On threefolds admitting a bielliptic curve as abstract complete intersection

(de Gruyter 2001

On threefolds admitting a bielliptic curve as abstract complete intersection

A. Del Centina and A. Gimigliano*

(Communicated by A. Sommese)

Abstract.We study smooth projective varietiesXJP^Nof dimension 3, such that there are two very ample invertible sheavesL,MonX, and there exist two sections ofL,Mwhich intersect along a bielliptic curveC. We give a classi®cation of such threefoldsXunder some hypotheses on the degree ofCwith respect to the two embeddings given byL,M.

Key words.Threefolds, polarizations, bielliptic curves, special varieties.

2000 Mathematics Subject Classi®cation. 14J30, 14C20

Introduction

The question of classifying projective varieties which possess hyperplane sections with special properties is a classical one in Algebraic Geometry (e.g. see [7], [11], [21], [13]).

In particular a problem that has been widely studied also in recent times is that of varieties with hyperelliptic, bielliptic or trigonal curve-sections (e.g. see [25], [6], [5], [12], [22], [4], [2], [9], [10]).

A natural generalization of this kind of problem is to classify projective varieties having particular curves C as intersection of sections of di¨erent very ample line bundles, according to the following de®nition:

De®nition 1.LetXbe a smooth, irreducible scheme of dimensiond, de®ned over an algebraically closed ®eldk of characteristic zero. LetL1;. . .;Lr be very ample line bundles on X. We say that a subscheme VJX, of dimension dÿr, is an abstract complete intersection of L1;. . .;Lr in X, ab.c.i. for short, if IVHOX is globally generated byrsectionsA1AH⁰…X;L1†;. . .;ArAH⁰…X;Lr†.

*Both authors have been partially supported by MURST in the framework of the National Project ``Geometria Algebrica, Algebra Commutativa ed Aspetti Computazionali''; for the second author: ``Lavoro svolto con il ®nanziamento dell'Univ. di Bologna. Finanziamento Speciale alle Strutture''.

A classi®cation of the possibilities forXwhend ˆ3 andCis a smooth hyperelliptic curve is given in [8], which is what inspired us for the present paper.

To be precise, in this paper we consider the case of triples…X;L;M†such that:

…† Xis a smooth irreducible scheme withdimX ˆ3,LandMare two very ample line bundles such that there is an irreducible, smooth, bielliptic curve CHX which is an ab.c.i. inXof two smooth, irreducible sectionsAAjMjandBAjLj.

We will always assume that M0L, in fact when MˆL we have that C is a curve-section of X (in the embedding given by L) and this case has already been studied in [10]. Moreover, in view of [8], we assume that C is not hyperelliptic (hence, in particular, we assume that for the genusg…C†ofCwe haveg…C†>2).

In order to introduce our results and to give some examples of the varieties we are concerned with, let us introduce some notation. LetFV denote the restriction of a sheaf F on Xto a subscheme VJX. We de®ne dAˆL_A², dBˆM_B², of course we have alsodAˆML² anddBˆLM². Without loss of generality we can always supposedAddB.

Then a ®rst example of this kind of varieties is o¨ered by:

Example 1. Let XGP³ and consider a (canonical) bielliptic plane quartic curve CHHof genus 3, whereHis a plane inP³. Of courseCis the complete intersection ofHand a quartic surface, hence if we putLˆO…4†,MˆO…1†we are in the situ- ation of…†, and heredAˆ16,dBˆ4.

We will be able to describe the triples…X;L;M†as in…†when either dAd18 or dBc8; see the statements of Theorems A, B and C.

Notice that ifdAd19 (Theorem A) thenXis a ®bration over a curve (either elliptic or bielliptic); this fact allows us to extend our classi®cation to the casedimXd4, see the statement of Theorem A⁰.

The case d_Aˆ18 described in Theorem B seems to be a threshold, in fact for d_Ac18 more kinds of varieties satisfying the condition in…†do appear.

Example 2. Let XGP³ and consider a (canonical) bielliptic curve C of genus 4 which is the complete intersection of a cone Lover a plane (smooth) cubic curve and a quadric not passing through the vertex ofL. Hence ifLˆO…3†,MˆO…2†we have a situation as in…†withd_Aˆ18,d_Bˆ12.

We remark that for dAˆ18, we cannot a½rm that all the varieties listed in Theorem B actually possess a curveCas in…†. On the other hand, we can see that there are examples of threefolds as in…†with 9cdBcdAc17:

Example 3. Let p:X!P³ be the blowing up of P³ at a point P, then PicXGZhH;Ei, where H is the strict transform of a generic plane in P³ (not through P) and E is the exceptional divisor. We have that MˆOX…2HÿE† and LˆOX…3HÿE†are very ample onXand we can choose sectionsA;Bof them (see Example 2) such that their intersection is a bielliptic curve C of genus 4. In this case we havedBˆ11,dAˆ17.

Example 4. Let XJP⁶ be a double covering p:X!Y of the rational normal threefoldYGP¹P²!P⁵, rami®ed along a divisor of typeOY…2;2†(Xis a Fano threefold with PicXGZ², see e.g. [19]). Then degX ˆ6 and X can be viewed as obtained by taking a cone over Y from a point in P⁶ and intersecting it with a quadric not passing through its vertex. LetMGp…O_Y…1;1††andLGp…O_Y…1;2††.

We have OY…a;b† O_Y²…1;1† ˆa‡2b, hence OY…1;2† O_Y²…1;1† ˆ5 and the generic intersection OY…1;2† OY…1;1† is an elliptic normal curve G5 in P⁴. Our curve C, an ab.c.i. of M, L, will be a double covering of G5, hence it will be a (canonical) bielliptic curve inP⁵.

Here we havedBˆ10 anddAˆ16 (sinceOY…a;b†²OY…1;1† ˆb²‡2ab).

The main tool we will use in the paper is adjunction theory, via the classi®cation of varieties of small degree in [16], [17] and [18], the results in [20] and those in [9] about surfaces with bielliptic curve sections, also taking into account the new results in [1].

