DocumentaMath. 23 The Abel{Jacobi Map for a Cubic Threefold and Periods of Fano Threefolds of Degree 14 A. Iliev and D. Markushevich Received: October 15,1999 Revised: January17,2000 CommunicatedbyThomasPeternell Abstract. The Abel{Jacobimaps of the families of elliptic quintics andrationalquarticslyingonasmoothcubicthreefoldarestudied. It isprovedthattheirgeneric(cid:12)beristhe 5-dimensionalprojectivespace for quintics, and a smooth 3-dimensional variety birational to the cubicitselfforquartics. Thepaperisacontinuationoftherecentwork of Markushevich{Tikhomirov,who showedthat the (cid:12)rst Abel{Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c = 0;c = 2 1 2 obtained by Serre’s construction from elliptic quintics, and that the factorizing map from the moduli space to the intermediate Jacobian is (cid:19)etale. The above result implies that the degree of the (cid:19)etale map is 1, hence the moduli component of vector bundles is birational to the intermediateJacobian. Asanapplication,itisshownthatthegeneric (cid:12)ber of the period map of Fano varieties of degree 14 is birational to the intermediate Jacobian of the associated cubic threefold. 1991 Mathematics Subject Classi(cid:12)cation: 14J30,14J60,14J45 Introduction Clemens and Gri(cid:14)ths studied in [CG] the Abel{Jacobi map of the family of linesonacubicthreefoldX. TheyrepresenteditsintermediateJacobianJ2(X) astheAlbanesevarietyAlbF(X)oftheFanosurfaceF(X)parametrizinglines on X and described its theta divisor. From this description, they deduced the Torelli Theorem and the non-rationality of X. Similar results were obtained by Tyurin [Tyu] and Beauville [B]. One can easily understand the structure of the Abel{Jacobi maps of some other familes of curves of low degree on X (conics, cubics or elliptic quartics), in reducing the problem to the results of Clemens{Gri(cid:14)ths and Tyurin. The (cid:12)rstnontrivialcasesarethoseofrationalnormalquarticsandofellipticnormal Documenta Mathematica 5 (2000)23{47 24 A. Iliev and D. Markushevich quintics. We determine the (cid:12)bers of the Abel{Jacobi maps of these families of curves, in continuing the work started in [MT]. Ourresultonellipticquinticsimpliesthatthemodulispaceofinstantonvector bundles of charge 2 on X has a component, birational to J2(X). We con- jecture that the moduli space is irreducible, but the problem of irreducibility stays beyond the scope of the present article. As far as we know, this is the (cid:12)rst example of a moduli space of vector bundles which is birational to an abelian variety, di(cid:11)erent from the Picard or Albanese variety of the base. The situation is also quite di(cid:11)erent from the known cases where the base is P3 or the 3-dimensional quadric. In these cases, the instanton moduli space is ir- reducible and rational at least for small charges, see [Barth], [ES], [H], [LP], [OS]. Remark, that for the cubic X, two is the smallest possible charge, but the moduli space is not even unirational. There are no papers on the geome- try of particular moduli spaces of vectorbundles for other 3-dimensionalFano varieties (for some constructions of vector bundles on such varieties, see [G1], [G2], [B-MR1], [B-MR2], [SW], [AC]). Theauthorsof[MT]provedthattheAbel{Jacobimap(cid:8)ofthefamilyofelliptic quintics lying on a general cubic threefold X factors through a 5-dimensional moduli component M of stable rank 2 vector bundles E on X with Chern X numbersc =0;c =2. Thefactorizingmap(cid:30)sendsanelliptic quinticC (cid:26)X 1 2 to the vector bundle E obtained by Serre’s construction from C (see Sect. 2). The (cid:12)ber (cid:30)(cid:0)1([E]) is a 5-dimensional projective space in the Hilbert scheme Hilb5n, and the map (cid:9) from the moduli space to the intermediate Jacobian X