Nonminimal rational curves on K3 surfaces

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(1)Article. Nonminimal rational curves on K3 surfaces. CORAY, Daniel, MANOIL, Constantin, VAISENCHER, Israel. Reference CORAY, Daniel, MANOIL, Constantin, VAISENCHER, Israel. Nonminimal rational curves on K3 surfaces. Enseignement mathématique, 1997, vol. 43, no. 3/4, p. 299-317. Available at: http://archive-ouverte.unige.ch/unige:12783 Disclaimer: layout of this document may differ from the published version..

(2) L'Enseignement Mathématique. CORAY, Daniel / Manoil, Constantin / VAINSENCHER, Israel. NONMINIMAL RATIONAL CURVES ON K3 SURFACES. Persistenter Link: http://dx.doi.org/10.5169/seals-63282 L'Enseignement Mathématique, Vol.43 (1997). PDF erstellt am: 1 déc. 2010. Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung des Konsortiums der Schweizer Hochschulbibliotheken möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag.. SEALS Ein Dienst des Konsortiums der Schweizer Hochschulbibliotheken c/o ETH-Bibliothek, Rämistrasse 101, 8092 Zürich, Schweiz [email protected] http://retro.seals.ch.

(3) NONMINIMAL RATIONAL CURVES ON K3 by Daniel CORAY,. SURFACES. Constantin MANOIL and Israel VAINSENCHER. Introduction The following assertion was made, in 1943, by B. Segre ([Se]): The gênerai quartic surface F contains a finite number (S) unicursal (Le., rational) curves of degré e Ah (for h— 1,2,3, ...).. c. >0. h. This was prompted by the opposite claim made by W. Fr. Meyer turn of the century ([Me], §3, pp.. (M) R. On. the. 1545-47):. generic (quartic surface). F4F. 4. there can lie no (rational curve). (m=1,2,...).". m. The degree. smooth makes. commonly used to mean: a rational curve of m, but it is not clear whether Meyer intended to limit his statement to rational curves. In fact, the argument he gives in support of his claim reasonable sensé for smooth curves: it takes 4m .+ conditions to that a quartic surface contains a smooth rational curve of degree m, notation. ip: P. R. m. was. 1. express but thèse. on. a. at. of. curves 3. 1. —>. P3P. dépend. on. 4m. constants. only.. Indeed,. a. parametrization. defined by four homogeneous polynomials of degree. 4(m+l) coefficients, which 3. are. arbitrary up. to. multiplication. by. m. dépends. a. common. 1. automorphisms of préserve the image of such a map. The independence of the conditions so exprès sed would need to be thoroughly examined. But, with this interprétation, assertion (M) does hold. scalar; and the. oo. and can be derived from. P. celebrated theorem of Max Noether, which is very well established (see [De], thm. 1.2): a generic quartic surface (in a spécifie a.

(4) sensé) has no other divisors than its hypersurface sections. genus of such divisors. Nevertheless, in. P3P. with. 3. never zéro.. Segre noticed, Meyer's statement is certainly false when. as. allowed. Indeed,. well-known that. gênerai quartic surface has 3200 tritangent planes. Each of them meets the surface in a quartic double points, which has géométrie genus zéro.. singularises 3. is. and the arithmetic. ;. It is. are. interesting. Mumford. and by. it is. compare with. to. Bogomolov. (see. similar statement proved. a. [M-M]). THEOREM (Bogomolov & Mumford).. contains. singular rational curve and. a. a. a. 1979 by. in. :. Every algebraic K3 surface over. C. pencil of singular elliptic curves.. Actually, they proved that a complète linear System of curves of minimal arithmetic genus (greater than one) on the surface contains at least one irre duciblerational member. For a smooth quartic in P which is a spécial case 3. ,. of K3 surface, this deals with the relatively uninteresting case where. h. =. 1.. Assertion (S) would be easy to establish if we knew that every complète linear System of curves of arithmetic genus greater than one on a K3 surface has. irreducible rational member. The main innovation of this paper is that, for a restricted family of K3 surfaces, we show the existence of singular rational irreducible members in some complète linear Systems which are not minimal. More precisely, we give a proof of Segre's assertion (S) for h — 2 and h — 3 (Theorem 3.4). In §4 we establish a similar resuit for K3 surfaces 4 in (Theorem 4.1). However, for reasons explained before Lemma 2.5, we an. P4P. hâve not been able to prove assertion (S) for S. > 3.. h. somewhat surprising. Sometimes the be very easy, and sometimes very hard. We shall try to. What happens in reality. CHOU A.. is. problem seems to explain hère why this is so. First we recall that the set of space curves of a given degree can be viewed as a variety, by a construction usually attributed to Chow, though much of the idea goes back to Cayley ([Ca]). (See [Sh] for a brief, but enlightening discussion, and [H-P] for an almost exhaustive treatment.) N be a projective variety of dimension n. We In a few words, let V cP P^. dénote by. of ail points can in. (/i 0. prove (cf. ÇPNP. N ). nn. +. l. is. (P^) /2+l xV contains x. One closed set whose projection. space of P^ and consider the set ,K,x) such that every hyperplane h. the dual ,. .. •. [Sh],. .. Chap.. defined by. 1, a. §6) that. A. single équation. is. a. Fy. .. Ac. t. The form. Fy. is. called the. Cayley form associated with V, and its coefficients are the coordinates of the.

