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 17 LECTURES ABHINAV KUMAR 1 K3 Surfaces contd Remark Note that K3 surfaces can only be elliptic over P1 on a K3 surface however one can have many di erent elliptic brations though not every K3 surface has one 2 Enriques Surfaces Recall that such surfaces have X 0 KX 0 b2 10 b1 0 OX 1 A classical Enriques surface has pg 0 q 0 0 while a non classical En riques surface has pg 1 q 1 2 which can only happen in characteristic 2 We will discuss only classical Enriques surfaces 2 Proposition 1 For an Enriques surface X OX but X OX Proof Since pg 0 X OX By Riemann Roch OX K OX 1 K 2K OX 1 so h0 OX K h0 OX 2K 1 Since KX 2 OX KX OX h0 OX K 0 since K 0 and so h0 OX 2K 1 2 Since 2K 0 2K 0 i e X OX So the order of K in Pic X is 2 Note that Pic X NS X because Pic 0 X 0 since q 0 0 for classical Enriques surfaces Proposition 2 Pic X Z 2Z where the former object is the space of divi sors numerically equivalent to zero modulo linear or algebraic equivalence or similarly the torsion part of NS Proof Let L 0 By Riemann Roch L OX 12 L L K OX 1 Thus h0 L 0 or h2 L h0 K L 0 But both L and K L are 0 so 1 either L OX or L OX i e L Proposition 3 Let X be an Enriques surface Suppose char k 2 Then an e tale covering X of degree 2 of X which is a K3 surface and the fundamental group of X X is Z 2Z Proof KX is a 2 torsion divisor class Let fij Z 1 Ui OX be a cocycle 1 2 representing K in Pic X H X OX Since 2K 0 fij is a coboundary 1 2 LECTURES ABHINAV KUMAR so we can write is as fij2 ggji on Ui Uj gi Ui OX Now X X de ned locally by zi2 gi on Ui given by zzji fij This is e tale since char k 2 X X X 0 as well Since OX 2 OX 2 X is a K3 surface from the classi cation theorem Remark Over C in terms of line bundles take X s L S 2 1 where X L O K is a line bundle equipped with an isomorphism L 2 OX The map L X s x x X X L de nes a nowhere vanishing section of L which is trivial implying that L KX is trivial This implies that OX 2 and thus X is K3 Proposition 4 Let X be a K3 surface and i a xed point free involution s t it 2 then X is an gives rise to an e tale connected covering X X If char K Enriques surface Proof X X and since X OX X 0 X 0 and OX 1 OX 1 By classi cation X is an Enriques surface 2 Thus Enriques surfaces are quotients of K3 surfaces by xed point free invo lutions Example The smooth complete intersection of 3 quadrics in P5 is a K3 surface Let fi Qi x0 x1 x2 Q i x3 x4 x5 for i 1 2 3 where Qi Q i are homo geneous quadratic forms the fi cut out X a K3 surface Now let P5 P5 x0 x5 x0 x1 x2 x3 x4 x5 be an involution Note that X X Generically the 3 conics Qi 0 in P2 respectively the conics Q i 0 have no points in common implying that X has no xed points in X giving us an Enriques surface as above Theorem 1 Every Enriques surface is elliptic or quasielliptic Proof Exercise 3 Bielliptic surfaces This is the fourth class of surfaces with X 0 b2 2 OX 0 b1 2 KX 0 There are two cases 1 pg 0 q 1 0 the classical bielliptic hyperelliptic surface 2 pg 1 q 2 2 which only happens in positive characteristic In either case b1 2 s b22 1 dim Alb X so the Albanese variety is an elliptic curve Theorem 2 The map f X Alb X has all bers either smooth elliptic curves or all rational curves each having one singular point which is an ordinary cusp The latter case happens only in characteristic 2 or 3 ALGEBRAIC SURFACES LECTURE 17 3 Proof Let B Alb X b B a closed point F Fb f 1 b Then F 2 0 F K 0 pa f 1 f X B is an elliptic or quasi elliptic bration the latter only in characteristic 2 or 3 All the bers of f are irreducible if we had a reducible ber F ai Ei then the classes of F Ei and H the hyperplane section would give 3 independent classes in NS X implying that b2 3 by the Igusa Severi inequality a contradiction Similarly one can show that there are no multiple bers implying that all bers are integral If the general ber is smooth or any closed ber is smooth then f F 0 B B is a regular 1 form on X vanishes exactly where f is not smooth implying that it is a global section of 1X k whose zero locus is either empty or of pure codimension 2 A result of Grothendieck shows that the degree of the zero locus is c2 1X k c2 2 2b1 b2 0 implying that f is everywhere nonzero and f is smooth Remark If all bers of the Albanese map are smooth call it a hyperellip tic bielliptic surface If all bers of the Albanese map are singular call it a quasihyperelliptic quasibielliptic surface Next we nd a second elliptic bration Theorem 3 Let X be as above f X B Alb X a hyperelliptic or quasihyperelliptic bration Then another elliptic bration g X P1 Proof Idea Find an indecomposable curve C of canonical type s t C Ft 0 for all t B where Ft f t First note the following De nition 1 Let X be a minimal surface and D ni Ei 0 be an e ective divisor on X We say that D is a divisor or curve of canonical type if K Ei D Ei 0 for all i 1 r If D is also connected and the g c d of the integers ni is 1 then we say that D is an indecomposable divisor or curve of canonical type Theorem 4 Let X be a minimal surface with K 2 0 and K C 0 for all curves C on X If D is an indecomposable curve of canonical type on X then an elliptic or quasi elliptic bration f X …