J2(X), de(cid:12)ned by (cid:8) = (cid:9)(cid:14)(cid:30), is (cid:19)etale on the open set representing (smooth) elliptic quintics which are not contained in a hyperplane (Theorem 2.1). We improve the result of [MT] in showing that the degree of the above (cid:19)etale map is 1. Hence M is birational to J2(X) and the generic (cid:12)ber of (cid:8) is just X onecopyofP5 (seeTheorem3.2andCorollary3.3). ThebehavioroftheAbel{ Jacobi map of elliptic quintics is thus quite similar to that of the Abel{Jacobi map of divisors on a curve, where all the (cid:12)bers are projective spaces. But we provethatthesituationisverydi(cid:11)erentinthecaseofrationalnormalquartics, wherethe(cid:12)beroftheAbel{Jacobimapisanon-rational3-dimensionalvariety: it is birationally equivalent to the cubic X itself (Theorem 5.2). The (cid:12)rstnew ingredientof ourproofs,comparingto [MT], isanotherinterpre- tation of the vectorbundles E from M . We representthe cubic X as a linear X section of the Pfa(cid:14)an cubic in P14, parametrizing6(cid:2)6 matrices M of rank4, and realize E_((cid:0)1) as the restriction of the kernel bundle M 7! kerM (cid:26) C6 (Theorem2.2). ThekernelbundlehasbeeninvestigatedbyA.AdlerinhisAp- pendixto[AR]. WeprovethatitembedsXintotheGrassmannianG=G(2;6), and the quintics C 2(cid:30)(cid:0)1([E]) become the sections of X by the Schubert vari- eties (cid:27) (L) for all hyperplanes L (cid:26) C6. We deduce that for any line l (cid:26) X, 11 each (cid:12)ber of (cid:30) contains precisely one pencil P1 of reducible curves of the form C0 +l (Lemma 3.4). Next we use the techniques of Hartshorne{Hirschowitz [HH] for smoothing the curves of the type \a rational normal quartic plus one of its chords in X" (see Sect. 4) to show that there is a 3-dimensional family Documenta Mathematica 5 (2000)23{47 The Abel{Jacobi Map for a Cubic Threefold ::: 25 of suchcurvesin ageneric(cid:12)berof (cid:30) andthat the abovepencil P1 fora generic l contains curves C0+l of this type (Lemma 4.6, Corollary4.7). The other main ingredient is the parametrization of J2(X) by minimal sec- tions of the 2-dimensional conic bundles of the form Y(C2)=(cid:25)(cid:0)1(C2), where l (cid:25) :Blowup (X)(cid:0)!P2istheconicbundleobtainedbyprojectingXfroma(cid:12)xed l l line l, and C2 is a generic conic in P2 (see Sect. 3). The standard Wirtinger approach[B]parametrizesJ2(X)byreduciblecurveswhicharesumsofcompo- nentsofreducible(cid:12)bersof(cid:25) . Ourapproach,developedin[I]inamoregeneral l form, replaces the degree 10 sums of components of the reducible (cid:12)bers of the surfaces Y(C2) by the irreducible curves which are sections of the projection Y(C2)(cid:0)!C2 with a certain minimality condition. This gives a parametriza- tion of J2(X) by a family of rational curves, each one of which is projected isomorphically onto some conic in P2. It turns out, that these rational curves are normal quartics meeting l at two points. They form a unique pencil P1 in each(cid:12)berof the Abel{Jacobimapof rationalnormalquartics. Combiningthis withtheabove,weconcludethatthecurvesoftypeC0+lformauniquepencil in each (cid:12)ber of (cid:8), hence the (cid:12)ber is one copy of P5. In conclusion, we provide a description of the moduli space of Fano varieties V as a birationally (cid:12)bered space over the moduli space of cubic 3-folds with 14 the intermediateJacobianasa(cid:12)ber(seeTheorem5.8). Theinterplaybetween cubics and varieties V is exploited several times in the paper. We use the 14 Fano{Iskovskikh birationality between X and V to prove Theorem 2.2 on 14 kernel bundles, and the Tregub{Takeuchi one (see Sect. 1) to study the (cid:12)ber of the Abel{Jacobi map of the family of rational quartics (Theorem 5.2) and the relation of this family to that of normal elliptic quintics (Proposition 5.6). Acknowledgements. The authors are grateful to the referee for his remarks which allowed to improve the exposition. The second author acknowledges with pleasure the hospitality of the MPIM at Bonn, where he completed the work on the paper. 