(5) Chow point of V. If - instead of looking merely at varieties - one considers ail cycles of a given degree m and dimension n, one shows that the Chow points form a projective algebraic variety. This is called the Chow variety of P^. ail cycles in. Scholium. of degree m and dimension It. 1.. is. known. that. the. n. .. variety of space curves of of dimension 4m, whose gênerai. Chow. degree m has an irreducible component 7l m élément is the Chow point of a smooth rational curve (cf. ,. Moreover, any irreducible space curve with degree zéro belongs to it.. m. [Co2], Lemma 2.4). and. géométrie genus. We dénote by Tk the projective space (of dimension 34) parametrizing ail 3 For each value of m we can consider the incidence quartic surfaces in P .. ,. correspondence X m C 7Z m xTK consisting of where (7) dénotes the support of the -cycle Now a smooth quartic curve F of genus 0 quadric surface. From this it is easy to compute 1. 17. surfaces. FG. ail pairs. (7. F). with. whose Chow point in. 3 P3P. is. CF,. (7). is. contained in. 7Z. 7. G. a. unique. m. that it is contained in precisely. conditions coming from the Bézout theorem are independent; indeed, given any set of 16 points on F, there is a union of two quadrics through them which does not contain F.) Thus the incidence correspondence 2" 4 has dimension 16+ 17 =33 above some nonempty open oo. TKT. K. (The. -. 17. subset of 7^4.. But. if we look. 3. family of plane quartics in P we see that its dimension is 14 + 3 = 17 (one more than the dimension of K4 !). It can be shown that those having 3 double points form a family of dimension (14. -. +. 3). curve. 3. =. the. at. 14.. ,. for. Now,. a. F, it is enough to impose. quartic. surface to contain. such. a. singular. double points, simple points and the since this also represents 11 + 2-3 =4-4+ l intersections. Thus F is contained in oo- 20 surfaces Fe T K It follows that the incidence correspondence 4 has a component of dimension > 14 + 20 = 34 above the singular plane 11. 3. X. .. quartics.. 1. ). Hence not only than its. 4X. is. reducible, but. it has. component of larger dimension dimension over the generic point of the Chow variety 7£ 4 X. 4X. 4. a. !. ) In fact, equality holds. The référée suggests the following argument: as a complète intersection, any plane quartic is arithmetically Cohen-Macaulay of arithmetic genus 3. Therefore it imposes exactly h°(O (4)) =14 conditions on quartic surfaces. There is also a family of rational curves of degree 4 and arithmetic genus 1, namely the rational reduced and irreducible quartic curves which are complète intersections of two quadric 18 surfaces (and hâve a double point). Thèse curves are contained in surfaces of degree 4. oo As a matter of fact, this family of curves S C 7l has dimension 15. So, the fibres above 4 S span a 33-dimensional variety, which is also a component of Z 4 J. r(4))r. .. ..

(6) This explains why we can say that both Segre and Meyer were right, in some sensé. :. they referred to the images in Tk of différent components of X m. We proceed with an can also get. a. informai discussion of the. pretty clear picture. 2, for. h —. case. .. which one. :. ([No], §17) for a discussion of space curves of degree 8. Noether uses several criteria ) to establish that a gênerai smooth rational octic Y is contained in no more than two independent quartic surfaces ) F 4. Thus the incidence correspondent Zg has dimension SCHOLIUM. We refer to Max Noether. 2.. 2. 3. 32+1 =33 above. some nonempty open subset of IZs and could not possibly. surjectively onto Tk if it were irreducible. This Meyer's assertion for degree 8.. is. map. in. agreement with. However, the complète intersections of a quartic and a quadric hâve the right dimension (33, which is one more than dimT^g)- Our task will consist in showing that those with 9 double points form a subfamily of 7£g of dimension. —9=. 33. 24. Again, since thèse curves are contained in oo. we hâve to do with. a. 10. quartic surfaces,. component of larger dimension than the one above the. will then show that this component maps onto dense constructible subset of Tkgênerai point of. 7£g. .. We. a. yet another heuristic way to understand why the dimension is one more than normally expected In Kg one can obtain a curve Y with 9 double. Hère. is. :. points. by. imposing only. 8. nodes.. Indeed, any quadric passing through the 8 nodes and one more point of Y has 2-8 +1 = 17 intersections with Y Hence Y is contained in an irreducible .. quadric. <2o-. Since we are moving in 2g of arithmetic. hence. Qo,. on. C. 7Z%. genus. 9.. x. Tk, But. the Y. singularises are ordinary double points. Hence ninth singular point. 2. divisor rational. is. Y. Y. is. and. of type the. (4,4). assigned. automatically acquires. a. a smooth quartic F containing F, any other quartic surface F' The linear System IF'l has dimension 0. Indeed, residual cuts out curve V through 2 2 2 whence (F') = (0(4) - F) = -2, so that F' is an isolated divisor. p a (T) =o=> (F) Hence any other quartic through F belongs to the pencil generated by F and F. ). For. F. instance,. on. a. .. =-2;. 3. ) One cannot leave out the word 'gênerai'. Indeed one also finds some smooth rational curves of degree 8 on any smooth quadric, where they correspond to the divisors of type (1,7). 9 Thèse curves are therefore contained in oo (reducible) surfaces of degree 4. As a matter of fact, this family of curves S C T^g has dimension 24. So the fibres above S span a 33-dimensional variety, which is also a component of X% ..