In the followingGwill denote isomorphisms, while@will denote linear equivalence of divisors. For the notation not de®ned in the paper we refer to [15].

We would like to thank the referee for the substantial help in correcting mistakes and imperfections in the ®rst draft of the paper.

1 Preliminaries

Let us recall some useful results about bielliptic curves. The ®rst lemma will give us a bound for the degree of an embedded bielliptic curve (for a reference see [9], 1.5 and 1.6).

Lemma 1.1.Let C be a bielliptic curve of genus gd3which is birational to some non- degenerate curve of degree d in Pⁿ. Then it must be ddn‡gÿ1.In particular,no bielliptic plane curve is smooth,unless gˆ3.

For bielliptic curves inP³ we have the following result.

Lemma 1.2.Let C be a smooth bielliptic curve inP³such that either:

1. C is contained in a quadric surface,or 2. C is a complete intersection.

Then C is a complete intersection of a quadric and a cubic,i.e.,a canonical curve of genus4and degree6.

Proof.Case 1. If the quadric containingCis smooth andCis a divisor of type…a;b†, then C has degree a‡b and genus …aÿ1†…bÿ1†; by Lemma 1.1, this is possible, for non-hyperelliptic curves, only if …a;b† ˆ …3;3†. If the quadric containing Cis a cone, things do not change much; there are two possible cases according to whetherC contains the vertex of the cone or not. Taking into account the degree and genus formulae (e.g. see [15], p. 352) we get a contradiction with Lemma 1.1, except in the case that the curve is the complete intersection of the cone with a cubic not passing through its vertex.

Case 2. If C is a complete intersection of two surfaces of degrees a and b, then degCˆab, and the genus ofC, from the exact sequence

0!O_P3…ÿaÿb† !O_P3…ÿa†lO_P3…ÿb† !I_C!0;

isg…C† ˆab…a‡bÿ4†=2‡1. Again by Lemma 1.1 the only possibility is thataˆ2, bˆ3.

LetXbe as in…†. By the Hodge Index Theorem we have L_A²M_A²c…LAMA†²; L_B²M_B²c…LBMB†² from which we get (onX)

…L²M†M³c…LM²†²; …M²L†L³c…L²M†² i.e.,

dAM³cd_B²; dBL³cd_A²: …1:1†

Remark.From (1.1) andd_Add_Bwe trivially haveM³cd_B. Lemma 1.3.Let…X;M;L†and C be as in…†,then d_Ad6.

Proof. By Lemma 1.1 (since g…C†d3,nd2) we have dAd4, but the case dAˆ5 cannot occur since we should havenc3 and there are no smooth curves of degree 5 and genusd3 inP³orP². IfdAˆ4, then, by Lemma 1.1 again,Cmust be a smooth plane quartic, hence A should be a quartic surface in P³ and the situation is as in Example 1:XˆP³,LˆO_P³…1†,MˆO_P³…4†. In this case we would havedBˆ16, contradicting our hypothesis that d_Add_B (of course we can have this kind of situ- ation interchanging the roles ofLandM).

Lemma 1.4. Let …X;M;L† and C be as in …† and let h⁰…X;M† ˆn‡1, i.e., Membeds X intoPⁿ,then

d_Bdn‡g…C† ÿ2dn‡1:

Proof. The second inequality is trivial since g…C†d3. For the ®rst inequality, we know thatCHP^nÿ1, becauseCˆAVBis contained in a hyperplane sectionAofX;

in order to obtain the ®rst inequality by applying Lemma 1.1, it is enough to show thatCis non-degenerate inP^nÿ1. Suppose the contrary, then alsoBis contained in a hyperplane section, i.e., we can writejAj ˆ jB‡B⁰jwhereB⁰is e¨ective, and we get

B²…B‡B⁰† ˆL²Mˆd_Add_BˆLM²ˆB…B‡B⁰†²;

hence we should haveB³‡B²B⁰dB³‡2B²B⁰‡BB⁰², i.e., 0dBB⁰…B‡B⁰† ˆABB⁰, which is impossible sinceAandBare very ample andB⁰is e¨ective.

2 The casedAdddd18 The following holds:

Theorem A.Let…X;L;M†and C be as in…†with d_Ad19.Then either:

A.1. X is a scroll over C with respect to both polarizations,i.e., X is aP² bundle over C and on every ®ber F we have…F;L_F†G…F;M_F†G…P²;O_P²…1††.

A.2. X is aP²-bundle over an elliptic curve E and for every ®ber FGP² we have M_FGO_P²…1†(i.e.,…X;M†is a scroll),whileL_FGO_P²…2†.

A.3. X is a quadric bundle over an elliptic curve E and for every ®ber FGP¹P¹ we haveM_FGL_FGO_P¹_P¹…1;1†.

Proof.By [9] we know that the possibilities for…A;LA†are the following:

1. …A;LA†is a scroll on a bielliptic curveC, 2. …A;LA†is a conic bundle on an elliptic curveE.

In case 1, from [3], Theorem 5.5.3, we have that the ®ber bundle structure onA extends to one onX(in fact the only possible cases in which this does not happen are when A is a quadric, which is not our case). In particular this gives that X is a P²-bundle over the curveC. More speci®cally, [3], Theorem 7.9.5 gives that…X;M†

is a scroll as required.

By denoting with f a ®ber ofA, we have

O_P¹…1† ˆ …L_A†_f ˆ …L_F†_f ˆO_P²…a†j_f ˆO_P¹…a†;

henceaˆ1, i.e.,…X;L†is a scroll, as required.

In case 2, we proceed very much as in [8], case 3.3; we sketch here an outline of the reasoning. The conic bundle structurep:A!E is given by the Remmert±Stein factorization off_KA‡LA, and by [24], Propositions 3.1 and 3.2, the bundleKX‡L‡M is spanned with the only possible exception (in our case) thatXGP…E†, whereEis a rank 3 vector bundle onCandL,Mare of the formx_E‡L_i,iˆ1;2, wherex_Eis the tautological line bundle and theL_i's are pull backs of line bundles onC. In this case Xis a scroll with respect to both polarizations, and we are in case A.1 of our theorem.