1. Birational isomorphisms between V and V 3 14 There aretwo constructionsof birationalisomorphismsbetween a nonsingular cubic threefold V (cid:26)P4 and the Fano variety V of degree 14 and of index 1, 3 14 which is a nonsingular section of the Grassmannian G(2;6) (cid:26) P14 by a linear subspace of codimension 5. The (cid:12)rst one is that of Fano{Iskovskikh, and it gives a birational isomorphism whose indeterminacy locus in both varieties is an elliptic curve together with some 25 lines; the other is due to Tregub{ Takeuchi,and its indeterminacy locus isa rational quarticplus 16 lines on the side of V , and 16conicspassingthroughone point on the side of V . We will 3 14 sketch both of them. Theorem 1.1 (Fano{Iskovskikh). Let X = V be a smooth cubic threefold. 3 Then X contains a smooth projectively normal elliptic quintic curve. Let C be such a curve. Then C has exactly 25 bisecant lines l (cid:26) X, i = 1;:::;25, and i Documenta Mathematica 5 (2000)23{47 26 A. Iliev and D. Markushevich there is a unique e(cid:11)ective divisor M 2jO (5(cid:0)3C)j on X, which is a reduced X surface containing the l . The following assertions hold: i (i) The non-complete linear system j O (7(cid:0)4C) j de(cid:12)nes a birational map X (cid:26) : X ! V where V = V is a Fano 3-fold of index 1 and of degree 14. 14 Moreover (cid:26)=(cid:27)(cid:14)(cid:20)(cid:14)(cid:28) where (cid:27) :X0 !X is the blow-up of C, (cid:20):X0 !X+ is a(cid:13)opoverthepropertransformsl0 (cid:26)X0 ofthel ,i=1;:::;25,and(cid:28) :X+ !V i i is a blowdown of the proper transform M+ (cid:26)X+ of M onto an elliptic quintic B (cid:26)V. The map (cid:28) sends the transforms l+ (cid:26)X+ of l to the 25 secant lines i i m (cid:26)V, i=1;:::;25 of the curve B. i (ii) The inverse map (cid:26)(cid:0)1 is de(cid:12)ned by the system jO (3(cid:0)4B)j. The excep- V tional divisor E0 =(cid:27)(cid:0)1(C)(cid:26)X0 is the proper transform of the uniquee(cid:11)ective divisor N 2jO (2(cid:0)3B)j. V For a proof , see [Isk1], [F], or [Isk-P], Ch. 4. Theorem 1.2 (Tregub{Takeuchi). Let X be a smooth cubic threefold. Then X contains a rational projectively normal quartic curve. Let (cid:0)be such a curve. Then (cid:0) has exactly 16 bisecant lines l (cid:26)X, i=1;:::;16,and there is a unique i e(cid:11)ectivedivisor M 2jO (3(cid:0)2(cid:0))jonX,which isareducedsurfacecontaining X the l . The following assertions hold: i (i) The non-complete linear system j O (8(cid:0)5(cid:0)) j de(cid:12)nes a birational map X (cid:31) : X ! V where V is a Fano 3-fold of index 1 and of degree 14. Moreover (cid:31) = (cid:27)(cid:14)(cid:20)(cid:14)(cid:28), where (cid:27) : X0 ! X is the blowup of (cid:0), (cid:20) : X0 ! X+ is a (cid:13)op over the proper transforms l0 (cid:26) X0 of l , i = 1;:::;16, and (cid:28) : X+ ! V is a i i blowdown of the proper transform M+ (cid:26) X+ of M to a point P 2 V. The map (cid:28) sends the transforms l+ (cid:26)X+ of l to the 16 conics q (cid:26)V, i=1;:::;16 i i i which pass through the point P. (ii) The inverse map (cid:31)(cid:0)1 is de(cid:12)ned by the system jO (2(cid:0)5P)j. The excep- V tional divisor E0 =(cid:27)(cid:0)1((cid:0))(cid:26)X0 is the proper transform of the unique e(cid:11)ective divisor N 2jO (3(cid:0)8P)j. V (iii) For a generic point P on any nonsingular V , this linear system de(cid:12)nes 14 a birational isomorphism of type (cid:31)(cid:0)1. Proof. For (i), (ii), see [Tak], Theorem 3.1, and [Tre]. For (iii), see [Tak], Theorem 2.1, (iv). See also [Isk-P], Ch. 4. 