(7) About finiteness. 1.. starting point was that a complète linear System of arithmetic genus g on a smooth quartic is also of dimension g. Now, demanding that the System hâve a member with g ordinary double points should represent g independent conditions. Hence, for every positive integer h,. heuristic. Segre's. there should exist. a. finite number of rational irreducible curves in the complète. linear System eut out by the family of ail surfaces of degree h.. argument is unsatisfactory as it stands, because there are always inflnitely many reducible curves with the right number of nodes (unless h = ). For instance, for h = 2 the curve eut by the union of a tritangent plane and any one of the inflnitely many biîangent planes also has nine double points. This. 1. In. Segre did try to. his. claim, but in. totally unconvincing way. There is no discussion of the irreducibility of the relevant incidence correspondences. What is more, the argument dépends on Computing self intersectionnumbers on a rather unspecified System of quartic surfaces, ail of which might in particular be singular. However, this part of Segre's argument can be replaced by the following lemma. We always work over an algebraically closed ground field k of characteristic 0. fact,. LEMMA 1.1.. justify. a. carnes any one-dimensional (non-constant) algebraic System of irreducible rational ciwves, whether smooth or singular. No K3 surface. A smooth rational curve. Proof.. zéro, and hence. (F). 2. =. F. -2. Therefore. aK3 surface. on. F. is. arithmetic genus not even numerically équivalent has. other irreducible curve. This yields the assertion for smooth curves. As for the singular case, a proof is sketched in [G-G] (Lemma 4.2), but we supply more détails. Let B be a parameter curve for an algebraic System of. to any. curves on. a. K3 surface X, whose generic member. is. irreducible and rational.. Without loss of generality. B is irreducible and even smooth, since we are free to replace it by its normalization. Let J c X x B be the subvariety of codimension corresponding to the algebraic System. Then, by [Sh] (Chap. 1, 1. §6, Thm.. and observe. Let. J. 8),. be. is. that p. irreducible. We dénote is. p:. dominant, unless the family. generic point of Byjissumption, the fibre F of r). by. a. J is. -> X the first. projection. constant.. B. over the ground field k,. J. above. so. that k(rj) = k(B).. rational over the algebraic closure k(rj) of k(rj). So, by a resuit which goes back to Hilbert and Hurwitz, is birationally équivalent over k(rj) to a smooth conic. Now, k is algebraically closed; so, if r is a variable then k(t) is aCx field {cf. [La], Thm. 6). 7?. 77. is. T?1T. ?1.

(8) As the function field of some curve, k(rj) is an algebraic extension of k(t) hence it is also C\ ([La], pp. 376-377). So, every conic defined over k(rf) has points defined over this field and is birationally équivalent to This. ;. P^.. shows that £(77X1^) is isomorphic to k(rj)(t). &-isomorphisms. Hence. there. is. Therefore we hâve the following. .. :. a. birational. équivalence. <p:. B. x. P. 1 >. J. Consider. the. 1. >X. Since q is dominant, composite rational map q = p o cp: B x P and X projective, we know (cf. [Sh], Chap. 3, §5, Thm. 2) that g* embeds the regular differentials (of any rank) on X into those on B x Since X is a K3 surface, we note that cj^ is trivial, and hence h°((Jx) = 1 1. Pl.P. .. On applying. g*. we see that. Indeed, if we dénote by p\ respectively, we hâve. h°(B. x. P^c^xpi). the. and pi. 0.. But this. projections from. is. BxPl. .. impossible. to B. and P. 1. :. On. the. other hand,. coherentsheaves the a contradiction.. H°(F\uj f i). H°(F\. i(-2)). =0,. and. for. quasi global section functor commutes with tensor products; =. opi(-2))0. p. Remark. Lemma 1.1 does not imply that a given K3 surface cannot contain infinitely many smooth rational curves; see [SwD], §5, for an example.. 2.. About existence. Finiteness statements are useless if they are not accompanied by some form In the présent of existence assertion. After ail, zéro is also a finite number section we show the existence of irreducible rational curves, of degree 8 or !. at. 12,. least on some smooth quartic. surfaces. For degree. 8. there. is. a. very. elementary proof, and we give it first. Then we shall proceed to the case of degree 12, which requires some more elaborate machinery. As mentioned above, on a quartic surface it is easy to find some reducible curves of degree 8 with nine double points by considering unions of two plane sections. Such curves are even infinité in number, but they do not lie on any smooth quadric. 4 ) That is why we start with a very explicit construction on 4. By the way, this may be one reason for working with the Chow variety rather than with scheme. Thèse degenerate cases hâve the same arithmetic genus, but they do not lie a Hilbert in n%. ).

(9) the smooth quadric. Thus. which. the image. is. of the standard Segre embedding. given by the équation. is. S. S. Let p:. LEMMA 2.1.. P. 1. -» P. 1. xPI. be. the map defined by. injective morphism, whose image is an irreducible rational curve Y of type (4.4). Under the standard Segre embedding, Y is the intersection of S with the quartic surface T defined by Then. is an. p. Of course, Une. £. =. T. is. a. with vertex. P. = (0. = o}. Howevei;. Y. is. cône. {X=Y-Z. smooth quartic surface of the form. F + H. 0. :. :. 0. 1). :. and has. a. also the intersection of G. =. 0. S. triple with. a. for some quadratic form. HÇC.Y.Z.W). unique singularity (at P), whose effect on the genus is the équivalent of nine double points. As for the last assertion, we state it in more gênerai form. Ail the assertions. Proof. are. easy to. verify.. Y. has. a. :. YCP3. complète intersection of two surfaces defined by F = 0, respectively G = 0. We assume thaï the surface defined by G — 0 is smooth, that Y is reduced, and that deg F > degG. Then ïhere exists a smooth suif ace among those with équation F + H G = 0, where Let. LEMMA 2.2.. be. the. •. deg H. by. —. deg F. —. deg G.. By a theorem of Bertini, the linear System determined by F and Proof ail polynomials of the form HG has no movable singularity in outside base locus. As H runs through the set of ail forms of the relevant degree, base locus is reduced to the points on Y — {F = G = o} P3P. its. the. 3. .. Now, if. P. is. a. singular point of. F-fH•G=oin. the. base. locus, we. dF{P) + H{P) dG(P) = 0. We can think of this as a System of four équations in one variable x= H(P) But the rank of the Jacobian matrix (F .G')p at P is equal to ruled out because the surface defined. see. that. .. f. lor2(ois.

