ON THE NÉRON-SEVERI GROUP OF SURFACES WITH MANY LINES

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1 ON THE NÉRON-SEVERI GROUP OF SURFACES WITH MANY LINES SAMUEL BOISSIÈRE AND ALESSANDRA SARTI Abstract. For a binary quartic form φ without multiple factors, we classify the quartic K3 surfaces φ(x, y) = φ(z, t) whose Néron-Severi group is (rationally) generated by lines. For generic binary forms φ, ψ of prime degree without multiple factors, we prove that the Néron-Severi group of the surface φ(x, y) = ψ(z, t) is rationally generated by lines. 1. Introduction The study of the Néron-Severi group NS(S) of a given surface S is interesting for understanding its geometry, but it is not an easy task in general. A first step is to compute its Picard number ρ(s) := rk NS(S). A second one is to give a family of generators of NS(S) over Z. To this purpose, it is very useful to find first a nice family of generators of NS(S) Z Q. If one already knows the value of the determinant of NS(S), this can help deducing a family of generators. If not, the study of the rational generators gives non trivial information for the value of the discriminant. Let φ be a binary quartic form without multiple factors. After a suitable linear change of coordinates, we may assume that φ is of the form: φ(x, y) = yx(y x)(y x) for C\0, 1}. Naturally associated to φ are the K3 surface S φ : φ(x, y) = φ(z, t) and the elliptic curve E φ : t = φ(1, y). Remark 1.1. Observe that if φ, φ are the forms corresponding to, and is one of the values, 1, 1, 1 1, 1, 1 then there is a linear isomorphism S φ = Sφ. The interplay between the geometry of the K3 surface S φ and the arithmetic of the elliptic curve E φ has been studied by many authors. Of particular interest is the link between the value of the Picard number ρ(s φ ) and the existence of a complex multiplication on E φ. The following result is classical (see [Kuw95] and references therein): 0 if E φ has a complex multiplication, ρ(s φ ) = 19 otherwise. We pursue the study by giving numerical conditions for the Néron-Severi group of S φ to be rationally generated by lines: 1991 Mathematics Subject Classification. 14J18,14J19. Key words and phrases. Néron-Severi group, Picard number, lines on surfaces. 1

2 SAMUEL BOISSIÈRE AND ALESSANDRA SARTI Notation Definition. Let S P 3 C be a smooth surface of degree d 3. If L is a line contained in S, by the genus formula the self-intersection of L in S is L = d +, so the class of L in NS(S) is not a torsion class. We denote by LC(S) the sublattice of the torsion-free part of NS(S) generated by the classes of the lines contained in S. For a generic surface S, it is well-known that LC(S) = 0. If not, these classes are natural candidates as generators of NS(S) and we say that NS(S) is rationally generated by lines if rk LC(S) = ρ(s), that is LC(S) Z Q = NS(S) Z Q. The most famous examples of surfaces whose Néron-Severi group is rationally generated by lines are certain Fermat surfaces (see [Shi81]). The surfaces we study here are a natural generalization of them. We prove ( ): Theorem 1.. The Néron-Severi group of S φ is rationally generated by lines exactly in the following cases: (1) / Q; () 1,, 1, 1+i 3 }; (3) Q \ 1,, 1, 1+i 3 } and ρ(s φ ) = 19. Looking now for a set of generators of the Néron-Severi group, we prove ( 3): Theorem 1.3. The Néron-Severi group of S φ is generated by lines only in case (). Generalizing the construction, one can consider two binary forms φ, ψ of degree d without multiple factors and the associated surface Sφ,ψ d : φ(x, y) = ψ(z, t). One can prove that ρ(sφ,ψ d ) (d 1) + 1 with equality for d prime and φ, ψ generic (see [Sas68]). We prove ( 4): Theorem 1.4. For d prime and φ, ψ generic, the Néron-Severi group of S d φ,ψ is rationally generated by lines. In Theorem 1. we do not consider the quartics Sφ,ψ 4 for φ ψ since, although ρ(sφ,ψ 4 ) = 18 (see again [Kuw95]), Proposition 4.1 below says that their 16 lines generate an intersection matrix of rank 10, so such surfaces do not enter in our context. We thank the referee for helpful suggestions and comments.. Proof of Theorem 1. The result follows from the following proposition: Proposition.1. If 1,, 1, 1+i 3 rk LC(S φ ) = 19. }, then rk LC(S φ ) = 0, otherwise Proof of Theorem 1.. Assuming Proposition.1, we prove Theorem 1.. The key argument is that if E φ has a complex multiplication, then its j-invariant is algebraic over Q (see [Sil94]). Since j(e φ ) = 56(1 + ) 3 ( 1), j(e φ ) Q if and only if Q. Then: - If / Q, E φ has no complex multiplication so ρ(s φ ) = 19 and by Proposition.1, rk LC(S φ ) = 19. This proves (1). - If 1,, 1, 1+i 3 }, by Proposition.1 we have rk LC(S φ ) = 0 so ρ(s φ ) = 0. This proves (). - If Q \ 1,, 1, 1+i 3 }, then ρ(s φ ) 19, 0} and rk LC(S φ ) = 19. This gives (3).