WhenK_X‡L‡Mis spanned,pis induced by a morphismP:X !E(given by the Remmert±Stein factorization off_KX‡L‡M). LetFbe a general ®ber ofP, thenF is a smooth surface and, by [23], Corollary 1.5.2, …F;A_F† ˆ …F;M_F† is one of the following:

a) (P²;O_P²…1†);

b) (P²;O_P²…2†);

c) …Fe;‰s‡bfŠ†, wheresis a section of minimal degrees²ˆ ÿeandfis a ®bre.

In case a) it follows from [27, Claim p. 194] that every ®ber is isomorphic to …P²;O_P²…1††, i.e., that…X;M†is a scroll overP¹and we are in case A.2.

In case b) we have thatMFGO_P²…t†for sometd1, and recalling that…A;LA†is a conic bundle necessarily we havetˆ1. So up to interchanging the roles ofLand Mwe are again in case A.2.

In case c), sinceMF is very ample, we must have …s‡bf† sd1, i.e.,bd1‡e, and, for the same reason, if LFGas‡bf, we must have a>0 and ÿea‡bd1.

SinceAis a conic bundle we haveMFLF ˆ ÿea‡b‡baˆ2 which implieseˆ0 andaˆbˆbˆ1, i.e.,L_FˆM_FGO_P1P¹…1;1†and…X;L†,…X;M†are both quadric

®brations, i.e., we are in case A.3.

If we restrict to threefolds of minimal degree, Theorem A yields the following result.

Proposition 2.1.Let…X;L;M†and C be as in…†and suppose that…X;M†is a three- fold of minimal degree.Then there are only three possible cases:

i) X is a quadric in P⁴ and LGOX…3†, MˆOX…1† (here dAˆ18, dBˆ6, g…C† ˆ4).

ii) XGP³as in Example1:LˆO…4†,MˆO…1†(here dAˆ16,dBˆ4,g…C† ˆ3).

iii) XGP¹P²andLˆOX…3;1†,MˆOX…1;1†(here dAˆ15,dBˆ7,g…C† ˆ3).

Proof.Let …X;M†be a threefold of minimal degree (i.e., a threefold of degreenÿ2 inPⁿ), henceAAjMj ˆ jO_X…1†jis a surface of minimal degree. IfAGP², then we are in case ii), so let PicAˆhs;fi: then we haveL_AGO_A…as‡bf† and suppose thatC@as‡bf is bielliptic. SinceLis very ample we haveb>aeanda>1, where eˆ ÿs²; we also havead3 since otherwiseCwould be rational or hyperelliptic.

From Theorem A, we have that L_A²ˆ ÿa²e‡2abˆa…2bÿae†c18. Hence, by easy computations, we get

2ae‡2c2bcae‡18 a :

Ifeˆ0, thenAGP¹P¹, i.e., a quadric surface, so, by Lemma 1.2, we get that LAGOA…3;3†andL_A²ˆ18, hence we are in case i) (see also Theorem B).

Ife>0, from the above inequalities it follows that we can only haveeˆ1,aˆ3, bˆ4. In this case we should have LAGOA…3s‡4f†, so our problem is to deter- mine if there is a very ample invertible sheafL on Xsuch thatLAGOA…3s‡4f†.

Since …A;MA† is a scroll, we have MAGOA…s‡kf† andXJPⁿ withnˆ2k‡1.

Moreover, PicX ˆhH;Fi, where H AjMj and Fis a ®ber, so we will have LG O_X…aH‡bF†. FromHA@s‡kf andFA@f, we getaˆ3 andbˆ4ÿ3k.

IfLGO_X…3H‡ …4ÿ3k†F†, then

L³ˆa³H³‡3a²bH²Fˆ81ÿ27k and

d_BˆLM²ˆ3k‡1:

From the ®rst equality we getkc2, hence eitherkˆ1 and…A;M_A† ˆ …F₁;O_A…s‡f††, but this contradicts the very ampleness ofM, orkˆ2 andXis embedded by Min

P⁵, soXis the Segre embedding of P¹P², withMGOX…1;1†and we are in case iii), sinceLGOX…3Hÿ2F†GOX…3;1†.

We can generalize the result in Theorem A to the case when dimXˆdd3;

namely, supposeXis as in De®nition 1, andCHX is an ab.c.i. ofL1;. . .;Ldÿ1. Let diˆdegLij_C (without loss of generality we can supposed1dd2d dddÿ1), let A1;. . .;Adÿ1 be sections of L1;. . .;Ldÿ1 which realize C as an ab.c.i. and suppose that all the varietiesS1ˆA2V VAdÿ1,Si2;...;ikˆ7_j0i2;...;ikAj, wherefi2;. . .;ikgH f2;. . .;dÿ1g andkˆ2;. . .;dÿ2, are smooth and irreducible. Then the following holds.

Theorem A⁰⁰⁰.Let …X;L1;. . .;Ldÿ1†and C be as above and suppose C to be a smooth irreducible bielliptic curve.Then if d1d19either:

A⁰.1. X is a scroll over C with respect to all the polarizations, i.e., X is a P^dÿ1 bundle over C and on every ®ber F we have…F;Lij_F†G…P^dÿ1;O_P^dÿ1…1††,or

A⁰.2. X is a P^dÿ1-bundle over an elliptic curve E, and for every ®ber FGP^dÿ1 we have: LijFGO_P^dÿ1…1†, for all iˆ2;. . .;dÿ1 (i.e., …X;Li† is a scroll), and L₁j_FGO_Pdÿ1…2†,or

A⁰.3. X is a quadric bundle over an elliptic curve E, and for every ®ber F we have …F;L_ij_F†G…F;M_ij_F†G…Q_dÿ1;O_Qdÿ1…1††,where Q_ris an r-dimensional hyperquadric inP^r‡1.