(10) smooth). If it not singular for any H. G. by P. =. is. is. 0. When the rank. is. 2. equal to. then there is no suitable. x; hence. .. unique solution and we get one linear condition in the affine space of the coefficients of H However, this occurs only at the finitely many singular points of F. Since a finite union of hyperplanes does not exhaust the space of parameters, we can choose H is. equal to. 1. ,. there. is. a. .. coefficients lie outside this union. For any such = 0 is smooth on the whole of F.. that its. so. F + H. G. As. further illustration, we show how. a. to. produce. //,. surface. the. example with nine. an. distinct double points. LEMMA 2.3.. Let p:. P. 1. —> P. 1. xPI. the map defined by. be. generically injecîive morphism, whose image is an irreducible rational curve Fof type (4, 4) with precisely 9 distinct ordinary double points. Under the standard Segre embedding, F is the intersection of S with Then. a. smooth quartic surface.. a. In view Proof singularités of. the p0:p. is. p. 0. :. A. fails. ]. -+P. 1. xP. 1. ,. of. Lemma. p.. To. this. 2.2,. main. the. effect,. thing. note. we. that. to a. do. is. to. polynomial. study map. defined by. injective when we hâve the following simultaneous equalities. to be. for two différent values. t. and. r. Therefore. we define. and. Then. po. fails. a{t) = fi(t) = over. C[t]. .. If. to 0. .. po. our assumptions,. r such that injective if, after fixing r, there exists t This involves studying the résultant R(r) of a(t) and f3(t) is generically injective then R(r) is not identically zéro. With be. it. is. a. polynomial of degree. <. 18,. whose roots. describe.

(11) In fact, the pairs of points that are mapped to the double points of F. of po. degree is equal to 18 if we work projectively and consider p instead. the. 9. R(r) = (r. In the présent case we obtain. 2. +. l)(r. 4. where. l)g(r),. +. décomposition into Q -irreducible factors; hence ail the roots are distinct. Furthermore, the degree is equal to 18, which means that the image of the point at infmity is smooth. (For the example of Lemma 2.1, one obtains which means that the whole singularity is concentrated at the image R(r) =. This. is. a. 1. ,. of the point. at. D. infinity.). The approach we hâve taken for thèse examples can also some gênerai statements. LEMMA (/i, v). on. a. 2.4.. The. serve to prove. :. reduced and irreducible. rational,. smooth quadric in. P3P. 3. are. parametrized. of bidegree. curves. irreducible quasi. by an. projecîivevariety Tl^^ C lZ m of dimension 2m —1, where m= jjl +v. A gênerai point on H^^ corresponds to an irreducible curve whose only singularities are distinct nodes. Any smooth quadric surface. Proof. isomorphic. is. rational irreducible curve of bidegree (/x, v) where p= (((/?o map p: P — P x. a. 1. a. on. P. P. to. xPI. 1. •. ,. •. pairs of homogeneous polynomials, respectively of degree. Pl.P .. Further,. image of consists of two. Pl,P. ». x. is. 1. 1. 1. 1. \i. the. and. i/, varying. 2i/+1 x independently. Thèse maps are parametrized by points of This dermes an incidence correspondence T, with base the open subset V 2iy+1 which 2^ +1 x of parametrizes those p which are generically injective and for which p(P ) is of bidegree (/i, z/). Indeed, the condition that pbe "many-to-one" is équivalent to the vanishing of some résultant polynomial (as in the proof of Lemma 2.3). P2P 2^ +1. P2i/+l.P. .. P2P. P2iy+lP. l. The argument given in [Co2], Lemma 2.4, shows that T is irreducible and that there is a correspondence between V and an irreducible subvariety TL^ M. of. 1Z 1. P. which dermes the same curves. IU. do. (2/i +. the. not modify + (2z/ +. image of. -3 =2m-. a. V. As the. as. map,. the. oo. 3. automorphisms of. dimension of V^^. v. is. equal to. K^. is nonempty. provided For n = v = A this is shown by Lemma 2.1, and the last assertion of the lemma follows from Lemma 2.3.. For. 1). the. 1). gênerai. precisely, take. 2. case,. < p < v. 1. ,. refer. Lemma 2.5 below. More and assume by induction the existence of a nodal,. we. to. [Ta],. as. in.

(12) irreducible, rational curve Y\ of bidegree (/i — \,v). Let Y^ be a line of type (1,0) avoiding the (/x — 2)(i/ - 1) nodes of Y\ We can also assume that it of thèse, in addition to meets Y\ in exactly v distinct points. Assign v .. 1. the nodes. of. connected,. This set of (/i desired.. Y\. as. —. .. l)(z/. —. 1). nodes makes. Fi. +Y2 virtually. We also know from Lemma 2.2 that any reduced curve of type on. smooth surface of degree /j,,. a. difficult. a. (/i, //) lies. fact that will play an important rôle later.. pursue such an explicit approach for the case where h = 3 because the smooth cubic surfaces are not ail alike. We therefore switch to It is. to. ,. déformation. the method of Tannenbaum [Ta]. His results, which are based on. theory, provide the existence - under précise conditions - of rational, reduced and irreducible curves, parametrized by an algebraic scheme. Unfortunately, they fail to apply on surfaces which are not rational. That is why we cannot. immediately generalize our results of degree higher than. to the. intersections of. a. quartic with surfaces. 3.. carries (for any positive Any smooth cubic surface FcP integer X) a rational, reduced and irreducible curve T\ of degree 3À having only nodes for singularities, which belongs to the linear System \X\\ eut out 3. LEMMA 2.5.. by ail surfaces of degree. Since. Proof. the. À. .. surface. F. Tannenbaum ([Ta], §2). The proof. is is. rational, we by induction. we consider the intersection. can on. use. the. results. of. À.. of F with its tangent plane at any point P that does not lie on any of the 27 lines. Then Tp is irreducible. If P is sufficiently gênerai then P has a node at P Indeed, one also obtains. For. À. =. 1. F/>. TPT. .. a. node with the plane sections of F that degenerate into the union of. smooth conic. Thus there are many ways to choose Tp 2 this oo -freedom in the rest of the proof.. and. a. ;. a. line. we shall use. Now suppose the resuit true for À We prove it for À+l. Thus we assume that there exists a rational, reduced and irreducible curve T\ G \X\\ 1} with p a (T\) = 3 A(A ~ +1 distinct nodes, as the genus formula shows. (Indeed, .. 22. F is embedded in. We apply is. a. P3P. 3. by its. anticanonical sheaf.). [Ta], Prop. 2.11, to the reduced curve. with Tp are. = T\ UF/>, where. TPT. P. intersection points among the nodes of 7, which therefore totals. sufficiently gênerai rational plane section. Then. of T\. Y. the. 3 À.

