼= O_S(αC)⊗π^∗M for some M ∈ Pic^βB. By Lemma 3.5(i) and the Leray spectral sequence we deduce thatH¹(S, D)∼=H¹(B, π_∗D) = 0for degree reasons.

We now show (ii) and (iii). Let d > g−1 be an integer. Since g > 1 we know that there is a nonempty open subset Vd ⊂ Pic^dB such that H¹(L) = 0for anyL∈ Vd. Given anyN ∈Pic^dBconsider the isomorphism φN : Pic⁰B →Pic^dB given by tensoring withN. Then we get a nonempty open subsetU_N :=φ⁻¹_N (V_d) ⊂Pic⁰B such that H¹(N⊗ P) = 0 for any P ∈ UN.

As above we have D ∼= O_S(αC)⊗π^∗M for some M ∈ Pic^βB and DP ∼=OS(αC)⊗π^∗(M ⊗ P). Since α> 1, Lemma 3.5(i) and the Leray

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spectral sequence imply thatH¹(S, D_P)∼=H¹(B, π_∗D_P). By Lemma 3.5(i) and (ii) we haveπ_∗D_P =M⊗Pwhenα= 1andπ_∗D_P = (M⊗P)⊕

α

L

i=2

(M⊗ L⁻ⁱ⊗ P)whenα>2. Hence, to prove (ii), we can choose P ∈ UD:=UM

whenα= 1 andP ∈ UD:=UM ∩

α

T

i=2

U_M_⊗L⁻ⁱ whenα>2.

Now to see (iii) consider, using additive notation for line bundles onB, the surjective morphismh₂ : Pic⁰B → Pic⁰B defined by h₂(P) = −2P. Hence, given any N ∈ Pic^dB, we deduce a surjective morphism ψN :=

φ_N ◦h₂: Pic⁰B→Pic^dB. ThereforeH¹(N−2P) = 0for anyP ∈ A_N :=

ψ_N⁻¹(Vd). Now by Proposition 3.6(v), we have H ∼= OS(aC)⊗π^∗MH for someM_H∈Pic^bB, whenceH−2DP ∼=OS((a−2α)C)⊗π^∗(M_H−2M−2P).

Sincea−2α>1, Lemma 3.5(i) and the Leray spectral sequence imply that H¹(S, H−2D_P)∼=H¹(B, π_∗(H−2D_P)). By Lemma 3.5(i) and (ii) we have π∗(H−2DP) = MH−2M −2P when a−2α= 1and π∗(H−2DP) = (M_H−2M −2P)⊕^a−2αL

i=2

(M_H−2M −iL −2P)whena−2α>2. Hence, to prove (iii), we can chooseP ∈ UH,D:=AMH−2M whena−2α= 1 and P ∈ UH,D:=AM_H−2M ∩

a−2α

T

i=2

AM_H−2M−iL whena−2α>2.

To prove (iv) note that H¹(D−f) = 0 by (i). LetF be any fiber of π.

We will be done if we prove that|D_|F| is base-point free. Now this follows by [8, Prop.2.3,I] since we haveα=D.F >2 = 2pa(F).

Similarly to see (v) note that, for any fiberF, we haveH¹(D−F) = 0 by (i) and|D−F|is base-point free by (iv). Let x, y∈S be two distinct points. Ifxandy belong to the same fiber F, we can separate them with sections in|D|since|D_|F|is very ample by [8, Thm.3.1] (because we have α=D.F >3 = 2pa(F)+1). Ifxandybelong to two different fibersFxand F_yrespectively, then to separate them just use the fact that|D−Fx|is base- point free. On the other hand suppose thatx∈S,y∈TxSanddϕD(y) = 0, wheredϕ_D is the differential of the morphismϕ_D:S→PH⁰(D). Arguing as above we deduce thaty must be tangent to Fx, contradicting the fact that|D_|F_x|is very ample.

Finally (vi) is a consequence of the fact that α = D.f and β−αn =

D.C.

Lemma 3.10. — Letπ:S →Bbe a smooth Weierstrass fibration with sectionC, general fiberf and fundamental line bundleLwithn= degL>

1andg=g(B). LetD₀∈PicSsuch that eitherD₀≡3C+βfwithβ >3n ifg= 0orD0≡2C+βf withβ>2n+ 2gifg>1. Then a general curve

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D ∈ |D0| is smooth irreducible and nonhyperelliptic. Moreover, if g >1, thenDis nontrigonal.