Proof. The proof works by complete induction ond. Fordˆ3 this is just Theorem A. When dd4, suppose that the result is known for everyd⁰cdÿ1 and consider the varietiesS_i2;...;ik. We can apply the result in [9] to the surface…S₁;L₁j_S1†, as we did at the beginning of the proof of Theorem A, in order to get that either…S₁;L₁j_S2†is a scroll on a bielliptic curveCor…S₁;L₁j_S2†is a conic bundle on an elliptic curveE.

By Theorem A, we get that, for anyi₂ˆ2;. . .;dÿ1, we can have three cases:

1. the threefolds…S_i2;L_i2j_Si2†and…S_i2;L₁j_Si2†are scrolls;

2. the threefolds…S_i2;L_i2j_Si2†are scrolls and…S_i2;L₁j_Si2†is a Veronese bundle (i.e., the ®bers are embedded as Veronese surfaces);

3. …S_i2;L_i2j_Si

2†and…S_i2;L₁j_Si

2†are all quadric bundles on an elliptic curve.

In cases 1 and 2, by using [3], Theorem 5.5.2, we can extend thePⁱ-bundle struc- ture fromSi2;...;iktoSi2;...;ik;ik‡1, and fromSi2;...;idÿ2 toXto get thatXis either as in A⁰.1 or as in A⁰.2 (in order to check what is the value afor which Li2j_FGO_P^dÿ1…a†one can proceed as in the proof of Theorem A).

In case 3 we can use [26], Proposition III (as in the analogous case in [10], Theorem A) to extend the quadric ®bration fromSi2;...;iktoSi2;...;ik;ik‡1, and fromSi2;...;idÿ2toXin order to get thatXis as in A⁰.3.

As we noticed in the introduction,dAˆ18 seems to be a threshold (as it is in the case of varieties with a bielliptic curve-section, [9], [10]), in fact in this case we have many possibilities for our threefolds, as the following shows.

Theorem B.Let…X;L;M†and C be as in…†with dAˆ18and C bielliptic.Then,if X is not as in Theorem A,it is one of the following:

B.1. XGP³as in Example2:LˆO…3†,MˆO…2†,

B.2. XGQ,where Q is a quadric hypersurface inP⁴andLˆO_Q…3†,MˆO_Q…1†, B.3. X is a Fano threefold of principal series withrˆPicXˆ1andp:X !P³ is a double covering with a sextic surface as discriminant divisor; Mˆ ÿK_XˆpO…1†

andLˆpO…3†,

B.4. X is a Fano threefold of principal series withrˆ2andMˆ ÿK_X, B.5. …X;M†is a conic bundle on a smooth surface,

B.6. …X;M†is a quadric bundle overP¹,

B.7. …X;M†is a scroll, either overP²,orP¹P¹,orF₁,

B.8. …X;M†is the blow up at two points of its reduction…Q;OQ…2††,Q as inB.2and the two points not lying on a line of Q,

B.9. …X;M†is the blow up at one point of its reduction…P…E†;2hÿpO_P¹…1††,where EˆO_P¹…1†lO_P¹…1†lO_P¹…1†,his the tautological bundle ofEand p:P…E† !P¹is the bundle projection.

Proof. Under our hypotheses it follows by [9], Theorem 3.5, that…A;LA† is either …P¹P¹;O_P1P¹…3;3†† or a double plane. Since also P¹P¹ has a double plane structure, the two cases can be treated together.

Let gˆg…C†, from Lemma 1.1 we have that d_Aˆ18dg‡2, hence gc16.

Moreover, since L_AGp…O_P²…3††, where p:A!P² is the double covering, again from [9] we get that pj_C is a 2:1 morphism onto an elliptic curve, hence the cardinality of p…C†VG, where G is the rami®cation curve of p, is exactly 2gÿ2.

Then, by Bezout, 3degGˆ2gÿ2, and so gÿ110 mod 3. Thus the only possible values for g are: 4, 7, 10, 13, and 16. Since K_AGp…O_P²…a††, with adÿ2, and degGˆ2…a‡3†, the values ofa corresponding to the ®ve possible values of g are, respectively, ÿ2,ÿ1, 0, 1, 2. Now, …X;M†is a threefold with a very ample divisor which is a double covering of P². From the classi®cation of such threefolds in [20], we get: cases B.1 and B.2 forgˆ4,aˆ ÿ2; cases B.3, B.4 forgˆ10,aˆ0; case B.5 forgˆ13;16,aˆ1;2 and cases B.6 to B.9 whengˆ7,aˆ ÿ1.

In order to prove the theorem we only have to show that the only two other cases which appear foraˆ ÿ1 in [20, Theorem 3.2], namely cases 3.2.3 and 3.2.5, cannot occur in our case.

In case 3.2.3,Xis as in Example 3, withMGO_X…3HÿE†. Then we should have LGO_X…2HÿE† by Lemma 1.2, but this is not possible because it would yield d_Bˆ17,d_Aˆ11.

In case 3.2.5, …X;M†G…P¹P²;O_P¹_P²…2;2††, and it cannot occur for degree reasons. In fact ifMˆO_P¹_P²…2;2†, say (2,2) for short, andLˆ …a;b†, then

18ˆdAˆML²ˆ …2;2†…a;b†…a;b† ˆ2b…b‡2a†;

i.e., 9ˆb…b‡2a†, whose only solutions are …0;3†, which does not correspond to a very ample divisor on XGP¹P², and …4;1†, which should imply that C is hyperelliptic.

3 The casedBcccc8 We have the following result.

Theorem C.Let…X;L;M†and C be as in…†with d_Bc8.If X is not as in Theorem A, then it is one of the following:

C.1. XGP³ andMˆO…1†,LˆO…4†.