(13) belongs to |X A +i| and we may assign 6 = 6-1 of the nodes, leaving out only one of the intersection points of T\ with T P In this way we obtain a flat family 2) relative to which thèse nodes are. nodes. Of course. Y. .. of assigned; and Y is virtually connected with respect to 2), in the sensé [Ta], Def. 2.12. Thus we can apply [Ta], Thm. 2.13, to deduce that a generic +1 nodes is irreducible. By the genus member of |X A +i| with 6=. 3^^. formula, such. rational.. is. curve. a. With this existence resuit we can now state the analogue of Lemma 2.4 for cubics.. LEMMA. FcP. 3. The. belonging. 3A,. —. m. 2.6.. to. rational, reduced and irreducible curves, of degree the linear system \X\\ on a smooth cubic surface. and having only nodes. for singularities. projective,equidimensional scheme Wa. Y. =. 6. p. a. V6. =. (Y). h°(Y)=p We. T\. =. a. A(A ~ 3. 1}. 99. (Y) + degY. dérive. (\X X \',Y),. of. of. \X\\. G. + (cf.. and. dimension. of. other. no. [Col], Lemma. dim|X A. -6. |. singular. has. precisely points. Further,. 1).. irreducible. smooth. a. which. lemma,. preceding. nodes. 1. existence. the. the. quasi. a. irreducible. We apply [Ta], Lemma 2.2, to the rational, reduced and. Proof curve. C 1Z m. parametrized by of dimension m—l. are. h°(Y). =. -1-. parametrizing reduced curves in \X\\ with precisely singularities which are flat déformations of Y in F curve of V {\X\\\Y) is irreducible.. p. a. -scheme. m-1,. =. {Y). nodes. 6. .. k. algebraic and. Of course,. no. other. gênerai. a. Ô. Let V6V. 6. x. 2). F. universal Cartier divisor of the flat family. Since smooth, we can regard 2J as a Weil divisor, and hence as an. C V§. is. be the. x F. incidence correspondence in this product. We prove that 2) is irreducible. Indeed, since 2) -^» V§ is flat, and every fibre is one-dimensional, it follows from [Hart], Chap. 3, Cor. 9.6, that every irreducible component 2) of 2) -. f. has. dimension equal. for every We. /.. to. dim. V$. +1.. But the generic fibre of. dénote. by. 2). ;. the. Now, is. ep. dense. irreducible, irreducible. Hence V$. is. subset. of. so /. =. (p(ïï)ï) =. V$. 1.. corresponding to the irreducible curves, and let V'6 = )- We now apply [H-P], Chap. 11, and conclude that there is an irreducible incidence §6, Thm. 11, to 2) correspondence T between V'6 and an irreducible subvariety WYW Y C TZ m which defines the same curves as 8 open. 2). 7. ;. ,. VB.V. .. Since every. curve parametrized by. dim V^. Taking ail possible irreducible. V§ Y G. is. reduced,. \X\\,. we get. we. hâve. dimWy =. irreducible varieties.

(14) parametrizing ail rational, reduced and irreducible curves belonging to \X\\ and having only nodes for singularities. We define W\ as the union of. MV. thèse varieties. WY. Remarks.. D. .. The schemes parametrizing ail curves of. 1). given géométrie. a. linear System on a rational surface, hâve been well examined (see [Ta], [Ha]). The new feature in Lemma 2.6 is that we "pull them up" to subschemes of the Chow variety lZ m. genus, in. a. .. We can think of. 2). a. smooth cubic surface. as. being. P2P. 2. with six points. blown up. Then, if we consider the effect of blowing-down on the curves of the linear system \X\\, we see that Lemma 2.5 has the following interesting conséquence: in the System of plane curves of degree 3À with six À-fold points, there are some rational, reduced and irreducible curves with only nodes as. further singularities.. Rational curves. 3.. rational. A Lp:. P. 1. —>. P. space. 3. curve. of. degree. quartics. on 8. is. given. as. in. $P^3$. the. image. of. defined by four homogeneous polynomials of degree. ,. 4-9 =. a. 8.. map Such. arbitrary coefficients; hence they are parametrized l Those maps which are generically injective and for which (p(P ) is by 35 of degree 8 correspond to an open subset UC a curve By (p GUwe mean that the coefficients of Lp are in U. maps dépend on. 36. 35 P35.P. .. P35.P. .. curve F is contained in at most one quadric Q. So, it will be convenient to consider the pair (F, Q) instead of F alone. For simplicity, we shall restrict to the case where Q is smooth. We dénote by Cq (resp., C c and Ck) the quasi-projective variety of smooth quadrics (resp., cubics and Such. a. quartics) in. P. 3 .. correspondences between quasi-projective varieties are algebraic and define closed subvarieties : a) the incidence correspondence G C IZ% x Cq parametrizing the rational LEMMA. The. 3.1.. curves of degree. 8. following. on smooth. quadrics. ;. incidence correspondence T C Ti% x Cq x Ck parametrizing the rational, reduced and irreducible curves of type (4, 4) on smooth quadrics which are eut out by smooth quartic surfaces; b). the. c). the. rational,. incidence correspondence H C 7Z% x Cq x Tk parametrizing the reduced and irreducible curves of type (4,4) on smooth quadrics..

