We show that if A is a d-dimensional abelian variety in a smooth quadric of
dimension 2d then d=1 and A is an elliptic curve of bidegree (2,2) on a
quadric. This extends a result of Van de Ven which says that A only can be
embedded in P^{2d} when d=1 or 2.Comment: 5 page

Garcia, Luis Fuentes

English

arXiv

Fuentes García, Luis

Numérisation de Documents Anciens Mathématiques

REND. SEM. MAT. UNIV. PADOVA, Vol. 111 (2004)A Note on Abelian Varieties Embedded in Quadrics.LUIS FUENTES GARCÍA (*)ABSTRACT - We show that if A is a d-dimensional abelian variety in a smoothquadric of dimension 2d then d41 and A is an elliptic curve of bidegree (2 , 2 )on a quadric. This extends a result of Van de Ven which says that A only canbe embedded in P 2d when d41 or 2 .1. Introduction.Let A be a d-dimensional abelian variety embedded in PN . It is wellknown that 2dGN . Moreover, in [8] Van de Ven proved that the equali-ty holds only when d41 or 2 .It is a natural question to study the possibilities for d when theabelian variety A is embedded in any other smooth 2d-dimensional vari-ety V . In particular, here we study the embedding in smooth quadrics.We obtain the following result:THEOREM 1.1. If A is a d-dimensional abelian variety in a smoothquadric of dimension 2d then d41 and A is an elliptic curve of bide-gree (2 , 2 ) on a quadric.We will use similar methods to Van de Ven’s proof. The calculation ofthe self intersection of A in the quadric and the Riemann-Roch theoremfor abelian varieties allow only the cases d41, 2 , 3 .(*) Indirizzo dell’A.: E.T.S. de Ingenieros de Caminos, Canales y Puertos,Universidad de A Coruña, Campus de Elviña, 15192 A Coruña, Spain. E-mail:luisfgHiccp.udc.esSupported by EAGER.Luis Fuentes García128The case d41 is the classical elliptic curve of type (2 , 2 ) containedin the smooth quadric of P 3 .When d42, A is an abelian surface in P 5 . We see that is the projec-tion of an abelian surface A 8%P 6 given by a (1 , 7) polarization. By a re-sult [7] due to R. Lazarsfeld, this is projectively normal and it is not con-tained in quadrics. Therefore, A is not contained in quadrics either.Finally, two results [5], [6] of J. N. Iyer allow us to discard the cased43.2. Proof of the Theorem.Let j : A %KQ be an embedding of a d-dimensional abelian varietyinto a 2d-dimensional smooth quadric, with dD1. The Chow ring of thesmooth quadric in codimension d is generated by cocycles a and b withthe relations a 24b 241, a . b40. Thus, A will be equivalent to aa1bbandA . A4a 21b 2 .(1)On the other hand, by the self-intersection formula ([4], pag 431) wehave A . A4 j* cd (NA , Q ). To obtain cd (NA , Q ), let us consider the normalbundle sequence:0 KTAK j * TQKNA , QK0Since the tangent bundle of an abelian variety is trivial, we see thatc(NA , Q ) 4 j *(c(TQ ) ). We compute the class of the tangent bundle of aquadric in the following lemma:LEMMA 2.1. Let i : Q %KPn11 be an n-dimensional smoothquadric in Pn11 . Thenc(TQ ) 4 (11H)n12 (112 H)21where H 4 i * H and H is a hyperplane in Pn11 .PROOF. We have an exact sequence:0 KTQK i * TP n11KNQ , P n11K0Since Q is a hypersurface NQ , P n11`OQ (Q) `OQ (2 H) and the totalclass of the normal bundle is c(NQ , P n11 ) 4112 H. On the otherA note on Abelian varieties etc. 129hand, it is well known that c(TP n11 ) 4 (11H)n12 . Now, from thesplitting principle the claim follows. rLet us apply this lemma to the previous situation. We obtainc(NA , Q ) 4 (11h)2d12 (112h)214 !k402d12g2d12kh h k !l40Q(22h)lwhere h4 j *H. In particular, the top class iscd4Fd h d , with Fd4 !k40d g2d12kh (22)(d2k) .Substituting this into the self-intersection formula, we have:A . A4Fd j*( j *Hd ) 4Fd Hd . j* A4Fd (aa1bb). Hd4Fd (a1b) .Combining this expression with (1) we obtain the following relationa 21b 24Fd (a1b)(2)or equivalently,ga2 Fd2h21 gb2 Fd2h24 Fd22.We are interested in bounding the degree of A , when (a , b) satisfy thisequation. Note that this is a circle of center g Fd2, Fd2h and radius Fdk2.Since deg (A) 4a1b , it is clear that the maximal degree is reachedwhen (a , b) 4 (Fd , Fd ), that is,deg (A) G2Fd .On the other hand, the abelian variety is embedded