C.2. XGQ,where Q is a quadric hypersurface inP⁴andLˆO_Q…3†,MˆO_Q…1†

(this is also caseB.2, since here d_Aˆ18).

C.3. XHP⁴ is a cubic hypersurface andLˆO_X…2†,MˆO_X…1†.

C.4. XHP⁵,XGP¹P²andLˆO_X…3;1†,MˆO_X…1;1†(see Proposition2.1).

C.5. XHP⁵ is a complete intersection of two hyperquadrics and LˆOX…2†, MˆOX…1†.

C.6. XHP⁵is a rational quadric bundle,MˆOX…1†andLˆOX…2AÿF†,where AAjMjand F is a ®ber.Here dBˆ8and dAˆ12.

CasesC.1toC.5actually occur.

Proof. We will work by considering d_Ac18 since the other cases are covered by Theorem A. By the remark in section 1, we have that if 8dd_B then M³c8. All the varieties of degreec8 are classi®ed in [16] and [17], taking into account also the missed case considered in [1], hence we have to check which are the ones that can possess a bielliptic curve as an ab.c.i. withdBc8 anddAc18. Notice that from …1:1†we also have that

dAM³cd_B²c64; …3:1†

which gives a better bound ond_Aas soon asM³d4.

We will proceed by examining the possibilities forXwith respect to the degreeM³ and the codimensionswith respect to the embedding given byM.

We notice that ifM³d4 the cases when…X;M†is a hypersurface inP⁴or a rational normal threefold are ruled out by Lemma 1.2 and by Proposition 2.1, respectively.

M³ˆ1. Here the only possibility is trivially case C.1 (see Proposition 2.1 and Example 1).

M³ˆ2. By Lemma 1.2, the only possibility is trivially C.2.

M³ˆ3. The only possibilities for a threefold of degree 3 are either a cubic hyper- surface inP⁴(and then by Lemma 1.2 we are in case C.3), or a rational normal scroll XHP⁵and then we are in case C.4 by Proposition 2.1.

M³ˆ4. The only possibility for X is to be the complete intersection of two quadric hypersurfaces inP⁵; in this case, by the Lefschetz theorem (e.g. see [15]), we have PicXGZhO_X…1†i, henceLGO_X…a†witha>1 andCis embedded byMas a complete intersection curve of type…2;2;a†inP⁴ with degreed_Bˆ4ac8. Then the only possibility isaˆ2 which corresponds to case C.5.

M³ˆ5. According to the classi®cation in [16], our threefold X can only have codimensionsˆ3;2.

sˆ3.XJP⁶ is a Del Pezzo threefold which is a section of the Grassmannian G…1;4†,MGO_X…1† and PicXGZM (see [19]). Moreover …A;M_A† is a Del Pezzo

surface andMAGÿKA, henceLAGM_A^a and then d_BˆM_AL_AˆaM²ˆ5ac8;

which impliesaˆ1. SoMGLandCshould be elliptic, a contradiction.

sˆ2.XJP⁵ is a rational quadric bundle, and AAjMj is a Del Pezzo surface of degree 1, i.e., it is isomorphic to the blow up of P² at eight points (and M_A is given by the linear system of the quartic curves passing at least doubly through one point and simply through the others). Since PicXGZhA;Fi, whereFis a ®ber (e.g., see [19], Theorem 1.4.3 and [16] 0.6)), let LGO_X…aA‡bF†and consider PicAG ZhE₀;E₁;E₂;. . .;E₈i, withM_AGO_A…4E₀ÿE₁ÿE₂ÿ ÿ2E₈†,Fj_A@E₀ÿE₈and LAˆOA…aAA‡bFA†ˆOA……4a‡b†E0ÿaE1ÿ ÿaE7ÿ…2a‡b†E8†. From M³ˆ5, M²OX…F† ˆMAOA…FA† ˆ2, by Lemma 1.4 anddBc8, we have

8dd_BˆLM²ˆaM³‡bO_X…F†M²ˆ5a‡2bd6:

By computing the genus g ofCas a divisor in jLAj we have that these inequalities only hold fordBˆ8,aˆ2,bˆ ÿ1,gˆ5,dAˆ12.

This situation corresponds to case C.6.

M³ˆ6. According to the classi®cation in [16], we can only havesˆ4;3;2.

sˆ4. One possibility is thatXGP¹P¹P¹(a Segre variety) andMGOX…1;1;1†.

LetLGOX…a1;a2;a3†, with 1ca3ca2ca1. We can rule out this case by computing dBˆLM². We consider plurihomogeneous coordinateshx0;x1;y₀;y₁;z0;z1ionX, two divisors injMjare given e.g. byx0y₀z0andx1y₁z1 and their intersection is given by the following six lines (given parametrically):

G1ˆ …a;b;0;1;1;0†; L1ˆ …a;b;1;0;0;1†;

G₂ˆ …0;1;a;b;1;0†; L₂ˆ …1;0;a;b;0;1†;

G₃ˆ …0;1;1;0;a;b†; L₃ˆ …1;0;0;1;a;b†:

So we have that G_i andL_i intersect a divisor of jLj in a_i points. Summing up we getd_Bˆ2…a₁‡a₂‡a₃†andd_Bc8 implies that…a₁;a₂;a₃† ˆ …2;1;1†(sinceL0M), but in this case the curveCwould be hyperelliptic (for any pointPin the ®rst factor, there are 2 points on C in the corresponding P¹P¹, so when P varies in P¹ it describes ag₂¹onC). Hence, as claimed, this case is not possible.