Proof. — By Lemma 3.9(iv) we know that|D0|is base-point free, whence Dis smooth and irreducible by Bertini’s theorems.

First consider the case g = 0. Note that f_|D gives a g¹₃ on D. If D is hyperelliptic thenD has a morphism D→P¹×P¹ which is (necessarily) birational onto its imageD. Therefore we get the contradiction

466n−263β−3n−2 =g(D)6pa(D) = 2.

Next consider the caseg >1, so that D₀² = 4β −4n>12, with equality only ifn=g= 1andβ= 4.

If equality holds andDis trigonal, using the2 : 1morphismπ_|D:D→B, we get a morphismD →B×P¹ which is (necessarily) birational onto its imageD. But this gives the contradiction8 =g(D)6pa(D) = 4.

To deal with the remaining cases, let A be a base-point free g_k¹ on D, withk= 2,3. By the above, we know that it cannot be k= 3, n=g = 1 andβ= 4.

Let F = Ker{H⁰(A)⊗ OS →A} and define E =F^∗. As is well known ([33]),E is a rank two vector bundle sitting in an exact sequence

(3.1) 0−→H⁰(A)^∗⊗ O_S −→ E −→N_D/S⊗A⁻¹−→0

and moreoverc₁(E) =Dandc₂(E) =k, so that∆(E) :=c₁(E)²−4c2(E) = D²−4k > 0. Therefore E is Bogomolov unstable ([33]), so that, if M is the maximal destabilizing subbundle (with respect to some fixed ample line bundleH onS), we have an exact sequence

(3.2) 0−→M −→ E −→ J_Z/S⊗L−→0

whereLis another line bundle onSandZ is a zero-dimensional subscheme ofS.

We now claim that these line bundles satisfy:

(i) D∼M +L;

(ii) k=M.L+ length(Z)>M.L>L²>0;

(iii) there exists an effective divisorZ₁onCof degreeM.L+L²−k>0 such thatA∼=L_|D(−Z1);

(iv) Lis base-component free and nontrivial;

(v) ifL²= 0then M.L=kandA∼=L_|D.

To see this claim note that computing Chern classes in (3.2) we get (i) and the equality in (ii). Since the destabilizing condition reads(M−L).H>0 and since(M−L)²= ∆(E) + 4 length(Z)>0, we see thatM−Lbelongs to the closure of the positive cone ofS.

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We want to prove that E is globally generated off a finite set.

To this end observe that we just need to prove thath⁰(N_D/S⊗A⁻¹)>

2g+ 1 = 2h¹(O_S) + 1, since then the mapψ:H⁰(E)→H⁰(N_D/S⊗A⁻¹) in (3.1) is nonzero and this gives thatE is globally generated off a finite set. Now the exact sequence

0−→ O_S −→ O_S(D)−→N_D/S−→0

shows thath⁰(N_D/S⊗A⁻¹)>h⁰(N_D/S)−k=g−1 +h⁰(OS(D))−k, since H¹(OS(D)) = 0by Lemma 3.9(i). Using Lemma 3.5 we findh⁰(OS(D)) = 2β−2n−2g+ 2, so that the desired inequality is satisfied andEis globally generated off a finite set.

Since E is globally generated off a finite set then so isL. It follows that L>0,L is base-component free and L² >0. Now the signature theorem [4, VIII.1] implies that (M −L).L >0 thus proving (ii). To see (iii) and (iv) note that ifM.L >0then the nefness ofLimplies thatH⁰(−M) = 0.

On the other hand if M.L = 0 then L² = D.L = 0, whence L ≡ 0 by the Hodge index theorem and thereforeD ≡M. ThenM.H =D.H >0, whence again H⁰(−M) = 0. Twisting (3.1) and (3.2) by −M we deduce thath⁰(L_|D⊗A⁻¹)>h⁰(E(−M))>1. This proves (iii) and (iv). Moreover it givesdeg(L_|D⊗A⁻¹)>0, whence, ifL²= 0, we get thatM.L>k. By (ii) it follows that M.L = k and therefore deg(L_|D⊗A⁻¹) = 0, whence L_|D∼=Aand also (v) is proved.