3 ON THE NÉRON-SEVERI GROUP OF SURFACES WITH MANY LINES 3 Remark.. In case (3) of Theorem 1., one can not be more precise since: When j(e φ ) Q (so Q), it is not clear whether E φ admits a complex multiplication or not. There is a dense and numerable set of Q such that ρ(s φ ) = 0 (see [Ogu]). Proof of Proposition.1. The description of the lines on S φ [Seg47]. We follow the presentation given in [BS07]. comes from Segre Case 1. If / 1,, 1, 1+i 3 }, the group of automorphisms of P 1 C permuting the set, 0, 1, } is the dihedral group D = id, s 1, s, s 1 s } and the surface S φ contains exactly the following 3 lines: vx = uy pt l z (u, v): l id (p): l s1 (p): vt = uz y = pt y = pz pt 1 u,v,0,1,} p 1, 1,i, i} p 1 1, i 1, i 1, 1 } l s (p): y = pz l s1s (p): y = pz + pt p 1, 1, i, i } p 1, 1, i, i } The intersection matrix of these 3 lines is easy to compute (we do not reproduce it here), and is independent of. One finds that its rank is 19, so rk LC(S φ ) = 19. Case. If 1,, 1 }, the surfaces are isomorphic to each other by Remark 1.1. The group of automorphisms is the dihedral group D 4 = D, r. The surface S φ contains exactly 48 lines: the 3 preceding ones and 16 other lines. For = 1 for example, these lines are: + pt l r (p): y = pz + pt l rs1 (p): y = pz l r 1(p): y = pz pt x = pz l s1 r(p): y = pt p 1+i, 1 i, 1+i, 1 i } p 1+i, 1 i The rank of the intersection matrix of the 48 lines is rk LC(S φ ) = 0., 1+i, 1 i } Case 3. If 1+i 3 }, the surfaces are isomorphic to each other by Remark 1.1. The group of automorphisms is the tetrahedral group T = r, s. The surface S φ contains exactly the following 64 lines: vx = uy l z (u, v): vt = uz l r (p): y = pz + pt l id (p): y = pt l r (p) : y = pz pt u,v,0,1,} p 1, 1,i, i}