Another possibility is thatXGP…T_P²†. ThenXcan also be viewed as a hyperplane section of the Segre variety of P²P²HP⁸. In this case (e.g., see [14]), PicXG Pic…P²P²†GZ² (where the isomorphism is given by the restriction map). With obvious notation, we have that MGO_X…1;1†. Let LGO_X…a;b†, we should have d_BˆLM² ˆ3a‡3bc8 which is impossible for positive values of…a;b†0…1;1†, hence also this case cannot occur.

sˆ3.Xis a Fano threefold with PicXGZ², which is a double coveringp:X!Y of the rational normal threefoldYGP¹P²!P⁵, rami®ed along a divisor of type OY…2;2† (see also Example 4). We have MGp…OY…1;1††. Let LGp…OY…a;b††, the inequalitydBˆLM²c8 impliesOY…a;b† O_Y²…1;1†c4, which is possible only when…a;b† ˆ …2;1†. But in this case the curveOY…a;b† OY…1;1†(onY) is a rational

normal quartic, hence C (which is a double covering of it, via p) would be hyper- elliptic. Thus also this case cannot occur.

sˆ2. We have two possibilities forX. The ®rst isXGP…E†, whereEis a rank 2 locally free sheaf onP²given by the exact sequence 0!O_P2!E!I_Y…4† !0,Y is a set of 10 general points in P², andMis the tautological sheaf onP…E†. In this caseAis isomorphic to the blow-up ofP²alongYandM_Ais associated to the linear system of quartic curves passing throughY. We have that PicXGZhA;Pi, where P denotes the divisor over a generic line in P² in the bundle structure of X, so A²Pˆ4,AP²ˆ1 andP³ˆ0 onX. LetLGO_A…aA‡bP), then we must have

8dd_BˆLM²ˆ6a‡4b (which, sincea>0, implies thatbc0) and

0cL³ˆ3a…2a²‡4ab‡b²†;

which yields adÿ2b‡ 2b²

2 .



p

These inequalities have integer solutions only for bˆ0;ÿ1. Ifbˆ0, the only possibility is that LˆM, that we do not consider. If bˆ ÿ1, thenac2 by the ®rst inequality, henceaˆ2 because foraˆ1 we would have L not very ample (this can be easily seen on L_A). Therefore we get d_Aˆ L²Mˆ6a²ÿ4a‡1ˆ17 and this is not possible becaused_Ac10 by …3:1†. Hence this case cannot occur.

The other possibility is that X is a complete intersection of type …2;3†. In this case, since PicXGZ, we have MGOX…1† andLGOX…b†. Then we should have that Cis a complete intersection of type …2;3;b†inP⁴; a simple computation (e.g., using the resolution of the ideal sheafIC) shows that such curves have genus 3b²‡1 (and degree 6b), hence they cannot be bielliptic by Lemma 1.1.

M³ˆ7. In this case, according to [16], we can only havesˆ5;4;3;2.

sˆ5.Xis the blowing upp:X!P³ ofP³ at one pointP(see also Example 3).

We have thatMˆO_X…2HÿE†. LetLˆO_X…aHÿbE†, we must haved_BˆM²Lˆ 4aÿbc8. Sinceadb and, for the very ampleness ofL, we must also havebd1, we get that either…a;b† ˆ …2;1†, and this yieldsLˆM, or …a;b† ˆ …2;2†in which case Lis not very ample (Lwould contract every line passing throughP). So this case cannot occur too.

sˆ4.Xis the blowing upp:X!P³ ofP³along an elliptic normal curveG. We have thatMˆO_X…3HÿE†whereHis the strict transform of a generic plane ofP³ andEis the exceptional divisor. IfAis a general element injMj, i.e.,Ais isomorphic to a smooth cubic surface containing G, let PicAGZhE0;E1;. . .;E6i. We can choose the generators of PicAin order to have that G@3E0ÿE1ÿE2ÿ ÿE5, henceMAGOA…9E0ÿ3E1ÿ ÿ3E6ÿG†GOA…6E0ÿ2E1ÿ ÿ2E5ÿ3E6†.

If LGOX…aHÿbE† (since PicXGZhH;Ei) we have LAGOA…3…aÿb†E0ÿ …aÿb†E1ÿ ÿ …aÿb†E5ÿaE6†, hencedBˆM²Lˆ5aÿ8bc8. On the other hand we must also have ad2b (since the ideal ofG is the complete intersection of two quadric forms). Hence we get 8dd_Bd2b, i.e., bc4 (recall also that b>0 to have very ampleness), moreover ad2b‡1 can satisfy 5aÿ8bc8 only for

…a;b† ˆ …3;1†, i.e., forLˆM, and we are not interested in this case. Thus we only have to consider …a;b† ˆ …2b;b†, bˆ1;2;3;4, but for these values L is not very ample (it is given by the generators of…IG†^b). So this case does not occur.

sˆ3. The ®rst possibility is that X is a scroll over an elliptic curve G. Let us consider AAjMj, which is a ruled surface on G with PicAGZhG₀;Fi and M_AˆO_A…G₀‡bF†. We must have M_A²ˆ2bÿeˆ7 and (for the very ampleness) bde‡3, hence either eˆ1, bˆ4 or eˆ ÿ1, bˆ3. Let L_AGO_A…aA‡bF†G O_A…aG₀‡ …ab‡b†F†, we must have …aA‡bF†G₀d3 hence ÿae‡ab‡bd3, moreover d_BˆL_AM_Ac8. If eˆ ÿ1,bˆ3 these two inequalities yield 3ÿ4ac bc8ÿ7a, while if eˆ1, bˆ4 they yield 3ÿ3acbc8ÿ7a. Both cases imply aˆ1, which is absurd sinceCwould be elliptic.

Another possibility is thatXis a quadric bundle overP¹ andAGAe, whereAe

is the blow-up of a Hirzebruch surface Fe at 9 points, with eˆ0;1;2 or 3. Then PicXGZhA;Fi, where F is a generic ®ber, and, with obvious notation, PicAG ZhC0;FA;E1;. . .;E9i. Let MAˆOA…2C0‡bFAÿE1ÿ ÿE6†, where bˆ4‡e, andLˆOA…aA‡bF†, so thatLAˆOA…2aC0‡ …b‡b†FAÿaE1ÿ ÿaE9†.