By the Hodge index theorem we now have

(3.3) L²(D²−4k)6L²(M−L)²6(L.(M−L))²= (M.L−L²)² and it is easily seen that (3.3) givesL²61and thatL²= 1holds precisely whenk= 3, g= 1 and eithern= 2, β= 6or n= 1, β = 5(recall that we have excluded the casek = 3, n=g = 1 and β = 4). Moreover, in both cases above withL²= 1, we have equality in (3.3), whenceM ≡3L and D ≡ 4L by (i) above. On the other hand, in both cases, 2 = f.D is not divisible by4.

Therefore L²= 0, M.L=k andA∼=L_|D by (v) above. HenceL.D=k and, sinceLis nef (by (iv) above), we must haveL.f = 0, whenceL.C= 1 and k = 2. But (iv) above also gives that L is effective, whence L ≡ f. From the exact sequence

0−→L−D−→L−→L_|D−→0

and Lemma 3.9(i) we get thath⁰(D, L_|D) =h⁰(S, L). By Proposition 3.6(v) and Lemma 3.5(i) we also know thath⁰(S, L) = h⁰(B,M) for some line

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bundleMof degree1onB. Sinceg>1we get the contradiction 2 =h⁰(A) =h⁰(D, L_|D) =h⁰(S, L) =h⁰(B,M)61.

4. Extending the morphism to the threefold

An elliptic surface has a surjective morphism onto a smooth curve with fibers elliptic curves. The goal of this section will be to give some sufficient conditions, both on S and on the singularities of X, to insure that this morphism extends to a threefold containing the elliptic surface as an ample divisor.

Extendability results for morphisms abound in the literature. We repro- duce here the one of [5, Thm.5.2.1] in a form that will be convenient for us.

Proposition 4.1. — Let X be a projective irreducible threefold with Cohen-Macaulay singularities, letLbe an ample line bundle onX and let A ∈ |L| be a normal divisor. Suppose that the restriction map PicX → PicAis an isomorphism and that there is a surjective morphismp:A→Y onto a projective varietyY such thatdimY 6dimA−1.

If there exists a very ample line bundle Lon Y such thatH¹(A, p^∗L − tL_|A) = 0 for everyt>1, thenpextends to a morphismp:X →Y.

Proof. — By hypothesis there is a line bundle H ∈ PicX such that H_|A ∼= p^∗L. We claim that the natural restriction map H⁰(X,H) → H⁰(A,H_|A)is surjective.

To this end it is of course enough to prove that H¹(H −L) = 0. Now, for eacht>1, we have an exact sequence

0−→ H −(t+ 1)L−→ H −tL−→p^∗L −tL_|A−→0,

whence, by hypothesis, we have that h¹(H −tL)6 h¹(H −(t+ 1)L) for everyt>1. Letω_X⁰ be a dualizing sheaf forX. Sinceh¹(H−jL) =h²(ω⁰_X⊗ (−H+jL)) = 0for largej by Serre vanishing, we get thath¹(H −tL) = 0 for everyt>1.

such thatD₁∩. . .∩D_m+1∩A=∅. Since Ais ample we have that either D1∩. . .∩Dm+1=∅ordimD1∩. . .∩Dm+1= 0. But in the latter case we get

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the contradiction0 = dimD₁∩. . .∩Dm+1>dimX−m−1 = dimA−m>1.

Therefore D1∩. . .∩Dm+1 = ∅ and |H| is base-point free and defines a morphismp:X →p(X)⊂PH⁰(H) such thatp_|A =p. Let us show that p(X) =Y. Of course we haveY =p(A) =p(A)⊆p(X). On the other hand if there existsx0∈Xsuch thatp(x0)6∈Y thenA∩p⁻¹(p(x0)) =∅, whence, asAis ample, we getdimp⁻¹(p(x0)) = 0, and thereforedimp⁻¹(p(x)) = 0 for a generalx∈X. Hencedimp(X) = dimX. Now D1∩. . .∩Dm+1=∅ implies that there are hyperplanesH_iinPH⁰(H)such thatH₁∩. . .∩Hm+1∩ p(X) = ∅, whence dimX = dimp(X) 6m 6 dimA−1 6dimX−2, a

contradiction.

Here is an effective way to apply this to elliptic surfaces.

Corollary 4.2. — Let X be a projective irreducible threefold with Cohen-Macaulay singularities, let L be a very ample line bundle on X and let S ∈ |L| be a smooth surface. Suppose that the restriction map PicX →PicSis an isomorphism and thatπ:S→Bis an elliptic fibration.

Suppose furthermore that H¹(S, KS +L_|S −f1−. . .−f_d(B)) = 0 for every set of distinct smooth fibersf_i’s.