(15) Let. Proof.. This that. is. UxP3 xCQ. closed subset of. a. :r)GgnK. <p(u. monomials in. (u. of. f). :. for. fo(p(u. :0G. (m. every. and. 0). :. 1 .. fx^-piii. (^(P. :. C. ). means. coefficients of ail. the. So,. Pl.P. gn#. 1. £*. Indeed,. x. fQ. must vanish (hère. r)). dénote the polynomial équations of g and K). This yields algebraic relations between x and the coefficients of (p, fQ, and /# and jfc. .. irreducible component of F. Call Uj its first projection. By [H-P] (Chap. 11, §6, Thm. II), there exists an irreducible correspondence Ti between Cq x Ck and an irreducible subvariety lZjj of 7Z% which defines the same curves as Uj We define T to be the union of the T\ Let Fj. be. any. i. .. .. Similarly,. define. we. G. {(ip.x. g) GUxP3 xCQ \x. =. H={. (^(P. G. 1. ). C. g}. .. ((f,x, Q. K) GUxP3x£Q xJ^|xG (^(P ) Cgn G and As before, x Cq, respectively // are closed subsets of U x U x x Cq x J^. Making use of irreducible components G/ of G, respectively Hf of //, we find irreducible correspondences Qi and 7Y/ between Cq, respectively Cq x J 7^ and irreducible varieties !Zw C 7^B, respectively G and H as the unions of thèse irreducible T^v,. C 7^B- Again, we define let. Further,. 1. P3P. P3P. .. 3. 3. t. components.. Finally, we note that T ',. G. ,. H,. and. as. closed subsets of quasi-projective. varieties, are quasi-projective. LEMMA 3.2. of dimension. It. is. a. correspondence. Tisa. quasi-projective variety. 34.. Define. Proof. Now, H. The incidence. tt 0. :. G->. >Cq. closed subset of. follows from [Sh] (Chap.. a. 1,. by. 7r. 0. (7, Q). =Q. (for 7G. quasi-projective variety, and §5,. Thm. 3) that. tv^H). is. 7^ 8. FKF. ). K. and. is. similarly. projective.. closed in. G,. and. hence also quasi-projective.. Since. smooth quadric Q is projectively normal and a linear équivalence class on Q is determined by the bidegree, every reduced and irreducible curve F of bidegree (4.4) is eut out, on g, by (at least) one irreducible quartic K. Let. be. a. the Chow. point of T, so that (j.Q.K) G H. Then the fibre of above (7. Q) contains K and ail the reducible quartics through g. Hence is of dimension 10. 7. tt. {. it.

(16) We also know from Lemma 2.2 that. of ail quartics through. over iï{J~). 7T. are. F. is. gênerai member of the linear system smooth. Hence n\ (H) — n{T) and the fibres of a. 10-dimensional.. also. Finally, by Lemma 2.4, ttootti maps onto Cq and ail the fibres of ttoltt^) hâve dimension 15. As Cq is irreducible of dimension 9, any irreducible 7 ) of maximal dimension has dimension 24. Further, the component of fibre of 7T over tï{T) is 10-dimensional, so T has dimension 34. correspondence J C 7Z\2 x £c x £# parametrizing the rational, reduced and irreducible curves which are the complète intersection of a smooth quartic and a smooth cubic surface, is a LEMMA. 3.3.. incidence. The. quasi-projective variety of dimension 34. The. Proof a. F. curve. argument. of degree. convenient. 12. is. much the same. as. for degree. cannot be contained in. 8.. than. more. For instance, one. cubic. C.. consider the pair (F, C) instead of F alone. For simplicity, we restrict to the case where C is smooth. Then Lemma 2.6 replaces Lemma 2.4, and the proof of Lemma 3.2 has to be modified mainly it. So,. is. to. for the actual computation of dimensions, which. is. as. follows.. The quasi-projective variety Ce of smooth cubic surfaces has dimension 19. And the dimension of the family of rational curves FG |X 4 is equal to 11, |. Lemma 2.6. Finally, a curve FCC lying on an irreducible quartic X, is also contained in ail the reducible quartics through C Hence the linear System of ail quartics through F has dimension 4, and a gênerai member is smooth. Putting everything together, we find 19 + 11+4 = 34 for the dimension. by. 5. ). .. of the incidence correspondence We now corne to the. J. .. proof of assertion (S) for. h. =. 2. and. h. =. 3 .. 3. carrying rational, reduced and 12, obtained as intersections with irreducible curves of degree 8, smooth quadrics, respectively cubics, form a constructible set of dimension 34 in Ck- A gênerai quartic in this set carries also some rational curves of THEOREM 3.4.. degree. 8. (resp.,. The smooth. 12 j. quartics in respectively. P3P. having only no de s for singularities.. Let p: T—> Ck be the projection map defined by p(7, Q,K) =K. Then every fibre of p is finite. Indeed, let us consider the first projection. Proof. 5. ) Strictly speaking, this is only a lower bound, since Lemma 2.6 does not count the curves having other singularities than nodes. But the proof of Theorem 3.4 shows that one has in fact equality..