1.3. Geometric description. We will brie(cid:13)y describe the geometry of the (cid:12)rst birational isomorphism between V and V following [P]. 3 14 Let E be a 6-dimensional vector space over C. Fix a basis e ;::: ;e for E, 0 5 then e ^e for 0 (cid:20) i < j (cid:20) 5 form a basis for the Plu(cid:127)cker space of 2-spaces i j in E, or equivalently, of lines in P5 =P(E). With Plu(cid:127)cker coordinatesx , the ij embedding of the GrassmannianG=G(2;E) in P14 =P(^2E) is preciselythe Documenta Mathematica 5 (2000)23{47 The Abel{Jacobi Map for a Cubic Threefold ::: 27 locus of rank 2 skew symmetric 6(cid:2)6 matrices 0 x x x x x 01 02 03 04 05 2 (cid:0)x01 0 x12 x13 x14 x15 3 (cid:0)x (cid:0)x 0 x x x M =6 02 12 23 24 25 7: 6 (cid:0)x (cid:0)x (cid:0)x 0 x x 7 6 03 13 23 34 35 7 6 (cid:0)x04 (cid:0)x14 (cid:0)x24 (cid:0)x34 0 x45 7 6 7 6 (cid:0)x05 (cid:0)x15 (cid:0)x25 (cid:0)x35 (cid:0)x45 0 7 4 5 There are two ways to associate to these data a 13-dimensional cubic. The Pfa(cid:14)an cubic hypersurface (cid:4) (cid:26) P14 is de(cid:12)ned as the zero locus of the 6(cid:2)6 Pfa(cid:14)an of this matrix; it can be identi(cid:12)ed with the secant variety of G(2;E), or else, it is the locus where M has rank 4. The other way is to consider the dual variety (cid:4)0 = G_ (cid:26) P14_ of G; it is also a cubic hypersurface, which is nothing other than the secant variety of the Grassmannian G0 = G(2;E_) (cid:26) P(^2E_)=P14_. As it is classically known, the generic cubic threefold X can be represented as a section of the Pfa(cid:14)an cubic by a linear subspace of codimension 10; see also a recent proof in [AR], Theorem 47.3. There are 15 essentially di(cid:11)erent ways to do this. Beauville and Donagi [BD] have used this idea for introducing the symplectic structure on the Fano 4-fold(parametrizinglines) of acubic 4-fold. Intheircase,onlyspecialcubics(adivisorialfamily)aresectionsofthePfa(cid:14)an cubic, sothey introducedthe symplectic structure onthe Fano4-foldsof these special cubics, and obtained the existence of such a structure on the generic one by deformation arguments. For any hyperplane section H\G of G, we can de(cid:12)ne rkH as the rank of the antisymmetricmatrix((cid:11) ),where (cid:11) x =0istheequationofH. So,rkH ij ij ij may take the values 2,4 or 6. If rkPH = 6, then H \G is nonsingular and for any p2P5 =P(E), there is the unique hyperplane L (cid:26)P5 =P(E), such that p q 2 H \G, p2 l () l (cid:26) L . Here l denotes the line in P5 represented by q q p q q 2G. (Thisisawaytoseethatthebaseofthefamilyof3-dimensionalplanes on the 7-fold H \G is P5.) The rankof H is 4if andonlyif H istangentto G atexactlyonepoint z,and in this case, the hyperplane L is not de(cid:12)ned for any p2l : we have for such p z p the equivalence p 2 l () x 2 H. Following Puts, we call the line l the x z center of H; it will be denoted c . H In the third case, when rkH = 2, H \G is singular along the whole Grass- mannian subvariety G(2;4)=G(2;E ), where E =ker((cid:11) ) is of dimension H H ij 4. We have x2H () l \P(E )6=?. x H This description identi(cid:12)es the dual of G with (cid:4)0 = fH j rkH (cid:20) 4g = fH j Pf(((cid:11) ))=0g, and its singular locus with fE g =G(4;E). ij H rkH=2 Now, associate to any nonsingular V = G\(cid:3), where (cid:3) = H \H \H \ 14 1 2 3 H \H , the cubic 3-fold V by the following rule: 4 5 3 (1) V =G\(cid:3) 7! V =(cid:4)0\(cid:3)_; 14 3 Documenta Mathematica 5 (2000)23{47 28 A. Iliev and D. Markushevich where (cid:3)_ =<H_;H_;H_;H_;H_ >, H_ denotes the orthogonalcomplement 1 2 3 4 5 i of H inP14_,andthe angularbracketsthe linearspan. OnecanprovethatV i 3 is also nonsingular. According to Fano, the lines l represented by points x 2 V sweep out an x 14 irreducible quartic hypersurface