Then πextends to a morphismπ:X →B.

Proof. — Setd=d(B)andF₁= 0,F_s=f₁+. . .+f_s−1for26s6d+1.

For eacht>1 andusuch that16u6dwe have exact sequences (4.1)

0−→KS+tL_|S−Fu+1−→KS+tL_|S−Fu−→(KS+tL_|S−Fu)_|f_u −→0.

We first claim thatH¹(K_S+L_|S−F_s+1) = 0for06s6d. Since the latter is true by hypothesis whend=s, we proceed by induction ond−s. Now whend−s>1 we have thatH¹(K_S +L_|S−F_s+2) = 0 by the inductive hypothesis andH¹((KS+L_|S−Fs+1)_|f_s+1) = 0, whence (4.1) with t= 1 andu=s+ 1 gives the claim.

Now let W = Im{H⁰(S, L_|S)→ H⁰(f_s, L_|f_s)}. Then W is very ample, whence dimW > 3. Hence we deduce, by [18, Thm.4.e.1], that for any N∈PicBand for anyt>1, the multiplication mapsµ_t,s:W⊗H⁰((K_S+ tL_|S+π^∗N)_|f_s)→H⁰((KS+ (t+ 1)L_|S+π^∗N)_|f_s)are all surjective.

Fort>1and16s6dconsider the restriction maps

ϕt,s:H⁰(KS+tL_|S−Fs)→H⁰((KS+tL_|S−Fs)_|f_s).

We want to prove, by induction ont, that they are all surjective. Fort= 1 this follows from (4.1) with u= s and the claim proved above. Now the

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commutative diagram

H⁰(KS+ (t−1)L_|S−Fs)⊗H⁰(L_|S) //

ϕt−1,s⊗r_fs

H⁰(KS+tL_|S−Fs)

ϕt,s

H⁰((KS+ (t−1)L_|S−Fs)_|f_s)⊗W ^µ^t,s //H⁰((KS+tL_|S−Fs)_|f_s) shows, by induction ont, thatϕt,s is surjective for everyt>1.

We now claim that H¹(KS+tL_|S−Fs+1) = 0fort>1and 06s6d.

Since the latter is true by Kodaira vanishing whens = 0, we proceed by induction ons. Now whens>1 we have thatH¹(KS+tL_|S−Fs) = 0by the inductive hypothesis, whence (4.1) with u=s and the surjectivity of ϕt,s gives the claim.

By definition ofdthere is a very ample line bundleLof degreedonBand we can obviously writeπ^∗L ∼f1+. . .+fd. ThereforeH¹(S, π^∗L−tL_|S) = 0 for everyt>1 by Serre duality and the last claim. Henceπ extends to a

morphismπ:X →Y by Proposition 4.1.

Remark 4.3. — The vanishing condition in the above corollary is satisfied in the case of smooth Weierstrass fibrations withn>1and in several other cases, even when N¹(S)∼= Z[C]⊕Z[f] with C.f >2. Suppose for example thatπ:S →B is a smooth Weierstrass fibration with sectionC, general fiber f and fundamental line bundle L with n = degL > 1 and g = g(B). Suppose furthermore that ρ(S) = 2. By Remark 3.8 we have L_|S ≡aC+bf. AlsoL_|Cis very ample onC∼=B, therefored(B)6L.C= b−an. ThenK_S+L_|S−f₁−. . .−f_d(B)≡aC+ (b+n+ 2g−2−d(B))f. Hence b+n+ 2g−2−d(B)>n+ 2g−2 +an>an+ 2g−1, therefore H¹(S, KS+L_|S−f1−. . .−f_d(B)) = 0by Lemma 3.9(i). The vanishing can be easily proved also in other cases, for example in the cases in 6.1, 6.2, 6.3, 6.4 (either withB∼=P¹ or withE sufficiently ample).

To apply Corollary 4.2 to elliptic surfaces we need to verify the hypothesis on the restriction map of Picard groups. It is here that the hypothesis on the rank becomes important.

Proposition 4.4. — LetX be a projective irreducible l.c.i. threefold, letL be an ample line bundle on X and let S ∈ |L| be a smooth surface withρ(S) = 2andκ(S) = 1. Then the restriction mapsPicX →PicS and N¹(X)→N¹(S)are isomorphisms.

