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RIMS-1743 K3 SURFACES of GENUS SIXTEEN by Shigeru RIMS-1743 K3 SURFACES OF GENUS SIXTEEN By Shigeru MUKAI February 2012 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan K3 SURFACES OF GENUS SIXTEEN SHIGERU MUKAI Abstract. The generic polarized K3 surface (S; h) of genus 16, that is, (h2) = 30, is described in a certain compactifeid moduli space T of twisted cubics in P3, as a complete intersection with respect to an almost homogeneous vector bundle of rank 10. As corollary we prove the unirationality of the moduli space F16 of such K3 surfaces. 1. Introduction Let Fg be the moduli space of polarized K3 surface (S; h) of genus g, 2 i.e.,(h ) = 2g ¡ 2. Fg is an arithmetic quotient of the 19-dimensional bounded symmetric domain of type IV, and a quasi-projective variety. For g · 10 and g = 12; 13; 18; 20, the generic (S; h) is a complete intersection in a suitable homogeneous space with respect to a suitable homogeneous vector bundle. As corollary the unirationality of Fg is proved for those values of g in [5, 6, 7]. In this article we shall describe 4 the generic member of F16 using the EPS moduli space T := G(2; 3; C ) of twisted cubics in P3. The EPS moduli space T is constructed by Ellingsrud-Piene-Str¿mme [2] as the GIT quotient of the tensor product C2 ­ C3 ­ V , V being a 4-dimensional vector space, by the obvious action of GL(2)£GL(3). T is a smooth equivariant compacti¯cation of the 12-dimensional homo- geneous space P GL(V )=P GL(2). A point t 2 T represents an equiva- lence class of 2£3 matrices whose entries belong to V . Its three minors de¯ne a subscheme Rt of the projective space P(V ). Rt is a cubic curve mostly and a plane with an embedded point in exceptional case. By construction there exists two natural vector bundles E; F of rank 3, 2, respectively, with det E' det F, and the tautological homomorphism E ­ V _ ¡! F on T , which induces linear embeddings (1) (S2V )_ ,! H0(E) and (S2;1V )_ ,! H0(F): Supported in part by the JSPS Grant-in-Aid for Scienti¯c Research (B) 22340007, (S) 19104001, (A)22244003 and (S)22224001. 1 2 S. MUKAI (See x2.) Here S2V is the second symmetric tensor product, and (2) S2;1V = ker[V ­ S2V ! S3V ] is the space of linear syzygies among second symmetric tensors. For two subspaces M ½ (S2V )_ and N ½ (S2;1V )_, we consider the common zero locus \ \ (3) (s)0 \ (t)0 ½ T : s2M t2N of global sections s 2 M ½ H0(E) and t 2 N ½ H0(F). The case dim M = dim N = 2 is most interesting. We denote the common zero locus (3) by SM;N in this case. Theorem 1.1. If M and N are general, then SM;N is a (smooth) K3 surface, and the restriction of H := c1(E) is a polarization of genus 16. For general M and N, SM;N is a complete intersection in T with respect to the vector bundle E ©2 © F ©2 of rank 10. Furthermore the following converse also holds: Theorem 1.2. Generic K3 surface of genus 16 is isomorphic to the complete intersection SM;N . A twisted cubic ( ¶ f11 f12 f13 (4) R : rank · 1; fij 2 V f21 f22 f23 in P(V ) = P3 is apolar to M if all minors of the matrix are perpendicular to M. Similarly R is apolar to N if all linear syzygies among the three minors are perpendicular to N. The K3 surface SM;N in Theorem 1.1 parametrizes all R which are apolar to both M and N. The totality of SM;N are parametrized by an open subset of a generic G(2; 12)-bundle P over the 16-simensional Grassmannian G(2; (S2V )_) which parametrizes M. By Theorem 1.1, we have the rational map (5) ª16 : P ¢ ¢ ¢ ! F16; (M; N) 7! (SM;N ;HjS); whose dominance is Theorem 1.2. Therefore, as bi-product, we have Corollary The moduli space F16 of polarized K3 surface of genus 16 is unirational. In order to prove the theorems, we we study a certain special case in 2 _ 2 _ detail. More explicitly, we consider the space M0 ½ (S V ) ' S (V ) spanned by two reducible quadratic forms q1 = XY; q2 = ZT , and T \ T T study the common zero locus M0 := (q1)0 (q2)0 of M0 in . M0 parametrizes all twisted cubics whose de¯ning quadratic