By Lemma 1.4 we have thatdBd7, so

7cd_BˆM_AL_Aˆ ÿ4ae‡2…b‡b† ÿ2abÿ9ac8:

From this condition we get

2…b‡e† ‡1

2e‡1 cac2…b‡e†

2e‡1 ; which yieldsbde‡1. FromdAddBwe must have:

7cd_AˆL_A² ˆ ÿ4a²e‡4a…b‡b† ÿ9a²:

Simple but tedious computations show that the former condition contradicts the latter one, for alleAf0;1;2;3g.

The last possibility is thatXGP…E†, whereEis a rank 2 locally free sheaf onP² given by the exact sequence 0!O_P²!E!I_Y…4† !0 with Ya set of 9 general points in P², andMis the tautological sheaf on P…E†(see also the similar case for M³ˆ6, sˆ2). We have that PicXGZhA;Pi, where P is the divisor over a generic line inP²in the bundle structure ofX, soA²Pˆ4,AP² ˆ1 andP³ˆ0 onX. IfLGOX…aA‡bP†, then we have

8ddBˆLM²ˆ7a‡4b

which, since a>0, implies that bc0. On the other hand for bˆ0 we must have aˆ1 which yieldsMˆL. So actually we haveb<0 andad2. SinceCis inP⁵, by Lemma 1.1 we get d_Bd5‡g…C† ÿ1d7, which, since d_Bcd_Acd_B²=M³ˆ7, implies that eitherd_Bˆd_Aˆ7, ord_Bˆ8 and 8cd_Ac9. From

dAˆL²Mˆ7a²‡8ab‡b²

it is just a computation to show that no values fora,b can give the required values ofdA,dB.

sˆ2. We have three possibilities forX. A ®rst one isXGP…E†, whereEis a rank 2 locally free sheaf on a smooth cubic surface SHP³ given by the exact sequence 0!OS!E!IY;S…2† !0 with Y a set of 5 general points on S, and M is the tautological sheaf on P…E† (see also the case above). So p:X !S is a scroll structure with respect toM, and PicXGZhA;F₀;. . .;F₆i, whereAAjMj, PicSG ZhE₀;E₁;. . .;E₆i(e.g. see [15]) andF_iˆp^ÿ1…E_i†. We have thatAis the blow up ofS atY, so ifE₇;. . .;E₁₁are its exceptional divisors and (with a slight abuse of notation) PicAGZhE₀;E₁;. . .;E₁₁i, we haveM_AGO_A…6E₀ÿ2E₁ÿ ÿ2E₆ÿE₇ÿ ÿE₁₁†.

LetLGO_X…aA‡bF₀ÿg₁F₁ÿ ÿg₆E₆†, thenL_AGO_A……6a‡b†E₀ÿ …2a‡g₁†E₁ ÿ ÿ …2a‡g₆†E₆ÿaE₇ÿ ÿaE₁₁†; we have d_BˆM_AL_Aˆ7a‡bÿ2P₆

iˆ1g_i and d_AˆL_A²ˆ7a²‡12ab‡b²ÿ4aP₆

iˆ1g_iÿaP₆

iˆ1g_i². By Lemma 1.4 we have 8dd_Bdg‡3, while from …3:1† we get d_Ac9. Moreover for the genus of C, we have

gˆ 6a‡bÿ1 2

ÿX⁶

iˆ1

2a‡g_i 2

ÿ5 a 2 ; which gives 2gˆdAÿaÿ3b‡P₆

iˆ1g_i‡2. From the bound ondAwe get 2a‡6bÿ2X⁶

iˆ1

g_id22ÿ4g while from the bounds ondB we have 3cgc5. Hence we get

7c5a‡

2a‡6bÿ2X⁶

iˆ1

g_i

c8;

which is clearly impossible forgˆ3 or 4, since ad1 and the part in parentheses is d22ÿ4g. Whengˆ5, which impliesdBˆ8, the bound above can be satis®ed only foraˆ1, but in this casedBˆ7‡6bÿ2P₆

iˆ1g_i, which cannot be eight. So also this case cannot occur.

A second possibility forsˆ2 is thatXis the blowing upp:X!Y of a smooth 3-foldYHP⁶, which is the complete intersection of three quadrics, at a pointPAY (i.e., Xis obtained by projectingYintoP⁵ fromP). Here PicXGZhH;Ei, where H is the strict transform of a generic hyperplane section ofY and E is the excep- tional divisor. We haveMGO_X…1†GO_X…HÿE†, and from…HÿE†³ ˆ7, together withH³ˆH …HÿE†²ˆ8 we getH²EˆHE²ˆ0 andE³ ˆ1. Now letLG O_X…aHÿbE†, since 0<d_Bc8 andd_A>0, we have:

0<dBˆ …HÿE†²…aHÿbE† ˆ8aÿbc8;

0<dAˆ …HÿE†…aHÿbE†²ˆ8a²ÿb²:

So bd8aÿ8 and b²<8a², hence 64a²ÿ32a‡64<8a², which is never true, and this case is impossible.

Eventually, the last possibility is that X is a cubic ®bration on P¹, where MGOX…1†, and this structure is given by the adjunction mapf_jKX‡Mj!P¹, with cubic surfaces S (in a P³) as generic ®bers. We have that …A;MA† is ®bered by elliptic curves on P¹ and no ®ber splits, see [16]. Hence also the ®bers ofX do not split and PicXGZhA;Si. We have A³ˆ7, A²Sˆ3, AS²ˆS³ˆ0. Let LG O_X…aA‡bS†, by Lemma 1.4, we haved_Bd6, so

6cd_BˆLM² ˆ7a‡3bc8 and, by the inequalities…1:1†and…3:1†we get

0<d_AˆL²Mˆa…7a‡6b†c9;

0<L³ˆ7a³‡9a²bˆa²…7a‡9b†c13:

From the ®rst inequalities we get b<0, ad2 (for bˆ0, aˆ1 we would have LˆM), from the third we get aˆ2 or aˆ3 and 7a‡9b>0. With aˆ2 the

®rst inequalities gives ÿ8c3bcÿ6, i.e.,bˆ ÿ2, but this contradicts 7a‡9b>0.