Proof. — Since S is smooth we have that dim Sing(X) 6 0 and X is normal, whencer_S : PicX→PicSis injective with torsion free cokernel by Lefschetz’s theorem [5, Cor.2.3.4]. Moreover the adjunction formulaKS =

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(K_X+L)_|S holds (see for example [1, Prop.2.3 and 2.4]). Sinceκ(S) = 1we have thatL_|S and K_X|S are numerically independent and sinceρ(S) = 2 we get that for any line bundle A ∈ PicS there are integers a, u, v with a>1 such thataA≡uL_|S+vK_X|S. Therefore we can writeaA∼uL_|S+ vKX|S +D with D ≡ 0. By [25, Thm.4.6] there is an integer m > 1 such thatmD∈Pic⁰S and by [5, Thm.2.3.1 and Thm.2.2.4] we have that Pic⁰X →Pic⁰S is an isomorphism, whence mD ∈ ImrS. Therefore also maA∈Imr_S, whenceA∈Imr_S, sinceCokerr_S is torsion free.

The mapN¹(X)→N¹(S)is now clearly surjective. To see its injectivity letM ∈ PicX such that M_|S ≡ 0. As above there is an integer m > 1 and a line bundleN ∈Pic⁰X such thatmM_|S ∼=N_|S, whencemM ∼=N,

thereforeM ≡0.

The previous two results, together with some facts already present in the literature, allow us to give our best version of an extension theorem for elliptic fibrations.

Theorem 4.5. — LetX be a projective irreducible threefold, letLbe an ample line bundle on X and let S ∈ |L| be a smooth surface having an elliptic fibrationπ:S →B with general fiber f. Thenπextends to a morphismπ:X →B if one of the following is satisfied:

(i) X is l.c.i. with rational singularities andg(B)>0.

(ii) Xis l.c.i. withQ-factorial terminal singularities, not a cone overS, Lis very ample, B ∼=P¹, κ(S) = 1and N¹(S)∼= Z[C]⊕Z[f] for some divisorCsuch thatC.f >1.

(iii) X is l.c.i., L is very ample, ρ(S) = 2, κ(S) = 1 and H¹(S, KS + L_|S −f₁−. . .−f_d(B)) = 0for every set of smooth distinct fibers fi’s.

(iv) X is smooth,B∼=P¹,N¹(S)∼=Z[C]⊕Z[f]for some divisorC and L_|S ≡aC+bf for anya, bwithb > ^aC.f₂ + 1 + _2aC.f¹ −_2C.f^aC². Proof. — Observe that (i) follows by [5, Thm.5.2.3] since, as in the beginning of the proof of Proposition 4.4, X is normal. Now (iii) is a consequence of Corollary 4.2 and Proposition 4.4. To see (iv) note that aC.f =L_|S.f >1. Therefore the assumed inequality onb is equivalent to (f.L_|S+ 1)²< L²_|S, whenceπextends by [45, Thm.1.4].

To prove (ii) we use first some adjunction theory to show that KX+L is nef.

By Proposition 4.4 we have that PicX →PicS and N¹(X)→ N¹(S) are isomorphisms, so that ρ(X) = 2. Since K_S ≡ ef for some e >1, we have thatXdoes not admit a surjective morphism onto a varietyY in such

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a way that a general fiberF intersects S in a curve γ withp_a(γ) = 0, for otherwise we would have that−2 =γ.(γ+KS) =eγ.f >0.

Now suppose thatK_X+Lis not nef, so thatK_Xis not nef and letτbe the nefvalue of(X, L)(see Definition 2.1). Thenτ >1and by Proposition 2.2 we deduce thatτ63and thatKX+ 3Lis nef and big. By Theorem 2.3 it follows thatKX+ 3Lis ample and thatτ 62.

To go further note that we cannot have KX ∼ −2L, for otherwiseKS ∼

−L_|S, giving the contradictionκ(S) =−∞. Furthermore let us show that X cannot have a surjective morphism ψ : X → Y with connected fibers onto a varietyY of dimensionm= 1,2in such a way thatK_X+ 2L∼ψ^∗L, for some ample line bundleL ∈PicY such thatψis defined by the linear system|q(KX+ 2L)|forq >>0.