W, which Fano calls the quartic da Palatini. W coincides with the union of centers of all H 2 V . One can see, that W is 3 singular along the locus of foci p of Schubert pencils of lines on G (cid:27) (p;h)=fx2Gjp2l (cid:26)hg 43 x which lie entirely in V , where h denotes a plane in P5(depending on p). The 14 pencils(cid:27) areexactlythe linesonV , soSingW isidenti(cid:12)edwith thebaseof 43 14 the family of lines on V , which is known to be a nonsingular curve of genus 14 26 for generic V (see, e. g. [M] for the study of the curve of lines on V , 14 14 and Sections 50,51of [AR] forthe study of SingW without any connectionto V ). 14 Theconstructionof the birationalisomorphism(cid:17) :V 99KV dependsonthe L 14 3 choice of a hyperplane L(cid:26)P5. Let (cid:30):V 99KW \L; x7!L\l ; :V 99KW \L; H_ 7!L\c : 14 x 3 H These two maps are birational, and (cid:17) is de(cid:12)ned by L (2) (cid:17) = (cid:0)1(cid:14)(cid:30): L Thelocus,onwhich(cid:17) isnotanisomorphism,consistsofpointswhereeither(cid:30) L or isnotde(cid:12)nedorisnotone-to-one. TheindeterminacylocusBof(cid:30)consists of all the points x such that l (cid:26) L, that is, B = G(2;L)\H \:::\H . x 1 5 For generic L, it is obviously a smooth elliptic quintic curve in V , and it 14 is this curve that was denoted in Theorem 1.1 by the same symbol B. The indeterminacy locus of is described in a similar way. We summarize the above in the following statement. Proposition 1.4. Any nonsingular variety V determines a unique nonsin- 14 gular cubic V by the rule (1). Conversely, a generic cubic V can be obtained 3 3 in this way from 15 many varieties V . 14 For each pair (V ;V ) related by (1), there is a family of birational maps (cid:17) : 14 3 L V 99K V , de(cid:12)ned by (2) and parametrized by points of the dual projective 14 3 space P5_, and the structure of (cid:17) for generic L is described by Theorem 1.1. L The smooth elliptic quintic curve B (resp. C) of Theorem 1.1 is the locus of points x2V such that l (cid:26)L (resp. H_ 2V such that c (cid:26)L). 14 x 3 H Definition 1.5. We will call two varieties V , V associated (to each other), 3 14 if V can be obtained from V by the construction (1). 3 14 1.6. Intermediate Jacobians of V , V . Both constructions of birational 3 14 isomorphisms give the isomorphism of the intermediate Jacobians of generic varieties V , V , associated to each other. This is completely obvious for the 3 14 second construction: it gives a birational isomorphism, which is a composi- tion of blowups and blowdowns with centers in nonsingular rational curves or Documenta Mathematica 5 (2000)23{47 The Abel{Jacobi Map for a Cubic Threefold ::: 29 points. According to [CG], a blowup (cid:27) : X~(cid:0)!X of a threefold X with a nonsingular center Z can change its intermediate Jacobian only in the case when Z is a curve of genus (cid:21) 1, and in this case J2(X~) ’ J2(X)(cid:2)J(Z) as principally polarized abelian varieties, where J2 (resp. J) stands for the in- termediate Jacobian of a threefold (resp. for the Jacobian of a curve). Thus, the Tregub{Takeuchibirationalisomorphismdoesnotchangetheintermediate Jacobian. Similar argument works for the Fano{Iskovskikh construction. It factors through blowups and blowdowns with centers in rational curves, and containsinitsfactorizationexactlyoneblowupandoneblowdownwithnonra- tionalcenters,whichareellipticcurves. So, wehaveJ2(V )(cid:2)C ’J2(V )(cid:2)B 3 14 for some elliptic curves C;B. According to Clemens{Gri(cid:14)ths, J2(V ) is irre- 3 ducible for every nonsingular V , so we can simplify the above isomorphism1 3 to obtain J2(V )’J2(V ); we also obtain, as a by-product, the isomorphism 3 14 C ’B. Proposition 1.7. Let V = V , X = V be a pair of smooth Fano varieties 14 3 related by either of the two