Ifaˆ3, thenÿ15c3bcÿ13, sobˆ ÿ5, but this is impossible sincedA>0.

M³ˆ8. The bounds M³cdBc8 imply dBˆ8 while the bounds dBcdAc d_B²=M³ imply d_Aˆ8. Moreover, d_BL³cd_A², gives L³c8. We can exclude that L³<8 by looking at all the cases we have seen before (we have considered all the polarized threefolds of degreec7), so we only have to study the case d_Bˆd_Aˆ M³ˆL³ˆ8.

Now, let AAjMj as always, we have that …A;L_A†is a surface of degree 8 with a bielliptic curve section. Such surfaces are classi®ed in [9], Theorem 4.1, with the exception of the elliptic conic bundles discovered in [1], and we will use these results to complete our proof.

We can easily check that under the degree assumptions above X cannot be a quadric bundle onP¹. In fact in this case, see e.g., [16], 0.6, the ®bers are all irre- ducible and PicXGZhA;Fi, whereFis a ®ber,A³ˆ8,A²F ˆ2 andAF²ˆF³ˆ0.

Hence, ifLGOX…aA‡bF†, we should haveL³ˆ8a³‡6a²bˆ2a²…4a‡3b† ˆ8 which is easily seen to be impossible.

Now we proceed as in the previous cases. According to [17] we can only have sˆ6;5;4;3;2.

sˆ6. In this case X is the double embedding of P³ into P⁹, i.e., XGP³ and MGO_P3…2†(see also Example 2). By Lemma 1.2 we should have LGO_P3…3†, but then we would haved_Bˆ12,d_Aˆ18.

sˆ5.Xis a hyperplane section of the Segre embedding ofP¹Q³intoP⁹, where Q³is a quadric hypersurface inP⁴. In this caseXwould be a quadric bundle, but we have just seen that this is impossible.

sˆ4. We have four possibilities forX. First, Xis a scroll on an elliptic curveE.

This would imply that alsoAis an elliptic scroll onE, then its irregularity would be q…A† ˆ1, but this is impossible by [9], Theorem 4.1.

Two other possibilities are that X is either the complete intersection of a hyper- quadric with a Segre variety V which is the embedding of P¹P³, or a double

covering of a hyperplane section of V. But in both cases X would be a rational quadric bundle and we have already excluded this possibility.

The last case is thatXGP…E†, whereEis a rank 2 locally free sheaf onP²given by the exact sequence 0!O_P2!L!I_Y…4† !0, Y is a set of 8 general points in P², andMis the tautological sheaf onP…E†. In this caseAis isomorphic to the blow-up of P² alongY andM_A is associated to the linear system of quartic curves passing through Y. We should have that …A;L_A† is a surface of degree 8 which appears in the classi®cation of [9], Theorem 4.1 (since its hyperplane section is a bielliptic curve), but this is possible only forM_AˆL_A, which impliesMˆL, so also this case cannot occur.

sˆ3. We have three possibilities for X. First, X could be a rational quadric bundle, but this is the case we have excluded.

The second case is thatp:X !Qis a scroll on a quadric surfaceQ. So PicXG ZhA;F1;F2i, whereAAjMjandFiˆp^ÿ1…Gi†, PicQGZhG1;G2i. We haveA²Fiˆ AF1F2ˆ1 andAF_i²ˆF_i²Fjˆ0,i;jˆ1;2. Let LGOX…aA‡bF1‡gF2†, then we must have

d_BˆLM²ˆ8a‡b‡gˆ8;

d_AˆL²Mˆ2a…4a‡b‡g† ‡2bgˆ8;

L³ˆa…8a²‡3a‡3b‡6bg† ˆ8:

By the third equalityamust divide 8 and it is easy to check that any such value of adoes not satisfy the ®rst and second equations inb,g.

The last possibility is that X is the complete intersection of three quadric hyper- surfaces in P⁶. In this case PicXGZ, then LGO_X…a† and L³ˆ8a²ˆ8 which yieldsLˆM.

sˆ2. X could be the complete intersection of a quadric and a quartic hyper- surface. Then, since LGOX…a† andL³ ˆ8a²ˆ8, we can exclude this case as we did above.

Another possibility is thatXis a Del Pezzo ®bration onP¹given by its adjunction mapf_jKX‡Mj:X !P¹. The generic ®ber offis a Del Pezzo surfaceSisomorphic to a complete intersection of two quadrics in P⁴. We have that PicXGZhA;Si, A³ˆ8,A²Sˆ4, AS²ˆS³ˆ0, and let LGOX…aA‡bS†. Hence: dAˆL²Mˆ 8a²‡8ab‡8a…a‡b† ˆ8, which is possible only for bˆ0, aˆ1, but this would implyLˆMonce more.

The last case to be considered is the one missed in [17], [18] (and hence also in [9]) which we mentioned at the beginning of the proof, namely whenXis such that…A;M_A† is a degree 8 conic bundle on an elliptic curve (see [1]). In this case, by working in a similar way as we did in the proof of Theorem A, case 2, we get thatXmust be as in Theorem A, but these cases have been excluded by our hypotheses.

To complete the proof of our theorem, we have only to notice that the existence of threefolds X as described in the ®rst four cases is obvious and the case C:5 occurs when C is a canonical bielliptic curve of genus 5. Unfortunately we have not been able to determine whether a threefoldXas in caseC:6 exists or not.

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Received 25 October, 2000; revised 15 January, 2001 and 21 February, 2001

A. Del Centina, Dipartimento di Matematica, UniversitaÁ di Ferrara, Via Machiavelli 35, I-44100, Ferrara, Italy

E-mail: [email protected]

A. Gimigliano, Dipartimento di Matematica, UniversitaÁ di Bologna, Piazza di Porta S. Donato 5, I-40126, Bologna, Italy

E-mail: [email protected]