MIT OpenCourseWare http ocw mit edu 18 727 Topics in Algebraic Geometry Algebraic Surfaces Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms ALGEBRAIC SURFACES LECTURE 15 LECTURES ABHINAV KUMAR 1 Elliptic Surfaces contd Assume we are over C Given f X C we can associate the functional in variant j C P1 c j fc and a homological invariant let c1 cn be points over which the bers are singular C C c1 cn and X f 1 C then we have the sheaf R1 f Z which is a homological invariant R1 f Z C C H 1 FC Z so it is a locally constant sheaf We get a representation G 1 C SL2 Z called the global monodromy The equivalence class of this representation is called the homological invariant The local monodromy at a point ci is the image of a loop around that point in SL2 Z We can write down the conjugacy 1 n class of the local monodromy for the bad bers for a In ber it is 0 1 J and G determine the elliptic bration up to isomorphism Given any elliptic bration we get a corresponding Jacobian bration which does have a section O C X Let X C be an elliptic bration with section and let the class of the ber be F For each singular ber let Fv v 0 Mv i v i where v 0 is the identity component Algebraic and numerical equivalence are the same for elliptic surfaces and KX f KC N where N is isomorphic to the normal bundle of the zero section O in X Note that OC N R1 f OX Thus KS 2g 2 F where the arithmetic genus of X h0 OX h1 OX h2 OX with KS 2 0 and deg N 02 For any section P we have 1 2g 2 P P K P 2 P K P 2 2g 2 so P 2 Let T be the subgroup of N S X generated by the zero section O and all irreducible components of bers called the trivial lattice 2 X C E K N S X T where K k C Here T is torsion free and the map is given by P E K P mod T We can obtain the Shioda Tate formula rk N S X 2 mv 1 rk E k where rk E k is the Mordell Weil rank 1 2 LECTURES ABHINAV KUMAR Remark An open arithmetic question is the rank of E K unbounded for K K C the function eld of a curve e g C P1 E Q x For Fp k the answer is yes Shafarevich Tate For E Q we can write down a nite list of the possible torsion Mazur s theorem 2 Kodaira dimension 0 If X is a surface with X 0 then K 2 0 pg 1 since K 2 0 would imply K ruled we will show this later Noether s formula says that 10 8q 12pg b2 2 where 2 q s Then 0 2pg and even implies that it is 0 or 2 Now the four possibilities based on combinations of b1 b2 are 0 10 are Enriques surfaces classical if pg q 0 0 0 and non classical if pg q 1 1 2 2 2 are bielliptic or hyperelliptic surfaces if pg q 0 1 0 and quasi hyperelliptic surfaces if pg q 1 2 2 0 22 are K3 surfaces with OX 0 q 0 pg 1 0 and 4 6 are abelian surfaces with OX 0 q 2 pg 1 0 Note that by the above formulae there is another possible combination b1 b2 2 14 q1 pg 1 0 Proposition 1 No surface exists with this combination of invariants Proof Since b1 2 2s the Picard variety of X has dimension 1 So a line bundle L0 on X di erence from OX but algebraically equivalent to 0 Since K 0 and pg 1 we get KX OX Applying Riemann Roch to L gives L 12 L L K 1 OX OX 1 Thus H 0 L 0 or H 0 L 1 H 2 L 0 But L 0 gives L OX a contradiction 2 1 Abelian surfaces These are smooth complete group varieties of dimen sion 2 over k For the general theory we refer to Mumford s Abelian Varieties and Birkenhaake Lange s Complex Abelian Varieties Over C these are com plex tori C2 equipped with a Riemann form Z This form is alternating bilinear and satis es the following two conditions R C2 C2 R satis es R iv iw R v w for v w C C The associated Hermitian form H v w R iv w i R v w is positive de nite This gives rise to a line bundle L on C2 which is the class of an ample line bundle L is called a polarization of the abelian variety and 3 h0 L disc Pf In general over any characteristic A Pic 0 A is the dual abelian variety and it is automatically reduced A polarization is an isogeny A A for example ALGEBRAIC SURFACES LECTURE 15 3 whenever we have an ample line bundle L on A the subgroup K L x A Tx L L is nite The map A A x Tx L L 1 gives an isogeny with nite kernel K L i e a polarization For a line bundle L 4 L Lg L2 L 2 deg L dim A 2 The polarization is principal if it has degree 1 There are two common examples of abelian surfaces with principal polarization 1 J C C a genus 2 curve Here we can explicitly write down equations for J C e g Cassels Flynn Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2 The image of C is J C under x x x0 where x0 C is a xed point is the theta divisor and it generates N S J C for a generic C We have C 2 C C K 2g 2 2 since K 0 for an abelian variety The theta divisor gives the principal polarization 2 OX C C2 1 so deg C OX C 2 1 2 Fix E2 with E1 E2 elliptic curves Then E1 E2 gives a principally polarized abelian variety with E1 E2 1 Ei2 0 E1 E2 2 2 1 The moduli space of principally polarized abelian varieties is made up of these two types of points with the second type forming the boundary There are lots of arithmetic questions here By the Mordell Weil theorem if A is de ned over a number eld K then A K is nitely generated for a number eld L containing K What is the structure of the torsion subgroup and what can you say about the rank Merel showed that given a positive integer d there is a constant Bd depending only on d s t for any number eld K of degree d over Q and any elliptic curve E …