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 11 Recall from last time that we de ned the group scheme PicX over k as well as the group scheme Pic0X which is the connected component of 0 i e OX in PicX and is a proper scheme over k Now let L be a line bundle in the class corresponding to the universal element L is a line bundle on X Pic 0X Choose a closed point x of X and let M L x Pic0X Then replace L by L p 2 M 1 0 0 and for every closed point a Pic so that we get L x Pic0X OPicX X the line bundle La LX a is algebraically equivalent to 0 Such an L is called a Poincare line bundle on X Pic 0X Given a choice of basepoint a it is unique up to isomorphism Now note further that the Zariski tangent space at 0 of Pic 0X is canonically isomorphic to H 1 X OX and Pic0X is a commutative group scheme If it is reduced then it is an abelian variety If char k 0 it is automatically reduced by a theorem of Grothendieck Cartier Theorem 1 Let X be a surface q h1 X OX its irregularity s the dimension of the Picard variety of X Let b1 be the rst Betti number h1et X Q Then b1 2s and 2q b1 2 q s lies between 0 and 2pg while 0 if char k 0 Proof Note that for relatively prime to p char k 0 1 1 Z Z b1 Het X Z Z a Pic X a 0 a Pic 0 X a 0 Z Z 2s where the second equality follows from Kummer theory on 0 Fm Fm 0 the second from the fact that Pic Pic 0 is nitely generated so the torsion group is nite and can be chosen larger than the size of the torsion group and the third because Pic 0 X is the underlying abelian group of the Picard variety of X The closed points of Pic0X Pic0X red so b1 2s Now 2 2q b1 2 q s 2dim TPic 0X 0 dim T Pic0X red 0 0 and 3 q s dim H 1 X OX dim i 1 Ker i where the i are the Bockstein homomorphisms de ned inductively by 4 1 H 1 X OX H 1 X OX i Ker i 1 coker i 1 1 2 2 Thus q s dim i 1 Im i h X OX pg In characteristic 0 proper group schemes of nite type are reduced so Pic0X is already an abelian variety 0 1 The Albanese Variety Let X be a smooth projective variety x0 X a xed closed point A pair A consisting of an abelian variety and a morphism X A s t x0 0 is called the Albanese variety of X For every morphism f X B s t B is an abelian variety and f x0 0 a unique homomorphism of abelian varieties g A B s t the diagram below commutes 5 f B g X A Note that a rigidity theorem for abelian varieties implies that any morphism as varieties g A B is of the form g a g a b where g A B is a homomorphism of abelian varieties and b g 0 B Thus we can formulate the de nition without the closed point x0 where we say that there exists a unique homomorphism g A B s t g f It is clearly unique if it exists For existence let X be a smooth projective variety and let P X be the re duced Picard variety of X and P X its dual abelian variety Then Pic0P X P X for an abelian variety Pic 0 is automatically reduced We have a univer sal Poincare line bundle L on X Pic0X and therefore on the reduced subscheme X P X Let X P X X P X be the switch y x x y L is a line bundle on P X X and therefore comes from the Poincare bundle on P X P X by a map X P X by the universal property of Pic 0P X One can check that this gives P X as the Albanese variety of X using general nonsense so Alb X exists and is unique up to unique isomorphism Further more it is dual to the Picard variety of X Note If X is a smooth projective curve then Pic 0 X is reduced and carries a principal polarization so P X P X Pic 0X is the Jacobian of X For a surface we showed that the dimension of the Albanese variety is q with equality holding 0 e g if char k 0 or if pg 0 If k C there is an explicit way to see the Albanese variety We have a map i H1 X Z H 0 X 1X de ned by i The image of i is a lattice in H 0 X 1X and the quotient is an abelian variety a priori a complex torus but a Riemann form exists It is Alb X and is functorial in X i e Y X 6 X Alb X Y Alb Y ALGEBRAIC SURFACES LECTURE 11 3 It follows that the image of X in Alb X generates the abelian variety else the subvariety that X generates inside Alb X would satisfy the universal property 0 X is not a point and if X Y is a surjection In particular if Alb X so is Alb X Alb Y Over C our construction gives us an isomorphism H1 X Z H1 Alb X Z so the inverse image under of any e tale covering of Alb X is connected All Abelian coverings are obtained in this way For now assume that char k 0 Proposition 1 Let X be a surface X Alb X the Albanese map Sup pose X is a curve C Then C is a smooth curve of genus q and the bers of are connected We rst prove the following lemma f j Lemma 1 Suppose factors as X T Alb X with f surjective Then j Alb T Alb X is an isomorphism Proof The functoriality of Alb gives a surjective morphism Alb X Alb T since X T is surjective along with a commutative diagram X 7 x Alb X f f T T Alb T Alb X j j f is the identity by the universal property so j must be an isomorphism