birational isomorphisms of Fano{Iskovskikh or of Tregub{Takeuchi. Then J2(X)’J2(V), V;X are associated to each other and related by a birational isomorphism of the other type as well. Proof. The isomorphism of the intermediate Jacobians was proved in the pre- vious paragraph. Let J2(V0) = J2(V00) = J. By Clemens-Gri(cid:14)ths [CG] or Tyurin [Tyu], the global Torelli Theorem holds for smooth 3-dimensional cu- bics, so there exists the unique cubic threefold X such that J2(X) = J as p.p.a.v. Let X0 and X00 be the unique cubics associated to V0 and V00. Since J2(X0)=J2(V0)=J =J2(V00)=J2(X00), then X0 ’X ’X00. Let now V0 and V00 be associated to the same cubic threefold X, and let J2(X)=J. Then by the above J2(V0)=J2(X)=J2(V00). Let X, V be related by, say, a Tregub{Takeuchi birational isomorphism. By Proposition 1.4, V contains a smooth elliptiic quintic curve and admits a bi- rational isomorphism of Fano{Iskovskikh type with some cubic X0. Then, as above, X ’ X0 by Global Torelli, and X, V are associated to each other by the de(cid:12)nition of the Fano{Iskovskikh birational isomorphism. Conversely, if we start from the hypothesis that X, V are related by a Fano{Iskovskikh bi- rational isomorphism, then the existence of a Tregub{Takeuchione from V to somecubicX0 isa(cid:14)rmedbyTheorem1.2,(iii). Hence,againbyGlobalTorelli, X ’X0 and we are done. 1Itisaneasyexercisetoseethatifanabelianvarietydecomposesintothedirectproductof twoirreducibleabelianvarietiesofdi(cid:11)erentdimensions,thensuchadecompositionisunique uptoisomorphism. Therefereepointed out tousthereferencetoShioda’scounterexample [Fac. Sc. Univ. Tokio 24, 11-21(1977)] of three nonisomorphic elliptic curves C1, C2, C3 such that C1(cid:2)C2 ’C1(cid:2)C3, which shows that the assumption of di(cid:11)erent dimensions is essential. Documenta Mathematica 5 (2000)23{47 30 A. Iliev and D. Markushevich 2. Abel{Jacobi map and vector bundles on a cubic threefold Let X be a smooth cubic threefold. The authors of [MT] have associated to everynormalellipticquinticcurveC (cid:26)X astablerank2vectorbundleE =E , C unique up to isomorphism. It is de(cid:12)ned by Serre’s construction: (3) 0(cid:0)!O (cid:0)!E(1)(cid:0)!I (2)(cid:0)!0; X C where I = I is the ideal sheaf of C in X. Since the class of C modulo C C;X algebraicequivalenceis5l,wherelistheclassofaline,thesequence(3)implies that c (E)=0;c (E)=2l. One sees immediately from (3) that detE is trivial, 1 2 and hence E is self-dual as soon as it is a vector bundle (that is, E_ ’E). See [MT, Sect. 2] for further details on this construction. Let H(cid:3) (cid:26) Hilb5n be the open set of the Hilbert scheme parametrizing normal X ellipticquinticcurvesinX,andM (cid:26)M (2;0;2)theopensubsetinthemoduli X spaceof vectorbundles onX parametrizingthose stablerank2vectorbundles which arise via Serre’s construction from normal elliptic quintic curves. Let (cid:30)(cid:3) : H(cid:3)(cid:0)!M be the natural map. For any reference curve C of degree 5 0 in X, let (cid:8)(cid:3) : H(cid:3)(cid:0)!J2(X), [C] 7! [C (cid:0)C ], be the Abel{Jacobi map. The 0 following result is proved in [MT]. Theorem 2.1. H(cid:3) and M are smooth of dimensions 10 and 5 respectively. They are also irreducible for generic X. There exist a bigger open subset H(cid:26) Hilb5n in the nonsingular locus of Hilb5n containing H(cid:3) as a dense subset and X X extensions of (cid:30)(cid:3);(cid:8)(cid:3) to morphisms (cid:30);(cid:8) respectively, de(cid:12)ned on the whole of H, such that the following properties are veri(cid:12)ed: (i) (cid:30) is a locally trivial (cid:12)ber bundle in the (cid:19)etale topology with (cid:12)ber P5. For every [E]2M, we have h0(E(1))=6, and (cid:30)(cid:0)1([E]) (cid:26)H is nothing but the P5 of zero loci of all the sections