(17) jt. q:. 7^. __>. We know from Lemma 3.1. 8>. algebraic. Hence, for any algebraic System of rational. Tis. that. l. push-forward q*p~ (K) describes an since curves on X, which cannot be of dimension > 1, by Lemma 1.1, K£CK is aK3 surface. Hence this algebraic System is finite. Moreover, T parametrizes only reduced curves of degree 8, which therefore do not belong. KeC K. the. ,. more than one quadric. Hence each Chow point 7 corresponds to a unique pair (7,0. Thus the fibre p~\K) contains only finitely many points.. to. irreducible component of top dimension 34. By the theorem on the dimension of the fibres (see [Sh], Chap. 1, §6, Thm. 7), we see that dim p(E) = dimE = 34. On applying Chevalley's theorem to the quasi-projective varieties T and Let. E. T. C. be. an. finite-type morphism p we also see that p(T) is constructible, is quasi Since K Le., a finite disjoint union of locally closed subsets V projective,so are the Vf. The same argument works for curves of degree 12, with the map p: J — Ck. Note that one gets, for a component E C J of maximal. CXC. K. and to the. ,. CKC. t. .. >. dimension,. Hence we obtain the same equality. Remarks. the. as. before.. D. As expected, singular points other than nodes do not affect. 1). dimensions of the relevant schemes. This. is. nodes impose the lowest number of conditions. because, roughly. speaking,. for decreasing the géométrie. genus. However, as is shown by Lemma 2.1, not ail curves in Theorem 3.4 hâve only nodes for singularities.. 7^-8. proof of Theorem 3.4, we could replace. In the. 2). xFqx. £>k. ,. where. Tq. dénotes. the. space. of ail. E. by its. closure. quadrics in. P. 3 .. E. in. Now,. projective variety. Hence p(Ë) is closed (cf. [Sh], Chap. 1, §5, Thm. 3) and p(E) = Ck- This would account in particular for the rational octics that lie on a quadratic cône, instead of a smooth quadric surface.. 7^B. x. Tq. is. a. Rational curves. 4.. Let. be. K3. surface. on K3 surfaces in. $P^4$. 4. (Le., not contained in any spanning hyperplane). The notation refers to the fact that such a surface is a smooth complète intersection of a quadric and a cubic threefold. We also write 2 for 5*2,3. a. P4P. S2S. the. 43-dimensional quasi-projective variety of ail 523. In the présent. section we prove:. 's. in. P4P. 4. (see. 3. Lemma 4.2)..

(18) THEOREM 4.1.. carrying rational intégral curves of intersections with smooth quadrics, form a constructible. The surfaces in. degree 12, obtained as set of. dimension. 43. in. £2,3. £2,3. .. consider the curves at issue as belonging to the intersection of two quadrics in P 4 This is a Del Pezzo surface (i.e., its anticanonical sheaf is ample). Hence it is not very différent from a cubic surface. In particular, it The idea. is to. .. rational and we can apply again the results of Tannenbaum. We write Va, for the quasi-projective variety of ail smooth intersections of. is. 4 two quadrics in P. Thus, V^. .. pencils of quadrics in. of rank. 2. -2). 2(15. an. open subset of the Grassmann variety of. 4 .. V 4has. LEMMA 4.2.. Proof. P. is. dimension 26; and. The dimension of. £2,3. has dimension. 43.. dimension of the Grassmann variety subspaces of the space of quadratic forms in 5 variables, to wit, Va,. is. the. = 26.. projective bundle over the of quadrics with fibre the projectivization of the space of cubic space forms modulo (linear) multiples of a quadric. Thus the fibre has dimension. Similarly, Pl4P. 3 (3(. |. 4. ). is. £2,3. an. open. subset. of. the. 14. -5-1=. 29.. D. Any smooth. 4. intersection of two quadrics P C carnes X) a rational, reduced and irreducible curve Y\. LEMMA 4.3.. P4P. (for any positive integer of degree m = 4À having only nodes for singularities, which belongs linear System \X\\ eut out by ail hypersurfaces of degree À.. the. to. Such curves are parametrized by an irreducible quasi-projective scheme of. dimension. Proof. m. —. 1 .. sheaf. Hence we can and. 4. by its anticanonical PGV4 is embedded in apply [Col], Lemma 1, and the proofs of Lemma 2.5. We simply note that. P4P. Lemma 2.6 carry over with minimal changes. In the présent paper we are especially interested in the case where. À. =. The lemma shows that there exist rational, intégral curves of degree m = on. some surfaces. SG. £2,3. .. They are obtained. as. 3. .. 12. intersections with smooth. quadrics and hâve only nodes for singularities.. Proof of Theorem 4.1. degree 12 in. P. 4 .. Let Gn be the Chow variety of rational curves of. As explained at the beginning of §3, we can work over an.