forms do not K3 SURFACES OF GENUS SIXTEEN 3 contain the term xy or zt, where (x : y : z : t) is a homogeneous coordinate P3 and (Z : Y : Z : T ) is the dual coordinate of P3;¤. P3 If N is general, then SM0;N is a quartic surface in which contains two quintic elliptic curves E1 and E2 with (E1:E2) = 3. In particular E we have Theorem 1.1. Moreover, the restriction of to SM0;N is an extension of three line bundles OS(E1); OS(E2) and OS(H ¡ E1), and ©2 the restriction of F contains OS(E1) as a subsheaf. These give us the following vanishing of higher cohomology groups which is the key of of the proof of Theorem 1.2. Proposition 1.3. If both M and N are general, then the restriction of E; F to S := SM;N are simple and satisfy i i i Ext (EjS; FjS) = H (S; EjS) = H (S; FjS) = 0; for all i > 0: After preparing some basic facts on the EPS moduli space T = 4 G(2; 3; C ) in x2 in x2, we ¯rst study the locus TQ of twisted cubics x T apolar to one reducible quadric in 3. We next study the locus B1;B2 of twisted cubics which have two skew lines as their bisecants in x4 and T x x the above M0 in 5. We prove Theorem 1.1 in 6 and Theorem 1.2 in x 7 using doubly octagonal K3 surfaces SM0;N . Notations and convention All varieties are considered over the complex number ¯eld C. The projective space P(V ) associated to a vec- tor space V is that in Grothendieck's sense. The Grassmann variety of s-dimensional subspaces of V is denoted by G(s; V ). The isomorphism class of G(s; V ) is denoted by G(s; n) when dim V = n. The dual vec- tor space and vector bundles are denoted by E is denoted by V _ and E_. Twisted cubic is used in the generalized sense of [2]. But the locus where twisted cubics are not curve is of su±ciently large codimension, and hence is never crucial in our argument. 2. Pair of vector bundles whose ranks differ by one Let (E; F ) be a pair of vector bundles on a scheme S such that (6) det E ' det F; rank E = rank F + 1: Let r be the rank of F . r homomrphisms f1; : : : ; fr 2 Hom(E; F ) gives rise the homomorphism r r f1 ^ ¢ ¢ ¢ ^ fr : ^ E ! ^ F ' det F which can be regarded as a global section of E by our assumption (6). Since f1 ^ ¢ ¢ ¢ ^ fr is symmetric with respect to f1; : : : ; fr, we have a liner map (7) SrHom(E; F ) ! Hom(^rE; ^rF ) ' H0(E): 4 S. MUKAI If g : E ! F is a homomorphism, then g(f1 ^ ¢ ¢ ¢ ^ fr) is a global section of F . Hence we have another linear map r 0 S Hom(E; F )­Hom(E; F ) ! H (F ); ((f1; : : : ; fr); g) 7! g(f1^:::^fr): Since Sr+1Hom(E; F ) lies in the kernel of this linear map, we have (8) Sr;1Hom(E; F ) ! H0(F ): Let V be a vector space and let G(r; r + 1; V ) be the GIT quotient of the tensor product Cr ­ Cr+1 ­ V by GL(r) £ GL(r + 1). There are two natural vector bundles E; F of rank r + 1, r, respectively, with det E' det F, and the tautological homomorphism (9) E ­ V _ !F on G(r; r + 1; V ). This has the following universal property. (*) If a homomorphism E ­ V _ ! F satis¯es (6) and if the induced linear map SrV ! H0(E) is surjective, then there exists a unique morphism © : S ! G(r; r + 1; V ) such that E ­ V _ ! F coincides with the pull-back of (9). This © will be denoted by ©E;F;V _ : S ! _ G(r; r + 1; V ), or ©E;F if V = Hom(E; F ). Remark 2.1. If E; F are vector bundles of rank r + 1; r, respectively. Then, putting L = (det E)¡1 ­ det F , we have Hom(E; F ) ' Hom(E ­ L; F ­ L) and det(E ­ L) ' det(F ­ L): Hence, the assumption (6) is not restrictive. In the sequel we apply the case r = 2; dim V = 4 to K3 surfaces of genus 16. G(2; 3; V ) is regarded as a subvariety of the Grassmannian 2 0 3 0 3 G(3;S V ) by R 7! H (P ; OP(2¡R)), where H (P ; OP(2¡R)) is the 3- dimensional space of quadratic forms vanishing on R. G(2; 3; V ) is also 2;1 a subvariety of another Grassmannian G(2;S V ) by R 7! SyzR, where 0 3 SyzR is the 2-dimensional space of linear syzygies among H (P ; OP(2¡ R)). Let SM;N ½ T = G(2; 3; V ) be as in the introduction for general 2-dimensional subspaces M and N.
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