of E(1). (ii)The(cid:12)bersof(cid:8)are(cid:12)niteunionsof thoseof(cid:30), andthemap(cid:9):M(cid:0)!J2(X) in the natural factorization (cid:8)=(cid:9)(cid:14)(cid:30) is a quasi-(cid:12)nite (cid:19)etale morphism. Now, we will give another interpretation of the vector bundles E . Let us C represent the cubic X = V as a section of the Pfa(cid:14)an cubic (cid:4)0 (cid:26) P14_ and 3 keep the notation of 1.3. Let K be the kernel bundle on X whose (cid:12)ber at M 2 X is kerH. Thus K is a rank 2 vector subbundle of the trivial rank 6 vector bundle EX = E (cid:10)C OX. Let i : X(cid:0)!P14 be the composition Pl(cid:14)Cl, whereCl:X(cid:0)!G(2;E) isthe classifyingmapof K(cid:26)E ,andPl:G(2;E),! X P(^2E)=P14 the Plu(cid:127)ckerembedding. Theorem 2.2. For any vector bundle E obtained by Serre’s construction start- ing from a normal elliptic quintic C (cid:26)X, there exists a representation of X as alinear section of (cid:4)0 suchthatE(1)’K_ andall theglobal sections of E(1)are the images of the constant sections of E_ via the natural map E_(cid:0)!K_. For X X generic X;E, sucha representationisuniquemodulo theaction of PGL(6)and the map i can be identi(cid:12)ed with the restriction v j of the Veronese embedding 2 X v :P4(cid:0)!P14 of degree 2. 2 Documenta Mathematica 5 (2000)23{47 The Abel{Jacobi Map for a Cubic Threefold ::: 31 Proof. Let C (cid:26) X be a normal elliptic quintic. By Theorem 1.1, there exists a V =G\(cid:3) together with a birational isomorphism X 99KV . Proposition 14 14 1.7 implies that X and V are associated to each other. By Proposition 1.4, 14 we have C = fH_ 2 X j c (cid:26) Lg = Cl(cid:0)1((cid:27) (L)), where (cid:27) (L) denotes H 11 11 the Schubert variety in G parametrizing the lines c (cid:26) P(E) contained in L. It is standard that (cid:27) (L) is the scheme of zeros of a section of the dualized 11 universal rank 2 vector bundle S_ on G. Hence C is the scheme of zeros of a section of K_ = Cl(cid:3)(S_). Hence K_ can be obtained by Serre’s construction from C, and by uniqueness, K_ ’E (1). C By Lemma 2.1, c) of [MT], h0(E (1)) = 6, so, to prove the assertion about C global sections, it is enough to show the injectivity of the natural map E_ = H0(E_)(cid:0)!H0(K_). The latter is obvious, because the quartic da Palatini is X not contained in a hyperplane. Thus we have E_ =H0(K_). For the identi(cid:12)cation of i with v j , it is su(cid:14)cient to show that 2 X i is de(cid:12)ned by the sections of O(2) in the image of the map ev : (cid:3)2H0(E(1))(cid:0)!H0(det(E(1))) = H0(O(2)) and that ev is an isomorphism. This is proved in the next lemmas. The uniqueness modulo PGL(6) is proved in Lemma 2.7. Lemma 2.3. Let Pf : P14 99K P14 be the Pfa(cid:14)an map, sending a skew- 2 symmetric 6(cid:2)6 matrix M to the collection of its 15 quadratic Pfa(cid:14)ans. Then Pf22 =idP14, the restriction of Pf2 to P14n(cid:4) is an isomorphism onto P14nG, and i=Pf j . 2 X Thus Pf is an example of a Cremona quadratic transformation. Such trans- 2 formations were studied in [E-SB]. Proof. Let(e ),((cid:15) )bedualbasesofE;E_ respectively,and(e =e ^e );((cid:15) ) i i ij i j ij the correspondingbasesof ^2E, ^2E_. Identify M in the sourceof Pf with a 2 2-formM = a (cid:15) . ThenPf canbegivenbytheformulaPf (M)= 1 M^ ij ij 2 2 2!4! M e123456, wPhere e123456 =e1^:::^e6, and stands for the contraction of tensors. Notice that Pf sends2-formsof rank6,4,resp. 2tobivectorsof rank 2 6,2, resp. 0. Hence Pf is not de(cid:12)ned on G0 and contracts (cid:4)0 nG0 into G. In 2 fact, the Pfa(cid:14)ansof a 2-formM of rank4 areexactly the Plu(cid:127)ckercoordinates of kerM, which implies i=Pf j . 