4 4 SAMUEL BOISSIÈRE AND ALESSANDRA SARTI l s (p): y = pz l r s(p): y = pz + pt l rs (p): y = pz + pt l sr (p): + pt y = pz l rsr (p) : + pt y = pt p,,i, i} l rsr s(p): l srs (p): x = pz + pt y = pz + pt y = pt l r srs(p): l rsrs (p): y = pz + pt y = pz p,,i, i } The rank of the intersection matrix of the 64 lines is rk LC(S φ ) = Proof of Theorem 1.3 As we explained in the Introduction, once one has found a nice family of rational generators of the Néron-Severi group, the next task is to get information on divisible classes. We call a divisor Λ = n i=1 α il i NS(S) m -divisible if the class of Λ in NS(S) is divisible by m ; for m = 1 we say also that the lines in Λ form an even set. Proof of Theorem 1.3. Cases (1) and (3). For / 1,, 1, 1+i 3 }, with the help of a computer program we obtain that the best choice of a family of 19 lines among the 3 generating rationally the Néron-Severi group gives a determinant of value 9. Denoting this lattice by M and its dual by M, the discriminant group is: M /M = (Z ) (Z 4 ) Z 8 hence we can have only m -divisible classes for m = 1,, 3. Denote by (M /M) the part of the discriminant group generated by the -torsion classes. We have (M /M) = (Z ) 5 hence rank(m /M) = 5. However, denoting by T the transcendental lattice of S φ, (NS(S φ ) / NS(S φ )) = (T /T ) has rank at most the rank of T, which is three: This shows that M NS(S φ ), and that there are at least two even sets of lines in the Néron Severi group. In particular there is no set of 19 lines generating NS(S φ ). Case () for 1,, 1 }. By Remark 1.1, the surfaces S φ are isomorphic to each other. The best choice of a family of 0 lines among 48 gives a determinant of value 6. Observe that a suitable permutation of the zeros of x 4 y 4 in P 1 C gives a cross-ratio equal to 1, so our surfaces are isomorphic to the Fermat quartic. It is then well-known that det NS(S φ ) = 64, so the lines generate the Néron-Severi group. Case () for 1+i 3 }. A computer program shows that the best choice of a family of 0 lines among the 64 contained in the surface, generating rationally the Néron-Severi group, gives a determinant of value 4 3. We show in Appendix B that det NS(S φ ) = 48 so the lines generate the Néron-Severi group.

5 ON THE NÉRON-SEVERI GROUP OF SURFACES WITH MANY LINES 5 4. Proof of Theorem 1.4 Since ρ(s d φ,ψ ) = (d 1) + 1 for d prime and φ, ψ generic, Theorem 1.4 follows from the following result: Proposition 4.1. It is rk LC(S d φ,ψ ) = (d 1) + 1. Proof of Proposition 4.1. We set S := S d φ,ψ. Let L be the line z = t = 0 and L be the line x = y = 0. The intersection S L is the set of zeros of φ, whereas S L is the set of zeros of ψ. If p L is a zero of φ and q L a zero of ψ, the line L p,q joining p and q is contained in S: this gives a family of d lines contained in S. The intersection matrix of this family is given by L = d + and L L = 1 if L and L intersect, 0 otherwise. Note that: (L p,q L p,q ) (p = p or q = q ). This implies that after ordering correctly the lines, the intersection matrix is the matrix M d := K d+,1,1,0 d (see the notation in Appendix A). Remark A.5 gives rk LC(S) = rk M d = (d 1) + 1. Appendix A. Some linear algebra Let a, b, c, d,... denote indeterminates. For d, let Ja,b d be the (d, d)-matrix defined by: a b ( Ja,b d :=... 1 ) = b + (a b) I d a b where I d denotes the identity (d, d)-matrix. The following lemma is clear: Lemma A.1. The following identities hold: J d a,b + J d a,b = J d a+a,b+b ; J d a,b J d a,b = J d aa +(d 1)bb,ab +a b+(d )bb. Let now Ka,b,c,d d be the (d, d )-matrix defined as the following (d, d)-blocks of (d, d)-matrices: Ka,b,c,d d := J d a,b J d c,d... J d c,d J d a,b The following lemma follows easily from Lemma A.1: Lemma A.. The following identity holds: K d a,b,c,d K d a,b,c,d = Kd α,β,γ,δ