Thenq(KX+ 2L)_|F_η∼0and by [5, Lemma3.3.2] we get(KX+ 2L)_|F_η∼0, whenceK_F_η+ 2L_|F_η ∼0 and therefore (F_η, L_|F_η)∼= (P¹,O_P1(1)). Now let F be any fiber ofψ. ThenF is a line and ifF ⊂Swe get the contradiction 06F.KS =F.(KX+L)_|S =F.(KX+L) =−L.F <0. Now consider the Fano varietyF(X)of lines contained inX, which is at least2-dimensional as the fibers ofψ are lines. LetΣY ⊆ F(X) be the irreducible subvariety of dimension two defined by the fibers of ψ. We have proved that the hyperplaneH ⊂P^N⁺¹=PH⁰(X, L)defining S does not contain any line in Σ_Y. As is well known, this implies that there is a point P ∈ P^N⁺¹ belonging to all lines inΣY, therefore X is a cone with vertexP. On the other hand, asS is smooth, we get that P 6∈S, thereforeX is a cone over S⊂P^N, a contradiction.

Now by Theorem 2.4 we have that either KX+ 2L is ample or there exists a surjective birational morphism φ : X → X⁰ onto a normal projective varietyX⁰ with the following two properties: a) φis the simultaneous contraction to distinct smooth points p⁰_i ∈ X⁰ of some divisors E_i ⊂ X−Sing(X)such thatEi ∼=P², OE_i(Ei)∼=O_P2(−1) andL_|E_i ∼=O_P2(1);

b) IfL⁰:= (φ_∗L)^∗∗ thenKX⁰+ 2L⁰ is ample.

We now claim that if such aφexists then it is an isomorphism.

As a matter of fact suppose thatφis not an isomorphism, so that there is at least one contracted divisorE_i. LetS⁰ =φ(S)∈ |L⁰|. AsS∩Sing(X) =∅ we have that S⁰ is certainly smooth outside all p⁰_i’s. On the other hand E_i²_|S =E_i.E_i.L=E_i|E_i.L_|E_i=−1, so thatp⁰_i is also a smooth point ofS⁰.

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ThereforeS⁰ is birational, but not isomorphic toS, whence κ(S⁰) = 1 and thereforeρ(S⁰)>2. But this gives the contradiction2 =ρ(S)>ρ(S⁰)+1>3.

Hence, in any case, K_X + 2L is ample, whence 1 < τ < 2 by [5, Lemma1.5.5]. In this context the first reduction(X,e L)e of(X, L)is defined (see Section 2) and the above discussion shows that in fact(X, L)∼= (X,e L).e Finally by Theorem 2.5 we get thatX has a surjective morphism onto a variety Y in such a way that a general fiber F intersects S in a curve γ withp_a(γ) = 0, case already excluded.

This proves that KX+L is nef and the base-point free theorem ([30, Thm.3.3]) gives that forq >>0 the line bundle q(KX+L) is base-point free. By [5, Lemma1.1.3] there exists a surjective morphism p : X → Y with connected fibers onto a normal projective variety Y in such a way thatK_X+L∼p^∗L, for some ample line bundleL ∈PicY andpis defined by the linear system|q(KX+L)|forq >>0.

We now claim thatY is a smooth curve. In fact letF be a general fiber of p, so that q(KX +L)_|F ∼ 0, whence, as above, −KF ∼ L_|F by [5, Lemma3.3.2]. If Y is a point we find that −KX ∼ L, so thatKS ∼ 0, a contradiction. On the other hand we cannot havedimY >2, for otherwise picking two general divisorsD1, D2∈ |q(KX+L)|we havedimD1∩D2= 1, giving the contradiction

0 =q²K_S² =q²(KX+L)².L=D1.D2.L >0.

ThereforeY is a smooth curve and the general fiber F of pis a smooth connected surface. The restriction p_|S : S → Y is clearly surjective and a general fiber f = F ∩S is therefore a smooth connected curve with f ∈ | −KF|, that is an elliptic curve. Thereforep_|S :S →Y is an elliptic fibration and, as the latter is unique, we deduce thatY =B, p_|S =π and

thatπextends toX, as required.

Remark 4.6. — The proof of the extension ofπunder hypothesis (ii) is inspired by [32]. In case X is smooth it has been proved first by Ionescu [21] (see also [12], [14]).

The above theorem gives that most threefolds studied in this article, provided that they have the appropriate singularities, are Mori fiber spaces.

Proposition 4.7. — LetX be a projective irreducibleQ-factorial ter- minal l.c.i. threefold, letLbe a very ample line bundle onX and letS∈ |L|

be a smooth surface having an elliptic fibrationπ : S → B with general fiberf. Suppose that κ(S) = 1, N¹(S)∼=Z[C]⊕Z[f] for some divisor C such thatC.f >1and eitherg(B)>0orB∼=P¹andX is not a cone over S. Thenπextends to a morphismπ:X→B andX is a Mori fiber space.