2 X In order to iterate Pf , we have to identify its source P(^2E_) with its target 2 P(^2E). We do it in using the above bases: (cid:15) 7! e . Let N = Pf2(M) = ij ij 2 b (cid:15) . Then each matrix element b = b (M) is a polynomial of degree 4 ij ij ij ij Pin (akl), vanishing on (cid:4)0. Hence it is divisible by the equation of (cid:4)0, which is the cubic Pfa(cid:14)an Pf(M). We can write b = ~b Pf(M), where ~b are some ij ij ij linearformsin(a ). Testingthemonacollectionofsimplematriceswithonly kl one variable matrix element, we (cid:12)nd the answer: Pf (M) = Pf(M)M. Hence 2 Pf is a birational involution. 2 Documenta Mathematica 5 (2000)23{47 32 A. Iliev and D. Markushevich Lemma 2.4. Let l(cid:26)V be a line. Then i(l) is a conic in P14, and the lines of 3 P5 parametrized by the points of i(l) sweep out a quadric surface of rank 3 or 4. Proof. The restriction of Cl to the lines in V is written out in [AR] on pages 3 170(for anon-jumpingline of K, formula (49.5))and171 (forajumping line). These formulas imply the assertion; in fact, the quadric surface has rank 4 for a non-jumping line, and rank 3 for a jumping one. Lemma 2.5. The map i is injective. Proof. Let (cid:4)~ be the naturaldesingularizationof (cid:4)0 parametrizingpairs(M;l), where M is a skew-symmetric 6(cid:2)6 matrix and l is a line in the projectivized kernel of M. We have (cid:4)~ = P(^2(E =S)), where S is the tautological rank 2 X vector bundle on G=G(2;6). (cid:4)~ has two natural projections p: (cid:4)~(cid:0)!G(cid:26)P14 and q : (cid:4)~(cid:0)!(cid:4)0 (cid:26) P14_. The classifying map of K is just Cl = pq(cid:0)1. q is isomorphic over the alternating forms of rank 4, so q(cid:0)1(V )’V . p is at least 3 3 bijective on q(cid:0)1(V ). In fact, it is easy to see that the (cid:12)bers of p can only 3 be linear subspaces of P14. Indeed, the (cid:12)ber of p is nothing but the family of matricesM whosekernelcontainsa(cid:12)xedplane,henceitisalinearsubspaceP5 of P14_, and the (cid:12)bers of pjq(cid:0)1(V3) are P5\V3. As V3 does not contain planes, the only possible (cid:12)bers are points or lines. By the previous lemma, they can be only points, so i is injective. Lemma 2.6. i is de(cid:12)ned by the image of the map ev : (cid:3)2H0(E(1)) (cid:0)! H0(det(E(1)))= H0(O(2)) considered as a linear subsystem of jO(2)j. Proof. Let (x = (cid:15) ) be the coordinate functions on E, dual to the basis (e ). i i i The x can be considered as sections of K_. Then x ^x can be considered i i j eitherasanelementx of^2E_ =^2H0(K_), orasasections of^2K_. For ij ij apointx2V ,thePlu(cid:127)ckercoordinatesofthecorrespondingplaneK (cid:26)E are 3 x x ((cid:23)) for a non zero bivector (cid:23) 2 ^2K . By construction, this is the same as ij x s (x)((cid:23)). This proves the assertion. ij Lemma 2.7. Let X(cid:0)~!(cid:4)0 \(cid:3) , X(cid:0)~!(cid:4)0 \(cid:3) be two representations of X as 1 2 linear sections of (cid:4)0, K ;K the corresponding kernel bundles on X. Assume 1 2 that K ’ K . Then there exists a linear transformation A 2 GL(E_) = GL 1 2 6 such that (cid:4)0\^2A((cid:3) ) and (cid:4)0\(cid:3) have the same image under the classifying 1 2 maps into G. The family of linear sections (cid:4)0\(cid:3) of the Pfa(cid:14)an cubic with the same image in G is a rationally 1-connected subvariety of G(5;15), generically of dimension 0. Proof. TherepresentationsX(cid:0)~!(cid:4)0\(cid:3) ,X(cid:0)~!(cid:4)0\(cid:3) de(cid:12)netwoisomorphisms 1 2 f :E_(cid:0)!H0(K ),f :E_(cid:0)!H0(K ). IdentifyingK ;K ,de(cid:12)neA=f(cid:0)1(cid:14)f . 1 1 2 2 1 2 2 1 Assume that (cid:3) = ^2A((cid:3) ) 6= (cid:3) . Then the two 3-dimensional cubics (cid:4)0 \(cid:3) 1 2 and (cid:4)0 \(cid:3) are isomorphic by virtue of the map f = f (cid:14)f(cid:0)1(cid:14)(^2A)(cid:0)1. By 2 2 1 construction,wehavekerM =kerf(M)foranyM 2(cid:4)0\(cid:3). Hence(cid:4)0\(cid:3)and Documenta Mathematica 5 (2000)23{47