6 6 SAMUEL BOISSIÈRE AND ALESSANDRA SARTI where: α = aa + (d 1)(bb + cc ) + (d 1) dd ; β = ab + a b + (d 1)(cd + c d) + (d )bb + (d 1)(d )dd ; γ = ac + a c + (d 1)(bd + b d) + (d )cc + (d 1)(d )dd ; δ = ad + a d + bc + b c + (d )(cd + c d + bd + b d) + (d ) dd. Set K d := K d 1,1,1,0. Its minimal polynomial µ Kd (t) is given by: Lemma A.3. µ Kd (t) = (t (d 1)) (t (d 1)) (t + 1). Proof. Note that: Applying Lemma A. one gets: K d (d 1)I d = K d d+,1,1,0; K d (d 1)I d = K d d+,1,1,0; K d + I d = K d,1,1,0. K d d+,1,1,0 K d d+,1,1,0 = K d (d 1), d+, d+, ; K d d+,1,1,0 K d,1,1,0 = K d,,,; K d d+,1,1,0 K d,1,1,0 = K d d+, d+, d+,; K d d+,1,1,0 K d d+,1,1,0 K d,1,1,0 = K d 0,0,0,0 = 0. For d 1, d 1, 1}, we denote by V () the eigenspace of K d associated to the eigenvalue. One computes: Lemma A.4. dim V (d 1) = 1; dim V ( 1) = (d 1) ; dim V (d 1) = (d 1). Proof. The first two results are a (quite long) direct computation. One deduces the third one using that K d is diagonalizable (Lemma A.3). Remark A.5. Since K,1,1,0 d = K d (1 )I d, the matrix K,1,1,0 d is invertible when 1 is not an eigenvalue of K d. By Lemma A.3 this is / d +, d +, }. For = d +, one has: rk K d d+,1,1,0 = d dim V (d 1) = (d 1) + 1. Appendix B. Results on Kummer surfaces We recall some classical facts from [Ino76, PŠŠ71, SI77, SM74]. If S is a K3 surface with Picard number 0, we denote by T S the transcendental lattice and Q S the intersection matrix of T S with respect to an oriented basis. Let Q be the set of positive definite, even integral matrices. The class [Q S ] Q/ SL (Z) is uniquely determined by S and det NS(S) = det Q S. For S φ, let σ be the involution (x : y : z : t) (x : y : z : t). Then the minimal resolution of S φ /σ is isomorphic to the Kummer surface Y := Km(E φ E φ ) and: Q Sφ = Q Y = 4Q A

7 ON THE NÉRON-SEVERI GROUP OF SURFACES WITH MANY LINES 7 where A := E φ E φ and Q A is the binary quadratic form associated to A as in [SM74]. For = 1+i 3, the group of automorphisms of the elliptic curve E φ fixing a point has order 6 (since j() = 0) so E φ = Cτ := C/Z + τz with τ = 1+i 3. By the construction of [SM74], for A = C τ C τ, one has Q A = ( 1 1 ) so Q Sφ = ( and det NS(S φ ) = det Q Sφ = 48. Moreover, observe that for A = C τ C τ with τ = i ( ) 4 3, one has Q A = so S 4 φ = Km(A ). Remark B.1. The same method has been used to compute the determinant of the Néron-Severi group of the Fermat quartic. References [BS07] Samuel Boissière and Alessandra Sarti, Counting lines on surfaces, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 6 (007), [Ino76] Hiroshi Inose, On certain Kummer surfaces which can be realized as non-singular quartic surfaces in P 3, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 3 (1976), no. 3, [Kuw95] Masato Kuwata, Elliptic fibrations on quartic K3 surfaces with large Picard numbers, Pacific J. Math. 171 (1995), no. 1, [Ogu] Keiji Oguiso, Picard numbers in a family of hyperkähler manifolds - A supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron, arxiv:math.ag/ [PŠŠ71] I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), [Sas68] Nobuo Sasakura, On some results on the Picard numbers of certain algebraic surfaces, J. Math. Soc. Japan 0 (1968), [Seg47] Beniamino Segre, On arithmetical properties of quartic surfaces, Proc. London Math. Soc. () 49 (1947), [Shi81] Tetsuji Shioda, On the Picard number of a complex projective variety, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, [SI77] T. Shioda and H. Inose, On singular K3 surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp [Sil94] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, [SM74] Tetsuji Shioda and Naoki Mitani, Singular abelian surfaces and binary quadratic forms, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp Lecture Notes in Math., Vol. 41. Samuel Boissière, Laboratoire J.A.Dieudonné UMR CNRS 661, Université de Nice Sophia-Antipolis, Parc Valrose, Nice address: samuel.boissiere@math.unice.fr Alessandra Sarti, Johannes Gutenberg Universität Mainz, Institut für Mathematik, Mainz, Germany address: sarti@mathematik.uni-mainz.de )