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Proof. — Since terminal singularities are rational [13, Thm.1], by The- orem 4.5π extends to π : X → B with general fiber F and by Proposi- tion 4.4 we have thatN¹(X)∼=Z[E]⊕Z[F]for a divisorEonX such that C=E_|S. Hence we can writeKX≡uE+vF andL≡aE+bF, for some integers u, v, aand b. Since L is very ample we get0 < L_|S.f =aC.f = aE_|S.F_|S =aE.F.(aE+bF) =a²E².F, so thata6= 0 andE².F 6= 0. Now 0 =KS.f = ((u+a)E+ (v+b)F).F.L=a(u+a)E².F implies thatu=−a and therefore−KX|F ≡aE_|F ≡ L_|F is ample, whence X is a Mori fiber

space.

We now study the fibers of the extended morphism.

Theorem 4.8. — Let S ⊂P^N be a smooth surface having an elliptic fibrationπ:S →Bwith general fiberf. LetX ⊂P^N⁺¹be a l.c.i. extension ofSgiven by a very ample divisorLonX. Suppose thatπ:S→Bextends to a morphismπ:X→B and thatN¹(S)∼=Z[C]⊕Z[f]for some divisor Csuch thatC.f >1. SetL_|S ≡aC+bf. Then

(4.2) (a, C.f)∈ {(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(3,3)}

and for the general fiberF ofπ, we haveL_|F ∼=−KF and settingd=K_F², F is one of the following:

(i) F∼=P². (ii) F∼=P¹×P¹.

(iii) F is isomorphic to the blow-up of 9−d distinct points in P² (no three collinear, no six on a conic) for36d68.

Moreover if, in addition,X is locally factorial, then36d66 in (iii) and (a, C.f)6∈ {(1,7),(1,8),(1,9)} in(4.2).

Proof. — We have F_|F ≡ 0, whence ρ(X) > 2. On the other hand, by [5, Cor.2.3.4], PicX → PicS is injective with torsion free cokernel.

Therefore also N¹(X) → N¹(S) is injective with torsion free cokernel, whence N¹(X)→N¹(S)is an isomorphism, since ρ(S) = 2. Let[E] and [F] be the generators ofN¹(X), restricting respectively to[C] and[f] on S, so thatL≡aE+bF andaC.f =L_|S.f >3, givinga>1.

We now claim that a generalF ⊂P^N⁺¹ is a smooth Del Pezzo surface, that isOF(1)∼=OF(−KF).

A generalFis certainly smooth by Bertini’s theorem. MoreoverF∩S=f is a smooth irreducible elliptic curve, whence also F is irreducible and it follows from [5, Thm.8.9.3] thatF ⊂P^N+1embedded byL_|F is either a Del Pezzo surface or (F, L_|F)∼= (PE, ξ), where E is a rank two vector bundle on an elliptic curve andξis the tautological line bundle.

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Suppose we are in the latter case and let γ be a fiber of PE, so that KF ≡ −2ξ+eγ for some e ∈ Z. We have ξ = L_|F ≡ aE_|F, whence 1 =ξ.γ = aE_|F.γ givesa = 1 and ξ ≡ E_|F. Note that X is Gorenstein, whenceKX is a line bundle and we can writeKX≡uE+vF. Now

0 =KS.f = (KX+L)_|S.f = (u+ 1)C.f

therefore u = −1, giving the contradiction −2 = KF.γ = (KX)_|F.γ =

−E_|F.γ=−ξ.γ=−1.

This proves the claim and henceforth F is a smooth Del Pezzo surface with−KF ∼ L_|F ≡aE_|F. Now d =K_F² =L²_|F =L².F = L_|S.f =aC.f and by [43, Thm.8] we have that36d69.

By [43, Thm.8] we deduce that eithera= 3andF ∼=P², so thatC.f = 3 or a= 2 and F ∼=P¹×P¹, so thatC.f = 4or a= 1 and 3 6C.f 69.

Thus (4.2) is proved.

Moreover, again by [43, Thm.8], when a= 1, we have that eitherL_|F is divisible, leading toC.f = 9,F ∼=P² andC.f = 8,F ∼=P¹×P¹ or L_|F is not divisible.