in Q%P 2d11 . WhendD2, by Van de Ven’s Theorem, it spans P 2d11 . Furthermore, by theRiemann-Roch theorem for abelian varieties, we know that h 0 (OA (h) ) 44 deg (A)d!. Thus, we have the following inequality:deg (A) F2(d11) !Comparing the two bounds we see that a sufficient condition for the non-Luis Fuentes García130existence of the embedding j is FdE (d11) ! . Now,Fd4 !k40d g2d12kh (22)(d2k)G !k40d g2d12kh (2)dG2d 22d11423d11 .We see that (d11) !D23d11FFd when d417. A simple inductive argu-ment shows that this holds if dF17.If dG17, using the exact value of Fd , we see that (d11) !DFd forany dD3.We conclude that the unique possibilities are d42 or d43.First, suppose that A is an abelian surface contained in a quadric.F247 and we can check that the unique positive integer solution of theequation (2) is a4b47. Thus A must be an abelian surface of degree 14given by the polarization (1 , 7 ). Note that A%Q%P 5 is not linearlynormal, that is, it is the projection of a linearly normal abelian surfaceA 8%P6. The quadric Q can be lifted to a quadric containing the surface A 8 .Lazarsfeld proved in [7] that a very ample divisor of type (1 , d) withdF13 or d47, 8 , 9 is projectively normal. From this the following se-quence is exact:0 KH 0 (IA 8 , P 6 (2 ) ) KH 0 (OP 6 (2 ) ) KH 0 (OA 8 (2) ) K0 .Since h 0 (OP 6 (2 ) ) 4h 0 (OA 8 (2) ) 428, there are not quadrics containingthe abelian surface A 8 and we obtain a contradiction.Finally, suppose that d43. Now, F3424 4 (311) ! , so the degree ofthe abelian variety is exactly 2F3448. The line bundle OA (h) corre-sponds to a divisor of type (1 , 1 , 8 ) or (1 , 2 , 4 ). But J.N.Iyer prove in [5]that a line bundle of type (1 , R , 1 , 2d11) is never very ample. More-over, in [6] she studies the map defined by a line bundle of type (1 , 2 , 4 )in a generic abelian threefold. She obtains that it is birational but not anisomorphism onto its image. Note that the very ampleness is an opencondition for polarized abelian varieties (see [2]). It follows that a linearsystem of type (1 , 2 , 4 ) cannot be very ample on any abelian threefoldand this completes the proof.REMARK 2.2. In [1] C. Ciliberto and V. Di Gennaro obtain moregeneral results about subvarieties of low codimension. In particularthey give a bound for the degree of a d-dimensional abelian variety em-bedded on a smooth hypersurface of dimension 2d .A note on Abelian varieties etc. 131REMARK 2.3. The sequence Fd4 !k40d g2d12kh (22)(d2k) is related tothe Fine numbers. For a reference see [3].Acknowledgement. I thank K. Hulek for suggesting me this problemand for his advice and interest. I am grateful to the Institut für Mathe-matik of Hannover for its hospitality and especially to E. Schellhammerfor his patience. I want also thank E. Deutsch for his remark aboutbounding Fd and its relation with the Fine numbers.R E F E R E N C E S[1] C. CILIBERTO - V. DI GENNARO, Boundedness for low codimensional subvari-eties, to Appear in Proceedings of Conference Held at the ESI of Wien in Oc-tober 2000 (Editor Prof. Popov).[2] O. DEBARRE - K. HULEK - J. SPANDAW, Very ample linear systems on abelianvarieties, Math. Ann., 300 (1994), pp. 181-202.[3] E. DEUTSCH - L. SHAPIRO, A survey of the Fine numbers, Discrete Math., 241(2001), pp. 241-265.[4] R. HARTSHORNE, Algebraic Geometry, GTM, 52, Springer-Verlag, 1977.[5] J. N. IYER, Linear systems on abelian varieties of dimenson 2g11, Proc.Am. Math. Soc., 130, no. 4 (2002), pp. 959-962.[6] J. N. IYER, Line bundles of type (1 , R , 1 , 2 , R , 2 , 4 , R , 4 ) on abelian va-rieties, Int. J. Math., 12, no. 1 (2001), pp. 125-142.[7] R. LAZARSFELD, Projectivite normale des surface abeliennes, Redige par O.Debarre, prepublication no. 14, Europroj - C.I.M.P.A., Nice (1990).[8] A. VAN DE VEN, On the embedding of Abelian varieties in projective spaces,Ann. Mat. Pura Appl., IV. Ser., 103 (1975), pp. 127-129.Manoscritto pervenuto in redazione il 14 maggio 2003.

A note on abelian varieties embedded in quadrics

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A note on Abelian varieties embedded in quadrics

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