(19) This will be implied whenever we open set of reduced and irreducible curves. write a new correspondance. (For simplicity we shall use the same notation, F, for. curve and for its Chow point.). a. dénote. We. quadric. by. (resp.,. CQC. Q. threefolds. cubic). (resp.,. correspondences. we. C c. P. in. working. are. the. ). 4 .. with. following algebraic correspondences. quasi-projective varieties of smooth As in Lemma 3.1, the incidence "pulled. be. can. up". to. define. the. :. and. In. view of Lemmas 4.2 and 4.3, the dimension of H. 37. This. also the dimension of its image in. is. second projection. finite, since. are. a. curve. than one intersection of two quadrics.. directly from J. but we can do. TL,. to. F. g n. (m-. 1). =. Indeed, the fibres of the. .. cannot belong to more. Gn. G. equal to 26 +. is. (In fact, there. is. even. a. map that goes. without it.). compute the dimension of J we notice that H and J hâve the same image in Gn- Now, a gênerai élément in the image of Ti corresponds to a curve F of degree 12 with 13 distinct nodes and belongs to a pencil of quadrics. But a surface S G Sl3 is contained in a unique quadric. Hence an élément in the fibre of J above F détermines a quadric, say Q, in the To. ,. -System of quadrics through F and is then determined by the family of ail cubic threefolds containing F, provided we discount the reducible éléments 1. oc. that contain. Q.. more than 24 conditions are required for a cubic hypersurface to contain F. In fact it is enough to impose 11 simple points and the 13 double points, since this represents 2-13 + 11 = 3-12+1 intersections. On the other hand,. Therefore,. as a. 35-24=. (>). no. vector space, the family of cubics containing F has dimension 11.. After discounting, as in Lemma 4.2, the 5-dimensional vector space of reducible cubics containing g as a component, we are left with an oc 5 -System of surfaces SG £2.3 containing F and contained in Q. As Q varies in a 6. ). pencil, the fibre of. follows that. It. J J. above F has dimension is. of dimension. (>). (>). 37 +. 6. 5. +. I=6.. = 43. Now, let p be the ail the fibres of this map. projection map from J to 2 3 By Lemma 1.1 are finite. Since the dimensions are right, as is shown by Lemma 4.2, we conclude exactly as in the proof of Theorem 3.4. <S 2S. •. .. 6 ). The smoothness can be proved by an extension of Lemma 2.2, in which we replace the. divisors. in. P3P. 3. by. divisors. in. Q..

(20) REMARK.. Theorems 3.4 and 4.1, together with [C-S], Example 1.3, clearly imply the following statements :. 3. carrying reduced and irre duciblecurves of degree 8, respecîively 12, and géométrie genus 9 — 6 (0 < 6 < 9), respectively 19 — 6 (0 < 6 < \9), obtained as intersections with smooth quadrics, respectively cubics, and having 6 nodes, form a constructible set of dimension 34 in Ck THEOREM. 3.4'. smooîh quartics in. The. P3P. .. THEOREM 4.1. 7. The surfaces in 523. and géométrie genus smooth quadrics, form. 13 a. carrying intégral curves of degree 12 — 6 (0 < 6 < 13 ), obtained as intersections with constructible set of dimension 43 in 52.3 .. REFERENCES [Ca]. Cayley,. new analytical représentation of curves in space. Quart. of Pure and App. Math. 3 (1860), 225-236 (Collected Papers, vol. A. On. a. J.. 4,. 446-455).. Sernesi. Nodal curves. [C-S]. Chiantini,. [Col]. Preprint, p. CORAY, D. Points algébriques sur les surfaces Se. Paris 284 A (1977), 1531-1534.. L.. and E.. on. surfaces of gênerai type.. 15. [Co2] [De]. [G-G]. [Ha]. de. Del Pezzo.. C.. R.. Acad.. Enumerative geometry of rational space curves. Proc. London Math. Soc. 46 (1983), 263-287. Deligne, P. Le théorème de Noether. In: SGA 7 //, Lecture Notes in Mathematics no. 340, exposé 19, pp. 328-340. Springer (BerlinHeidelberg-New York), 1973.. Griffiths.. Two. applications of algebraic geometry to entire holomorphic mappings. In: The Chern Symposium 1979. Springer (New York), 1980, 41-74. HARRIS, J. On the Severi Problem. Invent. Math. 84 (1986), 445-461.. Green, M.. and. Ph.. [La]. Springer (New York), 1977. HODGE, W.V.D. and D. PEDOE. Methods of Algebraic Geometry, vol. 2. Cambridge Univ. Press (Cambridge), 1952. LANG, S. On quasi algebraic closure. Annals of Math. 55 (1952), 373-390.. [Me]. MEYER,. [M-M]. MUKAI. The uniruledness of the moduli space of curves of genus 11. In: Lecture Notes in Maths 1016 (1983), 334-353. NOETHER, M. Zur Grundlegung der Théorie der algebraischen Raumcurven. Abh. Akad. Wissenschaften zu Berlin (1882), 120 p.. [Hait] [H-P]. [No] [Se]. HARTSHORNE, R. Algebraic Geometry.. Fr. Flâchen vierter und hoherer Ordnung. In Wiss. 111.2.2 B, 1533-1779.. MORI,. W.. S.. :. Enzykl.. Math.. and S.. unicursal curves lying on the gênerai quartic surface. of Maths 15 (1944), 24-25. Oxford Quarterly. SEGRE, B. A. remark. on. J..

(21) SHAFAREVICH,. [Sh]. I.R.. Basic. Algebraic. (Berlin-Heidelberg-New York),. Geometry. (2. nd. édition).. 1977.. [SwD]. SwiNNERTON-DYER, H.P.F. Applications of algebraic theory. Pwc. Symp. AMS 20 (1971), 1-52.. [Ta]. TANNENBAUM, A. Families of algebraic curves with nodes. 41 (1980), 107-126. (Reçu. le. 31. juillet 1997 ; version révisée. Daniel Coray. Université Section. de. Genève. de. Mathématiques 2-4, rue du Lièvre CH-1211 Genève 24 Switzerland e-mail: coray @ibm.unige.ch. Constantin Manoil École d'lngénieurs 4, rue de la Prairie CH-1202 Genève. Switzerland e-mail: [email protected] Israël Vainsencher. Universidade Fédéral de Pernambuco Cidade Universitâria 50670-901 Recife - Pe. Brazil e-mail: [email protected]. Springer. reçue. geometry. le. 14. to. number. Compas. Math.. novembre 1997).

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