Suppose we are in the latter case. By [43, Thm.8] we get 36d68 and Fis the anticanonical embedding of the blow-up ofP²in9−dpoints. Note that, asF is smooth, the blown-up points cannot be infinitely near, there is no line containing three of them and there is no conic containing six of them.

This proves thatF is as in (i), (ii) or (iii).

For the rest of the proof suppose that X is also locally factorial.

IfF is as in (iii), there is a nonempty open subsetU ⊂B such that each fiber overU is as in (iii). This gives rise to an irreducible curveB⁰⊂ F(X), the Fano variety of lines contained inX. LetT be the union of all the lines in B⁰. By [19, Exa.6.19 and Prop.6.13] we have that T is a Weil divisor onX and by [20, Prop.II.6.11] it follows that T is also a Cartier divisor onX. HenceT ≡rE+sF for some integersrands. Therefore, using the convention that ^m_n

= 0ifm < n, we have 9−d+

9−d 2

+

9−d 5

= degT_|F =L_|F.T_|F =L.T.F =T_|S.F_|S

= (rC+sf).f =rd, giving36d66.

Finally we exclude the cases (a, C.f) ∈ {(1,7),(1,8),(1,9)} in (4.2).

If (a, C.f) = (1,7) we have d = aC.f = 7, contradicting what we have just proved. Now assume that either (a, C.f) = (1,9), so that F ∼= P² or (a, C.f) = (1,8) and F ∼= P¹×P¹. Let k(B) be the quotient field of

(22)

B and k(B) its algebraic closure, so that the base changes of X, X_k(B) to k(B) and X_k(B) to k(B) are defined. Now either X_k(B) ∼= P²_k(B) or X_k(B)∼=P¹_k(B)×P¹_k(B), and, as in [41, Proof of 3.5.2, page 162], we have PicXk(B)∼= (PicX_k(B))^G, whereG= Gal(k(B)/k(B))is the Galois group.

This implies that the canonical bundle KX_k(B) is r divisible for r = 3 in the caseF ∼=P² andr= 2in the caseF ∼=P¹×P¹. Therefore we can find a nonempty open subsetU ⊂B and a line bundle Lonπ⁻¹(U)such that L_|F_u ∼=O_P²(1)ifF_u∼=P²andL_|F_u ∼=O_P¹_×P¹(1,1)ifF_u∼=P¹×P¹on every fiberF_u overU. Since X is locally factorial,L extends to a line bundleL having the same restriction property as L on a general fiberF. But now L ≡αE+βF, for some α, β∈Zand therefore

L_|F ≡ L_|F ≡αE_|F ≡αL_|F ≡ −αKF

which easily gives a contradiction.

With this baggage of results we are now ready to prove our main theorems.

Proof of Theorem 1.3. — Immediate consequence of Theorems 4.5 and

4.8.

Proof of Corollary 1.4. — If (g, n) 6= (0,2) the Corollary follows from Proposition 3.6(iv), Remark 3.8, Theorem 1.3(iii) and Remark 4.3. Now suppose that(g, n) = (0,2), so thatSis aK3surface by Proposition 3.6(iii).

In particular any line bundle numerically equivalent to0onS isO_S. Since S is smooth we have that X is normal and dim Sing(X) 6 0, whence rS : PicX → PicS ∼=N¹(S) is injective with torsion free cokernel by [5, Cor.2.3.4]. LetA∈PicX be such thatA≡0. ThenA_|S ≡0, whenceA_|S ∼ 0and thereforeA∼0by the injectivity ofrS. Hence alsoPicX∼=N¹(X).

Thereforeρ(X)62. Now ifρ(X) = 2 then necessarily PicX =N¹(X)∼= N¹(S) = PicS, therefore the morphismπ:S→B extends to a morphism π:X →B by Corollary 4.2 and Remark 4.3. Now we get a contradiction by Theorem 4.8.

Henceρ(X) = 1. By [5, Thm.5.3.1] we also get that−KX ∈ |OX(1)|and

thath¹(OX) =h²(OX) = 0.

Remark 4.9. — In the case (ii) of Corollary 1.4, if X is smooth, the surface S must belong to some proper closed subset of the linear system

|L|: In fact, for a general S⁰ ∈ |L| we have1 = h²(OS⁰) > h²(OX) = 0, whencePicX ∼= PicS⁰ by a theorem of Moishezon [40, Thm.7.5].

We now consider the extendability ofK3 Weierstrass fibrations.